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**1. Equations, Inequalities, and Mathematical Modeling**

- 1.1 Graphs of Equations
- 1.2 Linear Equations in One Variable
- 1.3 Modeling with Linear Equations
- 1.4 Quadratic Equations and Applications
- 1.5 Complex Numbers
- 1.6 Other Types of Equations
- 1.7 Linear Inequalities in One Variable
- 1.8 Other Types of Inequalities

**2. ****Functions and Their Graphs**

- 2.1 Linear Equations in Two Variables
- 2.2 Functions
- 2.3 Analyzing Graphs of Functions
- 2.4 A Library of Parent Functions
- 2.5 Transformations of Functions
- 2.6 Combinations of Functions: Composite Functions
- 2.7 Inverse Functions

**3. ****Polynomial Functions**

- 3.1 Quadratic Functions and Models
- 3.2 Polynomial Functions of Higher Degree
- 3.3 Polynomial and Synthetic Division
- 3.4 Zeros of Polynomial Functions
- 3.5 Mathematical Modeling and Variation

**4. ****Rational Functions and Conics**

- 4.1 Rational Functions and Asymptotes
- 4.2 Graphs of Rational Functions
- 4.3 Conics
- 4.4 Translations of Conics

**5.****Exponential and Logarithmic Functions**

- 5.1 Exponential Functions and Their Graphs
- 5.2 Logarithmic Functions and Their Graphs
- 5.3 Properties of Logarithms
- 5.4 Exponential and Logarithmic Equations
- 5.5 Exponential and Logarithmic Models

**6. ****Trigonometry**

- 6.1 Angles and Their Measure
- 6.2 Right Triangle Trigonometry
- 6.3 Trigonometric Functions of Any Angle
- 6.4 Graphs of Sine and Cosine Functions
- 6.5 Graphs of Other Trigonometric Functions
- 6.6 Inverse Trigonometric Functions
- 6.7 Applications and Models

**7 ****Analytic Trigonometry**

- 7.1 Using Fundamental Identities
- 7.2 Verifying Trigonometric Identities
- 7.3 Solving Trigonometric Equations
- 7.4 Sum and Difference Formulas
- 7.5 Multiple-Angle and Product-to-Sum Formulas

**8 ****Additional Topics in Trigonometry**

- 8.1 Law of Sines
- 8.2 Law of Cosines
- 8.3 Vectors in the Plane
- 8.4 Vectors and Dot Products
- 8.5 Trigonometric Form of a Complex Number

**9 ****Systems of Equations and Inequalities**

- 9.1 Linear and Nonlinear Systems of Equations
- 9.2 Two-Variable Linear Systems
- 9.3 Multivariable Linear Systems
- 9.4 Partial Fractions
- 9.5 Systems of Inequalities
- 9.6 Linear Programming

**10 ****Matrices and Determinants**

- 10.1 Matrices and Systems of Equations
- 10.2 Operations with Matrices
- 10.3 The Inverse of a Square Matrix
- 10.4 The Determinant of a Square Matrix
- 10.5 Applications of Matrices and Determinants

**11 ****Sequences, Series, and Probability**

- 11.1 Sequences and Series 800
- 11.2 Arithmetic Sequences and Partial Sums 811
- 11.3 Geometric Sequences and Series 821
- 11.4 Mathematical Induction 831
- 11.5 The Binomial Theorem 841
- 11.6 Counting Principles 849
- 11.7 Probability

Contenido del libro en versión texto.

mas libros gratis en http:www.leeydescarga.com www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Algebra and Trigonometry Eighth Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C. Falvo The Pennsylvania State University The Behrend College Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Algebra and Trigonometry, Eighth Edition Ron Larson 2011, 2007 BrooksCole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Publisher: Charlie VanWagner Acquiring Sponsoring Editor: Gary Whalen Development Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang Associate Media Editor: Lynh Pham Marketing Manager: Myriah FitzGibbon For product information and technology assistance, contact us at Cengage Learning Customer Sales Support, 1 800 354 9706 Marketing Coordinator: Angela Kim For permission to use material from this text or product, submit all requests online at www.cengage.compermissions. Further permissions questions can be emailed to permissionrequestcengage.com Marketing Communications Manager: Katy Malatesta Content Project Manager: Susan Miscio Senior Art Director: Jill Ort Senior Print Buyer: Diane Gibbons Production Editor: Carol Merrigan Library of Congress Control Number: 2009930253 Text Designer: Walter Kopek Student Edition: Rights Acquiring Account Manager, Photos: Don Schlotman Photo Researcher: Prepress PMG ISBN 13: 978 1 4390 4847 4 ISBN 10: 1 4390 4847 9 Cover Designer: Harold Burch Cover Image: Richard EdelmanWoodstock Graphics Studio Compositor: Larson Texts, Inc. BrooksCole 10 Davis Drive Belmont, CA 94002 3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.comregion Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Contents A Word from the Author Preface vii chapter P Prerequisites 1 P.1 Review of Real Numbers and Their Properties 2 P.2 Exponents and Radicals 15 P.3 Polynomials and Special Products 28 P.4 Factoring Polynomials 37 P.5 Rational Expressions 45 P.6 The Rectangular Coordinate System and Graphs 55 Chapter Summary 66 Review Exercises 68 Chapter Test 71 Proofs in Mathematics 72 Problem Solving 73 chapter 1 Equations, Inequalities, and Mathematical Modeling 75 1.1 Graphs of Equations 76 1.2 Linear Equations in One Variable 87 1.3 Modeling with Linear Equations 96 1.4 Quadratic Equations and Applications 107 1.5 Complex Numbers 122 1.6 Other Types of Equations 129 1.7 Linear Inequalities in One Variable 140 1.8 Other Types of Inequalities 150 Chapter Summary 160 Review Exercises 162 Chapter Test 165 Proofs in Mathematics 166 Problem Solving 167 chapter 2 Functions and Their Graphs 169 2.1 Linear Equations in Two Variables 170 2.2 Functions 185 2.3 Analyzing Graphs of Functions 200 2.4 A Library of Parent Functions 212 2.5 Transformations of Functions 219 2.6 Combinations of Functions: Composite Functions 229 2.7 Inverse Functions 238 Chapter Summary 248 Review Exercises 250 Chapter Test 253 Cumulative Test for Chapters P2 254 Proofs in Mathematics 256 Problem Solving 257 iii www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com iv Contents chapter 3 Polynomial Functions 259 3.1 Quadratic Functions and Models 260 3.2 Polynomial Functions of Higher Degree 270 3.3 Polynomial and Synthetic Division 284 3.4 Zeros of Polynomial Functions 293 3.5 Mathematical Modeling and Variation 308 Chapter Summary 320 Review Exercises 322 Chapter Test 326 Proofs in Mathematics 327 Problem Solving 329 chapter 4 Rational Functions and Conics 331 4.1 Rational Functions and Asymptotes 332 4.2 Graphs of Rational Functions 340 4.3 Conics 349 4.4 Translations of Conics 362 Chapter Summary 370 Review Exercises 372 Chapter Test 375 Proofs in Mathematics 376 Problem Solving 377 chapter 5 Exponential and Logarithmic Functions 379 5.1 Exponential Functions and Their Graphs 380 5.2 Logarithmic Functions and Their Graphs 391 5.3 Properties of Logarithms 401 5.4 Exponential and Logarithmic Equations 408 5.5 Exponential and Logarithmic Models 419 Chapter Summary 432 Review Exercises 434 Chapter Test 437 Cumulative Test for Chapters 35 438 Proofs in Mathematics 440 Problem Solving 441 chapter 6 Trigonometry 6.1 6.2 6.3 6.4 6.5 6.6 443 Angles and Their Measure 444 Right Triangle Trigonometry 456 Trigonometric Functions of Any Angle 467 Graphs of Sine and Cosine Functions 479 Graphs of Other Trigonometric Functions 490 Inverse Trigonometric Functions 501 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Contents 6.7 Applications and Models 511 Chapter Summary 522 Chapter Test 527 Problem Solving 529 chapter 7 Analytic Trigonometry v Review Exercises 524 Proofs in Mathematics 528 531 7.1 Using Fundamental Identities 532 7.2 Verifying Trigonometric Identities 540 7.3 Solving Trigonometric Equations 547 7.4 Sum and Difference Formulas 558 7.5 Multiple Angle and Product to Sum Formulas 565 Chapter Summary 576 Review Exercises 578 Chapter Test 581 Proofs in Mathematics 582 Problem Solving 585 chapter 8 Additional Topics in Trigonometry 587 8.1 Law of Sines 588 8.2 Law of Cosines 597 8.3 Vectors in the Plane 605 8.4 Vectors and Dot Products 618 8.5 Trigonometric Form of a Complex Number 628 Chapter Summary 638 Review Exercises 640 Chapter Test 644 Cumulative Test for Chapters 68 645 Proofs in Mathematics 647 Problem Solving 651 chapter 9 Systems of Equations and Inequalities 653 9.1 Linear and Nonlinear Systems of Equations 654 9.2 Two Variable Linear Systems 665 9.3 Multivariable Linear Systems 677 9.4 Partial Fractions 690 9.5 Systems of Inequalities 698 9.6 Linear Programming 709 Chapter Summary 718 Review Exercises 720 Chapter Test 725 Proofs in Mathematics 726 Problem Solving 727 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com vi Contents chapter 10 Matrices and Determinants 729 10.1 Matrices and Systems of Equations 730 10.2 Operations with Matrices 744 10.3 The Inverse of a Square Matrix 759 10.4 The Determinant of a Square Matrix 768 10.5 Applications of Matrices and Determinants 776 Chapter Summary 788 Review Exercises 790 Chapter Test 795 Proofs in Mathematics 796 Problem Solving 797 chapter 11 Sequences, Series, and Probability 799 11.1 Sequences and Series 800 11.2 Arithmetic Sequences and Partial Sums 811 11.3 Geometric Sequences and Series 821 11.4 Mathematical Induction 831 11.5 The Binomial Theorem 841 11.6 Counting Principles 849 11.7 Probability 859 Chapter Summary 872 Review Exercises 874 Chapter Test 877 Cumulative Test for Chapters 911 878 Proofs in Mathematics 880 Problem Solving 883 Appendix A Errors and the Algebra of Calculus Answers to Odd Numbered Exercises and Tests Index A1 A9 A123 Index of Applications web Appendix B Concepts in Statistics web B.1 B.2 B.3 Representing Data Measures of Central Tendency and Dispersion Least Squares Regression www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com A Word from the Author Welcome to the Eighth Edition of Algebra and Trigonometry We are proud to offer you a new and revised version of our textbook. With this edition, we have listened to you, our users, and have incorporated many of your suggestions for improvement. 8th 4th 7th 3rd 6th 2nd 5th 1st In the Eighth Edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible. There are many changes in the mathematics, art, and design; the more significant changes are noted here. New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In Real Life, and In Careers. In Mathematics describes an important mathematical topic taught in the chapter. In Real Life tells students where they will encounter this topic in real life situations. In Careers relates application exercises to a variety of careers. New Study Tips and WarningCautions Insightful information is given to students in two new features. The Study Tip provides students with useful information or suggestions for learning the topic. The WarningCaution points out common mathematical errors made by students. New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic. New Side by Side Examples Throughout the text, we present solutions to many examples from multiple perspectivesalgebraically, graphically, and numerically. The side by side format of this pedagogical feature helps students to see that a problem can be solved in more than one way and to see that different methods yield the same result. The side by side format also addresses many different learning styles. vii www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com viii A Word from the Author New Capstone Exercises Capstones are conceptual problems that synthesize key topics and provide students with a better understanding of each sections concepts. Capstone exercises are excellent for classroom discussion or test prep, and teachers may find value in integrating these problems into their reviews of the section. New Chapter Summaries The Chapter Summary now includes an explanation andor example of each objective taught in the chapter. Revised Exercise Sets The exercise sets have been carefully and extensively examined to ensure they are rigorous and cover all topics suggested by our users. Many new skill building and challenging exercises have been added. For the past several years, weve maintained an independent website CalcChat.comthat provides free solutions to all odd numbered exercises in the text. Thousands of students using our textbooks have visited the site for practice and help with their homework. For the Eighth Edition, we were able to use information from CalcChat.com, including which solutions students accessed most often, to help guide the revision of the exercises. I hope you enjoy the Eighth Edition of Algebra and Trigonometry. As always, I welcome comments and suggestions for continued improvements. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Acknowledgments I would like to thank the many people who have helped me prepare the text and the supplements package. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to review the changes in this edition and to provide suggestions for improving it. Without your help, this book would not be possible. Reviewers Chad Pierson, University of Minnesota Duluth; Sally Shao, Cleveland State University; Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo, Jacksonville State University; C. Altay Ozgener, Manatee Community College Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr. Arun K. Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University; Michael J. McConnell, Clarion University; Martha Chalhoub, Collin County Community College; Angela Lee Everett, Chattanooga State Tech Community College; Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington County Community College; Ward Shaffer, College of Coastal Georgia; Carmen Thomas, Chatham University; Emily J. Keaton My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project. My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text. I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements. On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support. Also, a special thanks goes to R. Scott ONeil. If you have suggestions for improving this text, please feel free to write to me. Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly. Ron Larson ix www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Supplements Supplements for the Instructor Annotated Instructors Edition This AIE is the complete student text plus point ofuse annotations for the instructor, including extra projects, classroom activities, teaching strategies, and additional examples. Answers to even numbered text exercises, Vocabulary Checks, and Explorations are also provided. Complete Solutions Manual This manual contains solutions to all exercises from the text, including Chapter Review Exercises and Chapter Tests. Instructors Companion Website of instructor resources. This free companion website contains an abundance PowerLectureTM with ExamView The CD ROM provides the instructor with dynamic media tools for teaching college algebra. PowerPoint lecture slides and art slides of the figures from the text, together with electronic files for the test bank and a link to the Solution Builder, are available. The algorithmic ExamView allows you to create, deliver, and customize tests both print and online in minutes with this easy to use assessment system. Enhance how your students interact with you, your lecture, and each other. Solutions Builder This is an electronic version of the complete solutions manual available via the PowerLecture and Instructors Companion Website. It provides instructors with an efficient method for creating solution sets to homework or exams that can then be printed or posted. x www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Supplements xi Supplements for the Student Student Companion Website student resources. This free companion website contains an abundance of Instructional DVDs Keyed to the text by section, these DVDs provide comprehensive coverage of the coursealong with additional explanations of concepts, sample problems, and applicationsto help students review essential topics. Student Study and Solutions Manual This guide offers step by step solutions for all odd numbered text exercises, Chapter and Cumulative Tests, and Practice Tests with solutions. Premium eBook The Premium eBook offers an interactive version of the textbook with search features, highlighting and note making tools, and direct links to videos or tutorials that elaborate on the text discussions. Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found in Larsons text, and then enhances it to help you learn Algebra and Trigonometry more effectively. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Prerequisites P.1 Review of Real Numbers and Their Properties P.2 Exponents and Radicals P.3 Polynomials and Special Products P.4 Factoring Polynomials P.5 Rational Expressions P.6 The Rectangular Coordinate System and Graphs P In Mathematics Real numbers, exponents, radicals, and polynomials are used in many different branches of mathematics. The concepts in this chapter are used to model compound interest, volumes, rates of change, and other real life applications. For instance, polynomials can be used to model the stopping distance of an automobile. See Exercise 116, page 36. Darren McCollester Getty Images News Getty Images In Real Life IN CAREERS There are many careers that use prealgebra concepts. Several are listed below. Engineer Exercise 115, page 35 Financial Analyst Exercises 99 and 100, page 54 Chemist Exercise 148, page 44 Meteorologist Exercise 114, page 70 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 2 Chapter P Prerequisites P.1 REVIEW OF REAL NUMBERS AND THEIR PROPERTIES What you should learn Represent and classify real numbers. Order real numbers and use inequalities. Find the absolute values of real numbers and find the distance between two real numbers. Evaluate algebraic expressions. Use the basic rules and properties of algebra. Real Numbers Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as 4 3 5, 9, 0, , 0.666 . . . , 28.21, 2, , and 32. 3 Here are some important subsets each member of subset B is also a member of set A of the real numbers. The three dots, called ellipsis points, indicate that the pattern continues indefinitely. 1, 2, 3, 4, . . . Why you should learn it Real numbers are used to represent many real life quantities. For example, in Exercises 8388 on page 13, you will use real numbers to represent the federal deficit. Set of natural numbers 0, 1, 2, 3, 4, . . . Set of whole numbers . . . , 3, 2, 1, 0, 1, 2, 3, . . . Set of integers A real number is rational if it can be written as the ratio pq of two integers, where q 0. For instance, the numbers 1 1 125 0.3333 . . . 0.3, 0.125, and 1.126126 . . . 1.126 3 8 111 are rational. The decimal representation of a rational number either repeats as in 173 1 55 3.145 or terminates as in 2 0.5. A real number that cannot be written as the ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers 2 1.4142135 . . . 1.41 3.1415926 . . . 3.14 and are irrational. The symbol means is approximately equal to. Figure P.1 shows subsets of real numbers and their relationships to each other. Real numbers Example 1 Classifying Real Numbers Determine which numbers in the set Irrational numbers Rational numbers Integers Negative integers Natural numbers FIGURE 13, Noninteger fractions positive and negative 1 3 5 8 5, 1, , 0, , 2, , 7 are a natural numbers, b whole numbers, c integers, d rational numbers, and e irrational numbers. Solution a. Natural numbers: 7 b. Whole numbers: 0, 7 c. Integers: 13, 1, 0, 7 Whole numbers Zero P.1 Subsets of real numbers 1 5 d. Rational numbers: 13, 1, , 0, , 7 3 8 e. Irrational numbers: 5, 2, Now try Exercise 11. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 3 Review of Real Numbers and Their Properties Real numbers are represented graphically on the real number line. When you draw a point on the real number line that corresponds to a real number, you are plotting the real number. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure P.2. The term nonnegative describes a number that is either positive or zero. Origin Negative direction FIGURE 4 3 2 1 0 1 2 3 Positive direction 4 P.2 The real number line As illustrated in Figure P.3, there is a one to one correspondence between real numbers and points on the real number line. 53 3 2 1 0 2.4 0.75 1 2 3 3 Every real number corresponds to exactly one point on the real number line. FIGURE 2 2 1 0 1 2 3 Every point on the real number line corresponds to exactly one real number. P.3 One to one correspondence Example 2 Plotting Points on the Real Number Line Plot the real numbers on the real number line. a. 7 4 b. 2.3 c. 2 3 d. 1.8 Solution All four points are shown in Figure P.4. 1.8 74 2 FIGURE 2 3 1 0 2.3 1 2 3 P.4 a. The point representing the real number 74 1.75 lies between 2 and 1, but closer to 2, on the real number line. b. The point representing the real number 2.3 lies between 2 and 3, but closer to 2, on the real number line. c. The point representing the real number 23 0.666 . . . lies between 0 and 1, but closer to 1, on the real number line. d. The point representing the real number 1.8 lies between 2 and 1, but closer to 2, on the real number line. Note that the point representing 1.8 lies slightly to the left of the point representing 74. Now try Exercise 17. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 4 Chapter P Prerequisites Ordering Real Numbers One important property of real numbers is that they are ordered. Definition of Order on the Real Number Line a 1 If a and b are real numbers, a is less than b if b a is positive. The order of a and b is denoted by the inequality a b. This relationship can also be described by saying that b is greater than a and writing b a. The inequality a b means that a is less than or equal to b, and the inequality b a means that b is greater than or equal to a. The symbols , , , and are inequality symbols. b 0 1 2 FIGURE P.5 a b if and only if a lies to the left of b. Geometrically, this definition implies that a b if and only if a lies to the left of b on the real number line, as shown in Figure P.5. Example 3 4 3 FIGURE 4 2 a. 3, 0 2 1 0 1 1 1 , 4 3 1 1 d. , 5 2 1 c. Because 14 lies to the left of 3 on the real number line, as shown in Figure P.8, you can say that 14 is less than 13, and write 14 13. P.8 12 15 1 FIGURE c. a. Because 3 lies to the left of 0 on the real number line, as shown in Figure P.6, you can say that 3 is less than 0, and write 3 0. b. Because 2 lies to the right of 4 on the real number line, as shown in Figure P.7, you can say that 2 is greater than 4, and write 2 4. 1 3 0 b. 2, 4 Solution P.7 1 4 FIGURE Place the appropriate inequality symbol or between the pair of real numbers. 0 P.6 3 FIGURE 1 Ordering Real Numbers d. Because 15 lies to the right of 12 on the real number line, as shown in Figure P.9, you can say that 15 is greater than 12, and write 15 12. 0 Now try Exercise 25. P.9 Example 4 Interpreting Inequalities Describe the subset of real numbers represented by each inequality. a. x 2 x2 x 0 FIGURE 1 2 3 4 P.10 2 x 3 x 2 1 FIGURE P.11 0 1 2 3 b. 2 x 3 Solution a. The inequality x 2 denotes all real numbers less than or equal to 2, as shown in Figure P.10. b. The inequality 2 x 3 means that x 2 and x 3. This double inequality denotes all real numbers between 2 and 3, including 2 but not including 3, as shown in Figure P.11. Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 5 Review of Real Numbers and Their Properties Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b are the endpoints of each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open interval are not included in the interval. Bounded Intervals on the Real Number Line Notation Interval Type Closed a, b The reason that the four types of intervals at the right are called bounded is that each has a finite length. An interval that does not have a finite length is unbounded see below . WARNING CAUTION Whenever you write an interval containing or , always use a parenthesis and never a bracket. This is because and are never an endpoint of an interval and therefore are not included in the interval. a, b Open a, b Inequality Graph a x b x a b a b a b a b a x b x a x b a, b x a x b x The symbols , positive infinity, and , negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the unboundedness of an interval such as 1, or , 3. Unbounded Intervals on the Real Number Line Notation a, Interval Type a, Open Inequality x a Graph x a x a x a , b x b x b , b Open , Entire real line x b x b Example 5 x x Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. b. m is at least 3. c. All x in the interval 3, 5 Solution a. The statement c is at most 2 can be represented by c 2. b. The statement m is at least 3 can be represented by m 3. c. All x in the interval 3, 5 can be represented by 3 x 5. Now try Exercise 45. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 6 Chapter P Prerequisites Example 6 Interpreting Intervals Give a verbal description of each interval. a. 1, 0 b. 2, c. , 0 Solution a. This interval consists of all real numbers that are greater than 1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 41. Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Definition of Absolute Value If a is a real number, then the absolute value of a is a a, a, if a 0 . if a 0 Notice in this definition that the absolute value of a real number is never negative. For instance, if a 5, then 5 5 5. The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, 0 0. Example 7 Finding Absolute Values 2 2 3 3 a. 15 15 b. c. 4.3 4.3 d. 6 6 6 Now try Exercise 51. Example 8 Evaluate Evaluating the Absolute Value of a Number x for a x 0 and b x 0. x Solution a. If x 0, then x x and x x 1. x b. If x 0, then x x and x x x 1. x x Now try Exercise 59. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 Review of Real Numbers and Their Properties 7 The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a b, a b, Example 9 or a b. Law of Trichotomy Comparing Real Numbers Place the appropriate symbol , , or = between the pair of real numbers. a. 43 b. 1010 c. 77 Solution a. 4 3 because 4 4 and 3 3, and 4 is greater than 3. b. 10 10 because 10 10 and 10 10. c. 7 7 because 7 7 and 7 7, and 7 is less than 7. Now try Exercise 61. Properties of Absolute Values 2 1 0 2. a a 3. ab ab 4. a a, b b b 0 Absolute value can be used to define the distance between two points on the real number line. For instance, the distance between 3 and 4 is 7 3 1. a 0 1 2 3 4 P.12 The distance between 3 and 4 is 7. 3 4 7 7 FIGURE as shown in Figure P.12. Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is da, b b a a b. Example 10 Finding a Distance Find the distance between 25 and 13. Solution The distance between 25 and 13 is given by 25 13 38 38. Distance between 25 and 13 The distance can also be found as follows. 13 25 38 38 Distance between 25 and 13 Now try Exercise 67. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 8 Chapter P Prerequisites Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 2x 3, 5x, 4 , x2 2 7x y Definition of an Algebraic Expression An algebraic expression is a collection of letters variables and real numbers constants combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition. For example, x 2 5x 8 x 2 5x 8 has three terms: x 2 and 5x are the variable terms and 8 is the constant term. The numerical factor of a term is called the coefficient. For instance, the coefficient of 5x is 5, and the coefficient of x 2 is 1. Example 11 Identifying Terms and Coefficients Algebraic Expression 1 7 b. 2x2 6x 9 3 1 c. x4 y x 2 a. 5x Terms Coefficients 1 7 2x2, 6x, 9 3 1 4 , x , y x 2 1 7 2, 6, 9 1 3, , 1 2 5x, 5, Now try Exercise 89. To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression, as shown in the next example. Example 12 Evaluating Algebraic Expressions Expression a. 3x 5 b. 3x 2 2x 1 2x c. x1 Value of Variable x3 x 1 x 3 Substitute Value of Expression 33 5 312 21 1 23 3 1 9 5 4 3210 6 3 2 Note that you must substitute the value for each occurrence of the variable. Now try Exercise 95. When an algebraic expression is evaluated, the Substitution Principle is used. It states that If a b, then a can be replaced by b in any expression involving a. In Example 12 a , for instance, 3 is substituted for x in the expression 3x 5. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 Review of Real Numbers and Their Properties 9 Basic Rules of Algebra There are four arithmetic operations with real numbers: addition, multiplication, subtraction, and division, denoted by the symbols , or , , and or . Of these, addition and multiplication are the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively. Definitions of Subtraction and Division Subtraction: Add the opposite. a b a b Division: Multiply by the reciprocal. If b 0, then ab a b b . 1 a In these definitions, b is the additive inverse or opposite of b, and 1b is the multiplicative inverse or reciprocal of b. In the fractional form ab, a is the numerator of the fraction and b is the denominator. Because the properties of real numbers below are true for variables and algebraic expressions as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the first property states that the order in which two real numbers are added does not affect their sum. Basic Rules of Algebra Let a, b, and c be real numbers, variables, or algebraic expressions. Property Commutative Property of Addition: Commutative Property of Multiplication: Associative Property of Addition: Associative Property of Multiplication: Distributive Properties: Additive Identity Property: Multiplicative Identity Property: Additive Inverse Property: Multiplicative Inverse Property: Example abba ab ba a b c a b c ab c abc ab c ab ac a bc ac bc a0a a1a a a 0 1 a 1, a 0 a 4x x 2 x 2 4x 4 x x 2 x 24 x x 5 x 2 x 5 x 2 2x 3y8 2x3y 8 3x5 2x 3x 5 3x 2x y 8 y y y 8 y 5y 2 0 5y 2 4x 21 4x 2 5x 3 5x 3 0 1 x 2 4 2 1 x 4 Because subtraction is defined as adding the opposite, the Distributive Properties are also true for subtraction. For instance, the subtraction form of ab c ab ac is ab c ab ac. Note that the operations of subtraction and division are neither commutative nor associative. The examples 73 3 7 and 20 4 4 20 show that subtraction and division are not commutative. Similarly 5 3 2 5 3 2 and 16 4 2 16 4 2 demonstrate that subtraction and division are not associative. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 10 Chapter P Prerequisites Example 13 Identifying Rules of Algebra Identify the rule of algebra illustrated by the statement. a. 5x32 25x3 b. 4x 31 4x 31 0 c. 7x 1 1, 7x x 0 d. 2 5x2 x2 2 5x2 x2 Solution a. This statement illustrates the Commutative Property of Multiplication. In other words, you obtain the same result whether you multiply 5x3 by 2, or 2 by 5x3. b. This statement illustrates the Additive Inverse Property. In terms of subtraction, this property simply states that when any expression is subtracted from itself the result is 0. c. This statement illustrates the Multiplicative Inverse Property. Note that it is important that x be a nonzero number. If x were 0, the reciprocal of x would be undefined. d. This statement illustrates the Associative Property of Addition. In other words, to form the sum 2 5x2 x2 it does not matter whether 2 and 5x2, or 5x2 and x2 are added first. Now try Exercise 101. Properties of Negation and Equality Let a, b, and c be real numbers, variables, or algebraic expressions. Notice the difference between the opposite of a number and a negative number. If a is already negative, then its opposite, a, is positive. For instance, if a 5, then a 5 5. Property 1. 1 a a Example 17 7 2. a a 6 6 3. ab ab ab 53 5 3 53 4. ab ab 2x 2x 5. a b a b x 8 x 8 6. If a b, then a c b c. 1 2 7. If a b, then ac bc. 42 8. If a c b c, then a b. 1.4 1 75 1 1.4 75 9. If ac bc and c 3x 3 x 8 0, then a b. www.elsolucionario.net 3 0.5 3 2 16 2 4 x4 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 Review of Real Numbers and Their Properties 11 Properties of Zero The or in the Zero Factor Property includes the possibility that either or both factors may be zero. This is an inclusive or, and it is the way the word or is generally used in mathematics. Let a and b be real numbers, variables, or algebraic expressions. 1. a 0 a and a 0 a 3. 0 0, a a 2. a 0 4. 00 a is undefined. 0 5. Zero Factor Property: If ab 0, then a 0 or b 0. Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that b 0 and d 0. 1. Equivalent Fractions: 2. Rules of Signs: c a if and only if ad bc. b d a a a a a and b b b b b 3. Generate Equivalent Fractions: a ac , c b bc 4. Add or Subtract with Like Denominators: a c ac b b b 5. Add or Subtract with Unlike Denominators: In Property 1 of fractions, the phrase if and only if implies two statements. One statement is: If ab cd, then ad bc. The other statement is: If ad bc, where b 0 and d 0, then ab cd. 6. Multiply Fractions: 7. Divide Fractions: Example 14 a b c 0 a c ad bc b d bd ac d bd a c a b d b d ad c bc , c 0 Properties and Operations of Fractions a. Equivalent fractions: x 3 x 3x 5 3 5 15 c. Add fractions with unlike denominators: b. Divide fractions: 7 3 7 2 14 x 2 x 3 3x x 2x 5 x 3 2x 11x 3 5 35 15 Now try Exercise 119. If a, b, and c are integers such that ab c, then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factorsitself and 1such as 2, 3, 5, 7, and 11. The numbers 4, 6, 8, 9, and 10 are composite because each can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers in precisely one way disregarding order . For instance, the prime factorization of 24 is 24 2 2 2 3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 12 Chapter P P.1 Prerequisites EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. p of two integers, where q 0. q ________ numbers have infinite nonrepeating decimal representations. The point 0 on the real number line is called the ________. The distance between the origin and a point representing a real number on the real number line is the ________ ________ of the real number. A number that can be written as the product of two or more prime numbers is called a ________ number. An integer that has exactly two positive factors, the integer itself and 1, is called a ________ number. An algebraic expression is a collection of letters called ________ and real numbers called ________. The ________ of an algebraic expression are those parts separated by addition. The numerical factor of a variable term is the ________ of the variable term. The ________ ________ states that if ab 0, then a 0 or b 0. 1. A real number is ________ if it can be written as the ratio 2. 3. 4. 5. 6. 7. 8. 9. 10. SKILLS AND APPLICATIONS In Exercises 1116, determine which numbers in the set are a natural numbers, b whole numbers, c integers, d rational numbers, and e irrational numbers. 11. 12. 13. 14. 15. 16. 9, 72, 5, 23, 2, 0, 1, 4, 2, 11 5, 7, 73, 0, 3.12, 54 , 3, 12, 5 2.01, 0.666 . . . , 13, 0.010110111 . . . , 1, 6 2.3030030003 . . . , 0.7575, 4.63, 10, 75, 4 , 13, 63, 122, 7.5, 1, 8, 22 25, 17, 125, 9, 3.12, 12, 7, 11.1, 13 In Exercises 17 and 18, plot the real numbers on the real number line. 52 7 2 17. a 3 b 18. a 8.5 b 4 3 c d 5.2 8 c 4.75 d 3 In Exercises 1922, use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. 19. 21. 5 8 41 333 20. 22. 1 3 6 11 24. 3 7 2 6 1 5 0 4 1 3 2 25. 4, 8 3 27. 2, 7 26. 3.5, 1 16 28. 1, 3 5 2 29. 6, 3 8 3 30. 7, 7 In Exercises 31 42, a give a verbal description of the subset of real numbers represented by the inequality or the interval, b sketch the subset on the real number line, and c state whether the interval is bounded or unbounded. x 5 x 0 4, 2 x 2 1 x 0 41. 2, 5 31. 33. 35. 37. 39. x 2 x 3 , 2 0 x 5 0 x 6 42. 1, 2 32. 34. 36. 38. 40. In Exercises 4350, use inequality notation and interval notation to describe the set. In Exercises 23 and 24, approximate the numbers and place the correct symbol or between them. 23. In Exercises 2530, plot the two real numbers on the real number line. Then place the appropriate inequality symbol or between them. 2 3 1 0 43. 44. 45. 46. 47. 48. 49. 50. y is nonnegative. y is no more than 25. x is greater than 2 and at most 4. y is at least 6 and less than 0. t is at least 10 and at most 22. k is less than 5 but no less than 3. The dogs weight W is more than 65 pounds. The annual rate of inflation r is expected to be at least 2.5 but no more than 5. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.1 In Exercises 5160, evaluate the expression. 10 0 3 8 4 1 1 2 3 3 5 5 58. 33 57. 59. x 2, x2 x 1 60. , x1 79. 80. 81. 82. x 2 x 1 In Exercises 6166, place the correct symbol , , or between the two real numbers. 61. 62. 63. 64. 65. 66. BUDGET VARIANCE In Exercises 7982, the accounting department of a sports drink bottling company is checking to see whether the actual expenses of a department differ from the budgeted expenses by more than 500 or by more than 5. Fill in the missing parts of the table, and determine whether each actual expense passes the budget variance test. 3 3 44 5 5 66 2 2 a b 0.05b 2600 2 2 a 126, b 75 a 126, b 75 a 52, b 0 a 14, b 11 4 16 112 a 5 , b 75 2407.3 2400 2200 2025.5 2000 1853.4 1880.3 1800 1600 1722.0 1453.2 1400 1200 1996 1998 2000 2002 2004 2006 Year 72. a 9.34, b 5.65 In Exercises 7378, use absolute value notation to describe the situation. 73. 74. 75. 76. 77. Budgeted Actual Expense, b Expense, a 112,700 113,356 9,400 9,772 37,640 37,335 2,575 2,613 FEDERAL DEFICIT In Exercises 8388, use the bar graph, which shows the receipts of the federal government in billions of dollars for selected years from 1996 through 2006. In each exercise you are given the expenditures of the federal government. Find the magnitude of the surplus or deficit for the year. Source: U.S. Office of Management and Budget In Exercises 6772, find the distance between a and b. 67. 68. 69. 70. 71. Wages Utilities Taxes Insurance Receipts in billions of dollars 51. 52. 53. 54. 55. 56. 13 Review of Real Numbers and Their Properties The distance between x and 5 is no more than 3. The distance between x and 10 is at least 6. y is at least six units from 0. y is at most two units from a. While traveling on the Pennsylvania Turnpike, you pass milepost 57 near Pittsburgh, then milepost 236 near Gettysburg. How many miles do you travel during that time period 78. The temperature in Bismarck, North Dakota was 60F at noon, then 23F at midnight. What was the change in temperature over the 12 hour period 83. 84. 85. 86. 87. 88. Year Receipts Expenditures 1996 1998 2000 2002 2004 2006 1560.6 billion 1652.7 billion 1789.2 billion 2011.2 billion 2293.0 billion 2655.4 billion Receipts Expenditures In Exercises 8994, identify the terms. Then identify the coefficients of the variable terms of the expression. 89. 7x 4 91. 3x 2 8x 11 x 93. 4x 3 5 2 www.elsolucionario.net 90. 6x 3 5x 92. 33x 2 1 x2 94. 3x 4 4 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 14 Chapter P Prerequisites In Exercises 95100, evaluate the expression for each value of x. If not possible, state the reason. 95. 96. 97. 98. 99. 100. Expression 4x 6 9 7x x 2 3x 4 x 2 5x 4 x1 x1 x x2 a a a a Values x 1 b x 3 b x 2 b x 1 b x0 x3 x2 x1 a x 1 b x 1 a x 2 b x 2 In Exercises 101112, identify the rule s of algebra illustrated by the statement. 101. x 9 9 x 102. 2 12 1 1 103. h 6 1, h 6 h6 104. x 3 x 3 0 105. 2x 3 2 x 2 3 106. z 2 0 z 2 107. 1 1 x 1 x 108. z 5x z x 5 x 109. x y 10 x y 10 110. x3y x 3y 3x y 111. 3t 4 3 t 3 4 1 1 112. 77 12 7 712 1 12 12 5 119. 120. 5x 6 2 In Exercises 121 and 122, use the real numbers A, B, and C shown on the number line. Determine the sign of each expression. 0 121. a A b B A 122. a C b A C 0.0001 0.000001 b Use the result from part a to make a conjecture about the value of 5n as n approaches 0. 124. CONJECTURE a Use a calculator to complete the table. 1 n 10 100 10,000 100,000 5n b Use the result from part a to make a conjecture about the value of 5n as n increases without bound. 128. Because 9 A 0.01 5n 127. If a b, then EXPLORATION C B 0.5 125. If a 0 and b 0, then a b 0. 126. If a 0 and b 0, then ab 0. 4 114. 76 7 6 13 10 116. 11 33 66 4 118. 6 8 2x x 3 4 1 n TRUE OR FALSE In Exercises 125128, determine whether the statement is true or false. Justify your answer. In Exercises 113120, perform the operation s . Write fractional answers in simplest form. 3 16 113. 16 5 1 5 115. 8 12 6 1 117. 12 4 123. CONJECTURE a Use a calculator to complete the table. 1 1 , where a a b 0 and b 0. ab a b c c c , then . c c c ab a b 129. THINK ABOUT IT Consider u v and u v, where u 0 and v 0. a Are the values of the expressions always equal If not, under what conditions are they unequal b If the two expressions are not equal for certain values of u and v, is one of the expressions always greater than the other Explain. 130. THINK ABOUT IT Is there a difference between saying that a real number is positive and saying that a real number is nonnegative Explain. 131. THINK ABOUT IT Because every even number is divisible by 2, is it possible that there exist any even prime numbers Explain. 132. THINK ABOUT IT Is it possible for a real number to be both rational and irrational Explain. 133. WRITING Can it ever be true that a a for a real number a Explain. 134. CAPSTONE Describe the differences among the sets of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 Exponents and Radicals 15 P.2 EXPONENTS AND RADICALS What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify and combine radicals. Rationalize denominators and numerators. Use properties of rational exponents. Integer Exponents Repeated multiplication can be written in exponential form. Repeated Multiplication aaaa a Exponential Form a5 444 43 2x2x2x2x 2x4 Why you should learn it Exponential Notation Real numbers and algebraic expressions are often written with exponents and radicals. For instance, in Exercise 121 on page 27, you will use an expression involving rational exponents to find the times required for a funnel to empty for different water heights. If a is a real number and n is a positive integer, then an a a a. . .a n factors where n is the exponent and a is the base. The expression an is read a to the nth power. An exponent can also be negative. In Property 3 below, be sure you see how to use a negative exponent. Properties of Exponents T E C H N O LO G Y Let a and b be real numbers, variables, or algebraic expressions, and let m and n be integers. All denominators and bases are nonzero. Property a mn You can use a calculator to evaluate exponential expressions. When doing so, it is important to know when to use parentheses because the calculator follows the order of operations. For instance, evaluate 24 as follows. 1. Scientific: 4. a0 1, 2 yx 4 2 Graphing: 4 2. a ma n 32 am x7 an amn 3. an x4 1 1 an a a n 0 34 Example 324 36 729 x7 4 x 3 y4 1 1 y4 y x 2 10 1 5. abm am bm 5x3 53x3 125x3 6. amn amn y34 y34 y12 ENTER The display will be 16. If you omit the parentheses, the display will be 16. 4 7. b a m am bm 8. a2 a2 a2 www.elsolucionario.net x 2 3 1 y12 23 8 3 3 x x 22 22 22 4 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 16 Chapter P Prerequisites It is important to recognize the difference between expressions such as 24 and 24. In 24, the parentheses indicate that the exponent applies to the negative sign as well as to the 2, but in 24 24, the exponent applies only to the 2. So, 24 16 and 24 16. The properties of exponents listed on the preceding page apply to all integers m and n, not just to positive integers, as shown in the examples in this section. Example 1 Evaluating Exponential Expressions a. 52 55 25 Negative sign is part of the base. b. 5 55 25 Negative sign is not part of the base. 2 c. 2 d. 2 2 4 14 2 32 5 Property 1 44 1 1 446 42 2 46 4 16 Properties 2 and 3 Now try Exercise 11. Example 2 Evaluating Algebraic Expressions Evaluate each algebraic expression when x 3. a. 5x2 b. 1 x3 3 Solution a. When x 3, the expression 5x2 has a value of 5x2 532 5 5 . 32 9 1 b. When x 3, the expression x3 has a value of 3 1 1 1 x3 33 27 9. 3 3 3 Now try Exercise 23. Example 3 Using Properties of Exponents Use the properties of exponents to simplify each expression. a. 3ab44ab3 b. 2xy23 c. 3a4a20 5xy 3 2 d. Solution a. 3ab44ab3 34aab4b3 12a 2b b. 2xy 23 23x3 y 23 8x3y6 c. 3a4a 20 3a1 3a, a d. y 5x 3 2 0 52x 32 25x 6 2 y2 y Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 Example 4 Rarely in algebra is there only one way to solve a problem. Dont be concerned if the steps you use to solve a problem are not exactly the same as the steps presented in this text. The important thing is to use steps that you understand and, of course, steps that are justified by the rules of algebra. For instance, you might prefer the following steps for Example 4 d . 3x 2 y 2 y 3x 2 2 y2 4 9x Note how Property 3 is used in the first step of this solution. The fractional form of this property is a b m 17 Rewriting with Positive Exponents Rewrite each expression with positive exponents. a. x1 b. 1 3x2 c. 12a3b4 4a2b 2 2 d. 3xy Solution 1 x a. x1 Property 3 1 1x 2 x 2 2 3x 3 3 3 4 3 12a b 12a a2 c. 2 4a b 4b b4 The exponent 2 does not apply to 3. b. 2 2 d. 3xy . b a Exponents and Radicals m Property 3 3a5 b5 Property 1 32x 22 y2 Properties 5 and 7 32x4 y2 Property 6 y2 32x 4 Property 3 y2 9x 4 Simplify. Now try Exercise 41. Scientific Notation HISTORICAL NOTE The French mathematician Nicolas Chuquet ca. 1500 wrote Triparty en la science des nombres, in which a form of exponent notation was used. Our expressions 6x3 and 10x2 were written as .6.3 and .10.2. Zero and negative exponents were also represented, so x0 would be written as .1.0 and 3x2 as .3.2m. Chuquet wrote that .72.1 divided by .8.3 is .9.2m. That is, 72x 8x3 9x2. Exponents provide an efficient way of writing and computing with very large or very small numbers. For instance, there are about 359 billion billion gallons of water on Earththat is, 359 followed by 18 zeros. 359,000,000,000,000,000,000 It is convenient to write such numbers in scientific notation. This notation has the form c 10n, where 1 c 10 and n is an integer. So, the number of gallons of water on Earth can be written in scientific notation as 3.59 100,000,000,000,000,000,000 3.59 1020. The positive exponent 20 indicates that the number is large 10 or more and that the decimal point has been moved 20 places. A negative exponent indicates that the number is small less than 1 . For instance, the mass in grams of one electron is approximately 9.0 1028 0.0000000000000000000000000009. 28 decimal places www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 18 Chapter P Prerequisites Example 5 Scientific Notation Write each number in scientific notation. a. 0.0000782 b. 836,100,000 Solution a. 0.0000782 7.82 105 b. 836,100,000 8.361 108 Now try Exercise 45. Example 6 Decimal Notation Write each number in decimal notation. a. 9.36 106 b. 1.345 102 Solution a. 9.36 106 0.00000936 b. 1.345 102 134.5 Now try Exercise 55. T E C H N O LO G Y Most calculators automatically switch to scientific notation when they are showing large or small numbers that exceed the display range. To enter numbers in scientific notation, your calculator should have an exponential entry key labeled or EE EXP . Consult the users guide for your calculator for instructions on keystrokes and how numbers in scientific notation are displayed. Example 7 Evaluate Using Scientific Notation 2,400,000,0000.0000045 . 0.000031500 Solution Begin by rewriting each number in scientific notation and simplifying. 2,400,000,0000.0000045 2.4 1094.5 106 0.000031500 3.0 1051.5 103 2.44.5103 4.5102 2.4105 240,000 Now try Exercise 63 b . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 Exponents and Radicals 19 Radicals and Their Properties A square root of a number is one of its two equal factors. For example, 5 is a square root of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube root of a number is one of its three equal factors, as in 125 53. Definition of nth Root of a Number Let a and b be real numbers and let n 2 be a positive integer. If a bn then b is an nth root of a. If n 2, the root is a square root. If n 3, the root is a cube root. Some numbers have more than one nth root. For example, both 5 and 5 are square roots of 25. The principal square root of 25, written as 25, is the positive root, 5. The principal nth root of a number is defined as follows. Principal nth Root of a Number Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol n a. Principal nth root The positive integer n is the index of the radical, and the number a is the radicand. 2 a. The plural of index is If n 2, omit the index and write a rather than indices. A common misunderstanding is that the square root sign implies both negative and positive roots. This is not correct. The square root sign implies only a positive root. When a negative root is needed, you must use the negative sign with the square root sign. Incorrect: 4 2 Example 8 Correct: 4 2 and 4 2 Evaluating Expressions Involving Radicals a. 36 6 because 62 36. b. 36 6 because 36 62 6 6. c. 5 5 because 125 64 4 4 3 3 53 125 . 43 64 5 32 2 because 25 32. d. 4 81 is not a real number because there is no real number that can be raised to the e. fourth power to produce 81. Now try Exercise 65. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 20 Chapter P Prerequisites Here are some generalizations about the nth roots of real numbers. Generalizations About nth Roots of Real Numbers Real Number a Integer n Root s of a a, Example 4 81 3 81 3, a 0 n 0, n is even. n 4 a 0 or a 0 n is odd. n a 3 8 2 a 0 n is even. No real roots 4 is not a real number. a0 n is even or odd. n 0 0 5 0 0 n a Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called perfect cubes because they have integer cube roots. Properties of Radicals Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers. Property n am 1. n a m n a 2. 3. n a n b n b n ab ab , n Example 2 22 4 3 82 3 8 b 5 7 5 7 35 4 27 0 4 9 279 4 4 3 m n a mn a 4. 3 6 10 10 n 5. a a 3 2 3 n n 6. For n even, a a. 122 12 12 n 3 123 12 n an a. For n odd, A common special case of Property 6 is a2 a. Example 9 Using Properties of Radicals Use the properties of radicals to simplify each expression. a. 8 3 5 b. 2 3 3 x3 c. 6 y6 d. Solution a. 8 2 8 2 16 4 b. c. 3 3 d. 6 6 3 5 3 5 x x y y Now try Exercise 77. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 Exponents and Radicals 21 Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied. 1. All possible factors have been removed from the radical. 2. All fractions have radical free denominators accomplished by a process called rationalizing the denominator . 3. The index of the radical is reduced. To simplify a radical, factor the radicand into factors whose exponents are multiples of the index. The roots of these factors are written outside the radical, and the leftover factors make up the new radicand. WARNING CAUTION When you simplify a radical, it is important that both expressions are defined for the same values of the variable. For instance, in Example 10 b , 75x3 and 5x3x are both defined only for nonnegative values of x. Similarly, in Example 10 c , 4 5x4 and 5 x are both defined for all real values of x. Example 10 Simplifying Even Roots Perfect 4th power Leftover factor 4 48 4 16 a. 4 24 4 3 3 3 2 Perfect square Leftover factor 3x 5x 3x b. 75x3 25x 2 Find largest square factor. 2 5x3x Find root of perfect square. 4 5x4 5x 5 x c. Now try Exercise 79 a . Example 11 Simplifying Odd Roots Perfect cube Leftover factor 3 24 3 8 a. 3 23 3 3 3 3 2 Perfect cube Leftover factor 3a 3 2a3 3a 3 24a4 3 8a3 b. 3 3a 2a c. 3 40x6 Find largest cube factor. Find root of perfect cube. 5 5 3 8x6 3 3 5 2x 2 Find largest cube factor. 2x 2 3 Find root of perfect cube. Now try Exercise 79 b . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 22 Chapter P Prerequisites Radical expressions can be combined added or subtracted if they are like radicalsthat is, if they have the same index and radicand. For instance, 2, 32, and 122 are like radicals, but 3 and 2 are unlike radicals. To determine whether two radicals can be combined, you should first simplify each radical. Example 12 Combining Radicals a. 248 327 216 3 39 3 Find square factors. 83 93 Find square roots and multiply by coefficients. 8 93 Combine like terms. 3 b. 3 16x 3 54x 4 3 8 2 Simplify. 2x 3 27 x3 2x 2x 3x 2x 3 3 2 3x 3 2x Find cube factors. Find cube roots. Combine like terms. Now try Exercise 87. Rationalizing Denominators and Numerators To rationalize a denominator or numerator of the form a bm or a bm, multiply both numerator and denominator by a conjugate: a bm and a bm are conjugates of each other. If a 0, then the rationalizing factor for m is itself, m. For cube roots, choose a rationalizing factor that generates a perfect cube. Example 13 Rationalizing Single Term Denominators Rationalize the denominator of each expression. a. 5 b. 23 2 5 3 Solution a. b. 5 5 23 23 3 3 3 is rationalizing factor. 53 23 Multiply. 53 6 Simplify. 2 2 3 3 5 5 3 52 3 52 3 52 is rationalizing factor. 3 52 2 3 53 Multiply. 3 25 2 5 Simplify. Now try Exercise 95. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 Example 14 23 Rationalizing a Denominator with Two Terms 2 2 3 7 3 7 Exponents and Radicals Multiply numerator and denominator by conjugate of denominator. 3 7 3 7 23 7 33 37 73 7 7 23 7 32 7 2 Use Distributive Property. Simplify. 23 7 97 Square terms of denominator. 23 7 3 7 2 Simplify. Now try Exercise 97. Sometimes it is necessary to rationalize the numerator of an expression. For instance, in Section P.5 you will use the technique shown in the next example to rationalize the numerator of an expression from calculus. WARNING CAUTION Do not confuse the expression 5 7 with the expression 5 7. In general, x y does not equal x y. Similarly, x 2 y 2 does not equal x y. Example 15 5 7 2 Rationalizing a Numerator 5 7 5 7 Multiply numerator and denominator by conjugate of numerator. 5 7 2 5 2 7 2 25 7 Simplify. 57 25 7 Square terms of numerator. 2 1 25 7 5 7 Simplify. Now try Exercise 101. Rational Exponents Definition of Rational Exponents If a is a real number and n is a positive integer such that the principal nth root of a exists, then a1n is defined as n a, where 1n is the rational exponent of a. a1n Moreover, if m is a positive integer that has no common factor with n, then n a a mn a1nm m The symbol and n a m. a mn a m1n indicates an example or exercise that highlights algebraic techniques specifically used in calculus. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 24 Chapter P Prerequisites WARNING CAUTION Rational exponents can be tricky, and you must remember that the expression bmn is not n b is a real defined unless number. This restriction produces some unusual looking results. For instance, the number 813 is defined because 3 8 2, but the number 826 is undefined because 6 8 is not a real number. The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken. Power Index n n m b mn b b m When you are working with rational exponents, the properties of integer exponents still apply. For instance, 212213 21213 256. Example 16 Changing From Radical to Exponential Form a. 3 312 2 3xy5 3xy52 b. 3xy5 4 3 c. 2x x 2xx34 2x134 2x74 Now try Exercise 103. Example 17 T E C H N O LO G Y Changing From Exponential to Radical Form a. x 2 y 232 x 2 y 2 x 2 y 23 3 There are four methods of evaluating radicals on most graphing calculators. For square roots, you can use the square root key . For cube roots, you can use the cube root key 3 . For other roots, you can first convert the radical to exponential form and then use the exponential key , or you can use the xth root key x or menu choice . Consult the users guide for your calculator for specific keystrokes. 4 3 b. 2y34z14 2 y3z14 2 yz 1 1 a32 a3 c. a32 5 d. x 0.2 x15 x Now try Exercise 105. Rational exponents are useful for evaluating roots of numbers on a calculator, for reducing the index of a radical, and for simplifying expressions in calculus. Example 18 Simplifying with Rational Exponents 5 32 a. 3245 4 24 1 1 4 2 16 b. 5x533x34 15x5334 15x1112, c. 9 a3 a39 d. 125 3 e. 2x 1 a13 6 125 2x 1 3 a 13 43 0 Reduce index. 5 6 x 3 536 2x 512 5 14313 2x 1, x 1 2 Now try Exercise 115. The expression in Example 18 e is not defined when x 2 12 1 13 1 because 2 013 is not a real number. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 P.2 EXERCISES 25 Exponents and Radicals See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. In the exponential form an, n is the ________ and a is the ________. A convenient way of writing very large or very small numbers is called ________ ________. One of the two equal factors of a number is called a __________ __________ of the number. n a. The ________ ________ ________ of a number a is the nth root that has the same sign as a, and is denoted by n In the radical form a, the positive integer n is called the ________ of the radical and the number a is called the ________. When an expression involving radicals has all possible factors removed, radical free denominators, and a reduced index, it is in ________ ________. Radical expressions can be combined added or subtracted if they are ________ ________. The expressions a bm and a bm are ________ of each other. The process used to create a radical free denominator is known as ________ the denominator. In the expression bmn, m denotes the ________ to which the base is raised and n denotes the ________ or root to be taken. SKILLS AND APPLICATIONS In Exercises 1118, evaluate each expression. In Exercises 3138, simplify each expression. 11. a 3 3 55 12. a 2 5 13. a 330 14. a 23 322 b 3 3 32 b 4 3 b 32 3 2 b 35 53 31. a 5z3 32. a 3x2 3 15. a 4 3 b 4843 2 4 32 22 31 17. a 21 31 18. a 31 22 16. a 3 b 20 b 212 b 322 3x 3, x 2 6x 0, x 10 2x 3, x 3 20x2, x 12 34. a z33z4 7x 2 x3 r4 36. a 6 r 20. 84103 43 22. 4 3 24. 26. 28. 30. a b b b a 2 37. a x2y211 In Exercises 2330, evaluate the expression for the given value of x. 23. 25. 27. 29. 33. a 6y 22y02 35. a In Exercises 19 22, use a calculator to evaluate the expression. If necessary, round your answer to three decimal places. 19. 4352 36 21. 3 7 b 5x4x2 b 4x 30, x 3x 5 b 3 x 25y8 b 10y4 12x y3 b 9x y 4 3 3 4 b y y 7x2, x 4 5x3, x 3 3x 4, x 2 12x3, x 13 38. a 6x70, x 0 3 2 b 5x2z635x2z63 0 In Exercises 3944, rewrite each expression with positive exponents and simplify. 39. a x 50, x 5 40. a 2x50, x 0 41. a 2x 234x31 42. a 4y28y4 43. a 3n 44. a www.elsolucionario.net 32n x 2 xn x 3 xn b 2x 22 b z 23z 21 x 1 b 10 x3y 4 3 b 5 2 a b 3 b 2 b a 3 a a 3 b 3 b b http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 26 Chapter P Prerequisites In Exercises 4552, write the number in scientific notation. 10,250.4 46. 7,280,000 0.000125 48. 0.00052 Land area of Earth: 57,300,000 square miles Light year: 9,460,000,000,000 kilometers Relative density of hydrogen: 0.0000899 gram per cubic centimeter 52. One micron millionth of a meter : 0.00003937 inch 45. 47. 49. 50. 51. In Exercises 53 60, write the number in decimal notation. 53. 1.25 105 54. 1.801 105 3 55. 2.718 10 56. 3.14 104 57. Interior temperature of the sun: 1.5 107 degrees Celsius 58. Charge of an electron: 1.6022 1019 coulomb 59. Width of a human hair: 9.0 105 meter 60. Gross domestic product of the United States in 2007: 1.3743021 1013 dollars Source: U.S. Department of Commerce In Exercises 61 and 62, evaluate each expression without using a calculator. 61. a 2.0 1093.4 104 b 1.2 1075.0 103 2.5 103 b 5.0 102 6.0 108 62. a 3.0 103 In Exercises 63 and 64, use a calculator to evaluate each expression. Round your answer to three decimal places. 0.11 800 365 67,000,000 93,000,000 b 0.0052 63. a 750 1 64. a 9.3 10636.1 104 b 2.414 1046 1.68 1055 In Exercises 6570, evaluate each expression without using a calculator. 65. 66. 67. 68. a a a a 9 2713 3235 10032 1 13 69. a 64 125 70. a 27 13 3 27 b 8 b 3632 16 34 b 81 9 12 b 4 1 25 b 32 1 43 b 125 In Exercises 7176, use a calculator to approximate the number. Round your answer to three decimal places. 71. a 57 3 452 72. a 73. a 12.41.8 74. a 7 4.13.2 2 75. a 4.5 109 76. a 2.65 10413 5 273 b 6 125 b 2.5 b 53 b 133 32 23 133 3 6.3 104 b b 9 104 In Exercises 77 and 78, use the properties of radicals to simplify each expression. 5 2 5 77. a 78. a 12 3 5 96x5 b 4 3x24 b In Exercises 7990, simplify each radical expression. 79. a 20 3 16 80. a 27 81. a 72x3 82. a 54xy4 83. 84. 85. 86. 87. 88. a a a a a a b 89. a 3 16x5 4 3x 4 y 2 250 128 427 75 5x 3x 849x 14100x 348x 2 7 75x 2 3x 1 10x 1 b 780x 2125x 90. a x 3 7 5x 3 7 b 11245x 3 945x 3 3 b 128 b 75 4 182 b z3 32a4 b b2 b 75x2y4 5 160x 8z 4 b b 1032 618 3 16 3 3 54 b b 29y 10y In Exercises 9194, complete the statement with , , or . 113 3 91. 5 3 5 3 92. 93. 532 22 94. 532 42 11 In Exercises 9598, rationalize the denominator of the expression. Then simplify your answer. 95. 97. 1 3 96. 5 14 2 www.elsolucionario.net 98. 8 3 2 3 5 6 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.2 In Exercises 99 102, rationalize the numerator of the expression. Then simplify your answer. 99. 8 2 5 3 101. 3 100. In Exercises 103 110, fill in the missing form of the expression. Radical Form 103. 2.5 3 104. 64 105. 106. 3 216 107. 108. 4 81 3 109. 110. 121. MATHEMATICAL MODELING A funnel is filled with water to a height of h centimeters. The formula t 0.031252 12 h52, 0 h 12 2 3 7 3 102. 4 Rational Exponent Form 8114 14412 24315 1654 represents the amount of time t in seconds that it will take for the funnel to empty. a Use the table feature of a graphing utility to find the times required for the funnel to empty for water heights of h 0, h 1, h 2, . . . , h 12 centimeters. b What value does t appear to be approaching as the height of the water becomes closer and closer to 12 centimeters 122. SPEED OF LIGHT The speed of light is approximately 11,180,000 miles per minute. The distance from the sun to Earth is approximately 93,000,000 miles. Find the time for light to travel from the sun to Earth. EXPLORATION TRUE OR FALSE In Exercises 123 and 124, determine whether the statement is true or false. Justify your answer. In Exercises 111114, perform the operations and simplify. 2x232 111. 12 4 2 x x3 x12 113. 32 1 x x x 43y 23 112. xy13 512 5x52 114. 5x32 In Exercises 115 and 116, reduce the index of each radical. 4 32 115. a 6 x3 116. a 6 x 14 b 4 3x24 b In Exercises 117 and 118, write each expression as a single radical. Then simplify your answer. 117. a 32 118. a 243x 1 4 2x b 3 b 10a7b 119. PERIOD OF A PENDULUM The period T in seconds of a pendulum is T 2L32, where L is the length of the pendulum in feet . Find the period of a pendulum whose length is 2 feet. 120. EROSION A stream of water moving at the rate of v feet per second can carry particles of size 0.03v inches. Find the size of the largest particle that can be carried by a stream flowing at the rate of 34 foot per second. The symbol 27 Exponents and Radicals indicates an example or exercise that highlights 123. x k1 xk x 124. a n k a n k 125. Verify that a0 1, a 0. Hint: Use the property of exponents ama n amn. 126. Explain why each of the following pairs is not equal. a 3x1 3 x b y 3 y 2 y6 c a 2b 34 a6b7 d a b2 a 2 b2 e 4x 2 2x f 2 3 5 127. THINK ABOUT IT Is 52.7 105 written in scientific notation Why or why not 128. List all possible digits that occur in the units place of the square of a positive integer. Use that list to determine whether 5233 is an integer. 129. THINK ABOUT IT Square the real number 53 and note that the radical is eliminated from the denominator. Is this equivalent to rationalizing the denominator Why or why not 130. CAPSTONE a Explain how to simplify the expression 3x3 y22. b Is the expression x4 in simplest form Why 3 or why not algebraic techniques specifically used in calculus. The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 28 Chapter P Prerequisites P.3 POLYNOMIALS AND SPECIAL PRODUCTS What you should learn Polynomials Write polynomials in standard form. Add, subtract, and multiply polynomials. Use special products to multiply polynomials. Use polynomials to solve real life problems. The most common type of algebraic expression is the polynomial. Some examples are 2x 5, 3x 4 7x 2 2x 4, and 5x 2y 2 xy 3. The first two are polynomials in x and the third is a polynomial in x and y. The terms of a polynomial in x have the form ax k, where a is the coefficient and k is the degree of the term. For instance, the polynomial Why you should learn it has coefficients 2, 5, 0, and 1. Polynomials can be used to model and solve real life problems. For instance, in Exercise 106 on page 34, polynomials are used to model the cost, revenue, and profit for producing and selling hats. 2x 3 5x 2 1 2x 3 5 x 2 0 x 1 Definition of a Polynomial in x Let a0, a1, a2, . . . , an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form an x n an1x n1 . . . a1x a 0 where an 0. The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term. Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. In standard form, a polynomial is written with descending powers of x. David NotonMasterfile Example 1 Writing Polynomials in Standard Form Standard Form Degree Leading Coefficient 5x 7 4x 2 3x 2 9x 2 4 7 2 5 9 8 8 8x 0 0 8 Polynomial a. 4x 2 5x 7 2 3x b. 4 9x 2 c. 8 Now try Exercise 19. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polynomials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree of its terms. For instance, the degree of the polynomial 2x 3y6 4xy x7y 4 is 11 because the sum of the exponents in the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest degree term. Expressions are not polynomials if a variable is underneath a radical or if a polynomial expression with degree greater than 0 is in the denominator of a term. The following expressions are not polynomials. x 3 3x x 3 3x12 x2 5 x 2 5x1 x The exponent 12 is not an integer. The exponent 1 is not a nonnegative integer. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.3 Polynomials and Special Products 29 Operations with Polynomials You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms terms having the same variables to the same powers by adding their coefficients. For instance, 3xy 2 and 5xy 2 are like terms and their sum is 3xy 2 5xy 2 3 5 xy 2 2xy 2. WARNING CAUTION When an expression inside parentheses is preceded by a negative sign, remember to distribute the negative sign to each term inside the parentheses, as shown. x 2 x 3 x 2 x 3 Example 2 Sums and Differences of Polynomials a. 5x 3 7x 2 3 x 3 2x 2 x 8 b. 7x4 5x 3 x 3 7x2 2x2 x 3 8 Group like terms. 6x 3 5x 2 x 5 Combine like terms. 4x 2 x2 7x 4 3x4 7x 4 4x 4 x2 4x 2 3x 4 x2 3x 2 7x 2 3x 4x 2 4x 2 3x Distributive Property 4x 3x 2 Group like terms. 3x 4 4x2 Combine like terms. Now try Exercise 41. To find the product of two polynomials, use the left and right Distributive Properties. For example, if you treat 5x 7 as a single quantity, you can multiply 3x 2 by 5x 7 as follows. 3x 25x 7 3x5x 7 25x 7 3x5x 3x7 25x 27 15x 2 21x 10x 14 Product of First terms Product of Outer terms Product of Inner terms Product of Last terms 15x 2 11x 14 Note in this FOIL Method which can only be used to multiply two binomials that the outer O and inner I terms are like terms and can be combined. Example 3 Finding a Product by the FOIL Method Use the FOIL Method to find the product of 2x 4 and x 5. Solution F O I L 2x 4x 5 2x 2 10x 4x 20 2x 2 6x 20 Now try Exercise 59. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 30 Chapter P Prerequisites When multiplying two polynomials, be sure to multiply each term of one polynomial by each term of the other. A vertical arrangement is helpful. Example 4 A Vertical Arrangement for Multiplication Multiply x 2 2x 2 by x 2 2x 2 using a vertical arrangement. Solution x2 2x 2 Write in standard form. x 2x 2 Write in standard form. x 2x 2x x 2x 2 2x 2 2 4 3 2 2x3 4x2 4x 2xx2 2x 2 2x2 4x 4 2x2 2x 2 x 4 0x 3 0x 2 0x 4 x 4 4 Combine like terms. So, x 2 2x 2x 2 2x 2 x 4 4. Now try Exercise 61. Special Products Some binomial products have special forms that occur frequently in algebra. You do not need to memorize these formulas because you can use the Distributive Property to multiply. However, becoming familiar with these formulas will enable you to manipulate the algebra more quickly. Special Products Let u and v be real numbers, variables, or algebraic expressions. Special Product Sum and Difference of Same Terms u vu v u 2 v 2 Example x 4x 4 x 2 42 x 2 16 Square of a Binomial u v 2 u 2 2uv v 2 x 3 2 x 2 2x3 32 x 2 6x 9 u v 2 u 2 2uv v 2 3x 22 3x2 23x2 22 9x 2 12x 4 Cube of a Binomial u v3 u 3 3u 2v 3uv 2 v 3 x 23 x 3 3x 22 3x22 23 x 3 6x 2 12x 8 u v3 u 3 3u 2v 3uv 2 v 3 x 13 x 3 3x 21 3x12 13 x 3 3x 2 3x 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.3 Example 5 Polynomials and Special Products 31 Sum and Difference of Same Terms Find the product of 5x 9 and 5x 9. Solution The product of a sum and a difference of the same two terms has no middle term and takes the form u vu v u 2 v 2. 5x 95x 9 5x2 9 2 25x 2 81 Now try Exercise 67. Example 6 When squaring a binomial, note that the resulting middle term is always twice the product of the two terms. Square of a Binomial Find 6x 52. Solution The square of a binomial has the form u v2 u 2 2uv v 2. 6x 5 2 6x 2 26x5 52 36x 2 60x 25 Now try Exercise 71. Example 7 Cube of a Binomial Find 3x 2 3. Solution The cube of a binomial has the form u v3 u 3 3u 2v 3uv 2 v 3. Note the decreasing powers of u 3x and the increasing powers of v 2. 3x 23 3x3 33x 22 33x22 23 27x 3 54x 2 36x 8 Now try Exercise 73. Example 8 The Product of Two Trinomials Find the product of x y 2 and x y 2. Solution By grouping x y in parentheses, you can write the product of the trinomials as a special product. Difference Sum x y 2x y 2 x y 2x y 2 x y 2 22 x2 2xy y2 Sum and difference of same terms 4 Now try Exercise 81. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 32 Chapter P Prerequisites Application Example 9 An open box is made by cutting squares from the corners of a piece of metal that is 16 inches by 20 inches, as shown in Figure P.13. The edge of each cut out square is x inches. Find the volume of the box when x 1, x 2, and x 3. 20 2x 16 in. 16 2x x x Volume of a Box x x Solution The volume of a rectangular box is equal to the product of its length, width, and height. From the figure, the length is 20 2x, the width is 16 2x, and the height is x. So, the volume of the box is Volume 20 2x16 2xx 20 in. 320 72x 4x 2x 320x 72x 2 4x 3. x 16 2x 20 2x FIGURE P.13 When x 1 inch, the volume of the box is Volume 3201 7212 413 252 cubic inches. When x 2 inches, the volume of the box is Volume 3202 7222 423 384 cubic inches. When x 3 inches, the volume of the box is Volume 3203 7232 433 420 cubic inches. Now try Exercise 109. CLASSROOM DISCUSSION Mathematical Experiment In Example 9, the volume of the open box is given by Volume 320x 72x 2 4x 3. You want to create a box that has as much volume as possible. From Example 9, you know that by cutting one , two , and three inch squares from the corners, you can create boxes whose volumes are 252, 384, and 420 cubic inches, respectively. What are the possible values of x that make sense in this problem Write your answer as an interval. Try several other values of x to find the size of the squares that should be cut from the corners to produce a box that has maximum volume. Write a summary of your findings. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.3 P.3 EXERCISES Polynomials and Special Products 33 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 15, fill in the blanks. 1. For the polynomial an x n an1 x n1 . . . a1x a0, an 0, the degree is ________, the leading coefficient is ________, and the constant term is ________. 2. A polynomial in x in standard form is written with ________ powers of x. 3. A polynomial with one term is called a ________, while a polynomial with two terms is called a ________, and a polynomial with three terms is called a ________. 4. To add or subtract polynomials, add or subtract the ________ ________ by adding their coefficients. 5. The letters in FOIL stand for the following. F ________ O ________ I ________ L ________ In Exercises 68, match the special product form with its name. 6. u vu v u2 v2 7. u v2 u2 2uv v2 8. u v2 u2 2uv v2 a A binomial sum squared b A binomial difference squared c The sum and difference of same terms SKILLS AND APPLICATIONS In Exercises 914, match the polynomial with its description. The polynomials are labeled a , b , c , d , e , and f . a c e 9. 10. 11. b 1 2x3 d 12 2 f 3 x4 x2 10 A polynomial of degree 0 A trinomial of degree 5 A binomial with leading coefficient 2 3x2 x3 3x2 3x 1 3x5 2x3 x 12. A monomial of positive degree 2 13. A trinomial with leading coefficient 3 14. A third degree polynomial with leading coefficient 1 In Exercises 1518, write a polynomial that fits the description. There are many correct answers. 15. 16. 17. 18. A third degree polynomial with leading coefficient 2 A fifth degree polynomial with leading coefficient 6 A fourth degree binomial with a negative leading coefficient A third degree binomial with an even leading coefficient In Exercises 1930, a write the polynomial in standard form, b identify the degree and leading coefficient of the polynomial, and c state whether the polynomial is a monomial, a binomial, or a trinomial. 19. 14x 12 x 5 21. x2 4 3x4 23. 3 x6 20. 2x 2 x 1 22. 7x 24. y 25y2 1 25. 3 27. 1 6x 4 4x 5 29. 4x 3y 26. 8 t2 28. 3 2x 30. x 5y 2x 2y 2 xy 4 In Exercises 3136, determine whether the expression is a polynomial. If so, write the polynomial in standard form. 31. 2x 3x 3 8 3x 4 33. x 2 35. y y 4 y 3 32. 5x4 2x2 x2 34. x 2 2x 3 2 36. y4 y In Exercises 3754, perform the operation and write the result in standard form. 37. 38. 39. 40. 41. 42. 43. 44. 45. 47. 49. 51. 53. 6x 5 8x 15 2x 2 1 x 2 2x 1 t3 1 6t3 5t 5x 2 1 3x 2 5 15x 2 6 8.3x 3 14.7x 2 17 15.6w4 14w 17.4 16.9w4 9.2w 13 5z 3z 10z 8 y 3 1 y 2 1 3y 7 46. y 24y 2 2y 3 3xx 2 2x 1 48. 3x5x 2 5z3z 1 50. 4x3 x 3 1 x 34x 52. 2 3.5y2y 3 1.5t2 53t 54. 6y5 38 y 2x0.1x 17 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 34 Chapter P Prerequisites In Exercises 5562, perform the operation. 55. 56. 57. 58. 59. 60. 61. 62. Add 8 and 4. Add 2x 5 3x 3 2x 3 and 4x 3 x 6. Subtract x 3 from 5x 2 3x 8. Subtract t 4 0.5t 2 5.6 from 0.6t 4 2t 2. Multiply x 7 and 2x 3. Multiply 3x 1 and x 5. Multiply x2 2x 3 and x2 2x 3. Multiply x2 x 4 and x2 2x 1. 7x 3 2x 2 3x3 In Exercises 63100, multiply or find the special product. 63. 65. 67. 69. 71. 73. 75. 77. 79. 80. 81. 82. 83. 84. 85. 87. 89. 91. 93. 94. 95. 96. 97. 98. 99. 100. x 3x 4 64. x 5x 10 3x 52x 1 66. 7x 24x 3 x 10x 10 68. 2x 32x 3 x 2yx 2y 70. 4a 5b4a 5b 2x 3 2 72. 5 8x 2 3 x 1 74. x 2 3 2x y 3 76. 3x 2y 3 3 2 4x 3 78. 8x 32 x 2 x 1x 2 x 1 x 2 3x 2x 2 3x 2 x2 x 53x2 4x 1 2x2 x 4x2 3x 2 m 3 nm 3 n x 3y zx 3y z x 3 y2 86. x 1 y2 2r 2 52r 2 5 88. 3a 3 4b23a 3 4b2 2 2 1 3 90. 5 t 4 4 x 5 1 1 1 1 92. 3x 6 3x 6 5 x 35 x 3 2.4x 32 1.8y 52 1.5x 41.5x 4 2.5y 32.5y 3 5xx 1 3xx 1 2x 1x 3 3x 3 u 2u 2u 2 4 x yx yx 2 y 2 105. COST, REVENUE, AND PROFIT An electronics manufacturer can produce and sell x MP3 players per week. The total cost C in dollars of producing x MP3 players is C 73x 25,000, and the total revenue R in dollars is R 95x. a Find the profit P in terms of x. b Find the profit obtained by selling 5000 MP3 players per week. 106. COST, REVENUE, AND PROFIT An artisan can produce and sell x hats per month. The total cost C in dollars for producing x hats is C 460 12x, and the total revenue R in dollars is R 36x. a Find the profit P in terms of x. b Find the profit obtained by selling 42 hats per month. 107. COMPOUND INTEREST After 2 years, an investment of 500 compounded annually at an interest rate r will yield an amount of 5001 r2. a Write this polynomial in standard form. b Use a calculator to evaluate the polynomial for the values of r shown in the table. 212 r 3 4 412 5 5001 r 2 c What conclusion can you make from the table 108. COMPOUND INTEREST After 3 years, an investment of 1200 compounded annually at an interest rate r will yield an amount of 12001 r3. a Write this polynomial in standard form. b Use a calculator to evaluate the polynomial for the values of r shown in the table. r 2 3 312 4 412 12001 r3 c What conclusion can you make from the table 109. VOLUME OF A BOX A take out fast food restaurant is constructing an open box by cutting squares from the corners of a piece of cardboard that is 18 centimeters by 26 centimeters see figure . The edge of each cut out square is x centimeters. 101. 102. 103. 104. x yx y 5 x5 x x 5 2 x 3 2 x 26 2x 18 2x x In Exercises 101104, find the product. The expressions are not polynomials, but the formulas can still be used. x 26 cm 18 cm x x 26 2x 18 2x a Find the volume of the box in terms of x. b Find the volume when x 1, x 2, and x 3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.3 110. VOLUME OF A BOX An overnight shipping company is designing a closed box by cutting along the solid lines and folding along the broken lines on the rectangular piece of corrugated cardboard shown in the figure. The length and width of the rectangle are 45 centimeters and 15 centimeters, respectively. Polynomials and Special Products GEOMETRY In Exercises 113 and 114, find a polynomial that represents the total number of square feet for the floor plan shown in the figure. x 113. x 14 ft 45 cm 22 ft 15 cm x 35 a Find the volume of the shipping box in terms of x. b Find the volume when x 3, x 5, and x 7. 111. GEOMETRY Find the area of the shaded region in each figure. Write your result as a polynomial in standard form. 2x + 6 a b x+4 14 ft x x 2x 114. 12x 8x x 115. ENGINEERING A uniformly distributed load is placed on a one inch wide steel beam. When the span of the beam is x feet and its depth is 6 inches, the safe load S in pounds is approximated by 6x 9x c d 3x x+6 5x S6 0.06x 2 2.42x 38.71 2. x+1 3x + 10 x+2 112. GEOMETRY Find the area of the shaded region in each figure. Write your result as a polynomial in standard form. a b 4x 2 4x x 18 ft 4x When the depth is 8 inches, the safe load is approximated by S8 0.08x 2 3.30x 51.93 2. a Use the bar graph to estimate the difference in the safe loads for these two beams when the span is 12 feet. b How does the difference in safe load change as the span increases 3x 10x Safe load in pounds S 10x c d 4x + 2 x1 2x + 8 x1 4x + 2 x+4 1600 1400 1200 1000 800 600 400 200 6 inch beam 8 inch beam x x+4 2x + 8 www.elsolucionario.net 4 8 12 16 Span in feet http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 36 Chapter P Prerequisites 116. STOPPING DISTANCE The stopping distance of an automobile is the distance traveled during the drivers reaction time plus the distance traveled after the brakes are applied. In an experiment, these distances were measured in feet when the automobile was traveling at a speed of x miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time R was R 1.1x and the braking distance B was B 0.0475x 2 0.001x 0.23. a Determine the polynomial that represents the total stopping distance T. b Use the result of part a to estimate the total stopping distance when x 30, x 40, and x 55 miles per hour. c Use the bar graph to make a statement about the total stopping distance required for increasing speeds. 250 Reaction time distance Braking distance Distance in feet 225 200 175 150 125 100 75 EXPLORATION TRUE OR FALSE In Exercises 119 and 120, determine whether the statement is true or false. Justify your answer. 119. The product of two binomials is always a seconddegree polynomial. 120. The sum of two binomials is always a binomial. 121. Find the degree of the product of two polynomials of degrees m and n. 122. Find the degree of the sum of two polynomials of degrees m and n if m n. 123. WRITING A students homework paper included the following. x 32 x 2 9 Write a paragraph fully explaining the error and give the correct method for squaring a binomial. 124. CAPSTONE A third degree polynomial and a fourth degree polynomial are added. a Can the sum be a fourth degree polynomial Explain or give an example. b Can the sum be a second degree polynomial Explain or give an example. c Can the sum be a seventh degree polynomial Explain or give an example. 50 25 x 20 30 40 50 60 Speed in miles per hour GEOMETRY In Exercises 117 and 118, use the area model to write two different expressions for the area. Then equate the two expressions and name the algebraic property that is illustrated. x 117. 4 x 125. THINK ABOUT IT Must the sum of two seconddegree polynomials be a second degree polynomial If not, give an example. 126. THINK ABOUT IT When the polynomial x 3 3x2 2x 1 is subtracted from an unknown polynomial, the difference is 5x 2 8. If it is possible, find the unknown polynomial. 127. LOGICAL REASONING Verify that x y2 is not equal to x 2 y 2 by letting x 3 and y 4 and evaluating both expressions. Are there any values of x and y for which x y2 x 2 y 2 Explain. 1 x+4 x 118. a x a x+a www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.4 Factoring Polynomials 37 P.4 FACTORING POLYNOMIALS What you should learn Remove common factors from polynomials. Factor special polynomial forms. Factor trinomials as the product of two binomials. Factor polynomials by grouping. Why you should learn it Polynomial factoring can be used to solve real life problems. For instance, in Exercise 148 on page 44, factoring is used to develop an alternative model for the rate of change of an autocatalytic chemical reaction. Polynomials with Common Factors The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, then it is prime or irreducible over the integers. For instance, the polynomial x 2 3 is irreducible over the integers. Over the real numbers, this polynomial can be factored as x 2 3 x 3 x 3 . A polynomial is completely factored when each of its factors is prime. For instance x 3 x 2 4x 4 x 1x 2 4 Completely factored is completely factored, but x 3 x 2 4x 4 x 1x 2 4 Not completely factored is not completely factored. Its complete factorization is x 3 x 2 4x 4 x 1x 2x 2. Mitch WejnarowiczThe Image Works The simplest type of factoring involves a polynomial that can be written as the product of a monomial and another polynomial. The technique used here is the Distributive Property, ab c ab ac, in the reverse direction. ab ac ab c a is a common factor. Removing factoring out any common factors is the first step in completely factoring a polynomial. Example 1 Removing Common Factors Factor each expression. a. 6x 3 4x b. 4x 2 12x 16 c. x 22x x 23 Solution a. 6x 3 4x 2x3x 2 2x2 2x 3x 2 b. 4x 2 2x is a common factor. 2 12x 16 4x 2 43x 44 4 is a common factor. 4x 2 3x 4 c. x 22x x 23 x 22x 3 x 2 is a common factor. Now try Exercise 11. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 38 Chapter P Prerequisites Factoring Special Polynomial Forms Some polynomials have special forms that arise from the special product forms on page 30. You should learn to recognize these forms so that you can factor such polynomials easily. Factoring Special Polynomial Forms Factored Form Difference of Two Squares Example u 2 v 2 u vu v 9x 2 4 3x 2 2 2 3x 23x 2 Perfect Square Trinomial u 2 2uv v 2 u v 2 x 2 6x 9 x 2 2x3 32 x 32 u 2 2uv v 2 u v 2 x 2 6x 9 x 2 2x3 32 x 32 Sum or Difference of Two Cubes u 3 v 3 u vu 2 uv v 2 x 3 8 x 3 23 x 2x 2 2x 4 u3 v3 u vu2 uv v 2 27x3 1 3x 3 13 3x 19x 2 3x 1 One of the easiest special polynomial forms to factor is the difference of two squares. The factored form is always a set of conjugate pairs. u 2 v 2 u vu v Difference Conjugate pairs Opposite signs To recognize perfect square terms, look for coefficients that are squares of integers and variables raised to even powers. Example 2 In Example 2, note that the first step in factoring a polynomial is to check for any common factors. Once the common factors are removed, it is often possible to recognize patterns that were not immediately obvious. Removing a Common Factor First 3 12x 2 31 4x 2 3 12 3 is a common factor. 2x 2 31 2x1 2x Difference of two squares Now try Exercise 25. Example 3 Factoring the Difference of Two Squares a. x 22 y 2 x 2 yx 2 y x 2 yx 2 y b. 16x 4 81 4x 22 92 4x 2 94x 2 9 Difference of two squares 4x2 92x2 32 4x2 92x 32x 3 Difference of two squares Now try Exercise 29. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.4 Factoring Polynomials 39 A perfect square trinomial is the square of a binomial, and it has the following form. u 2 2uv v 2 u v 2 or u 2 2uv v 2 u v 2 Like signs Like signs Note that the first and last terms are squares and the middle term is twice the product of u and v. Example 4 Factoring Perfect Square Trinomials Factor each trinomial. a. x 2 10x 25 b. 16x 2 24x 9 Solution a. x 2 10x 25 x 2 2x5 5 2 x 52 b. 16x2 24x 9 4x2 24x3 32 4x 32 Now try Exercise 35. The next two formulas show the sums and differences of cubes. Pay special attention to the signs of the terms. Like signs Like signs u 3 v 3 u vu 2 uv v 2 u 3 v 3 u vu 2 uv v 2 Unlike signs Example 5 Unlike signs Factoring the Difference of Cubes Factor x 3 27. Solution x3 27 x3 33 Rewrite 27 as 33. x 3x 2 3x 9 Factor. Now try Exercise 45. Example 6 Factoring the Sum of Cubes a. y 3 8 y 3 23 Rewrite 8 as 23. y 2 y 2 2y 4 b. 3 x3 64 3 x3 43 3x 4x 2 4x 16 Factor. Rewrite 64 as 43. Factor. Now try Exercise 47. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 40 Chapter P Prerequisites Trinomials with Binomial Factors To factor a trinomial of the form ax 2 bx c, use the following pattern. Factors of a ax2 bx c x x Factors of c The goal is to find a combination of factors of a and c such that the outer and inner products add up to the middle term bx. For instance, in the trinomial 6x 2 17x 5, you can write all possible factorizations and determine which one has outer and inner products that add up to 17x. 6x 5x 1, 6x 1x 5, 2x 13x 5, 2x 53x 1 You can see that 2x 53x 1 is the correct factorization because the outer O and inner I products add up to 17x. F O I L OI 2x 53x 1 6x 2 2x 15x 5 6x2 17x 5 Example 7 Factoring a Trinomial: Leading Coefficient Is 1 Factor x 2 7x 12. Solution The possible factorizations are x 2x 6, x 1x 12, and x 3x 4. Testing the middle term, you will find the correct factorization to be x 2 7x 12 x 3x 4. Now try Exercise 57. Example 8 Factoring a Trinomial: Leading Coefficient Is Not 1 Factor 2x 2 x 15. Solution Factoring a trinomial can involve trial and error. However, once you have produced the factored form, it is an easy matter to check your answer. For instance, you can verify the factorization in Example 7 by multiplying out the expression x 3x 4 to see that you obtain the original trinomial, x2 7x 12. The eight possible factorizations are as follows. 2x 1x 15 2x 1x 15 2x 3x 5 2x 3x 5 2x 5x 3 2x 5x 3 2x 15x 1 2x 15x 1 Testing the middle term, you will find the correct factorization to be 2x 2 x 15 2x 5x 3. O I 6x 5x x Now try Exercise 65. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.4 Factoring Polynomials 41 Factoring by Grouping Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping. It is not always obvious which terms to group, and sometimes several different groupings will work. Example 9 Factoring by Grouping Use factoring by grouping to factor x 3 2x 2 3x 6. Another way to factor the polynomial in Example 9 is to group the terms as follows. Solution x 3 2x 2 3x 6 x 3 2x 2 3x 6 x3 2x2 3x 6 xx2 3 2x2 3 x2 3x 2 As you can see, you obtain the same result as in Example 9. x 2 3x 2 x 2 x2 x3 3x 2x2 6 Group terms. x2 Factor each group. 3 Distributive Property Now try Exercise 73. Factoring a trinomial can involve quite a bit of trial and error. Some of this trial and error can be lessened by using factoring by grouping. The key to this method of factoring is knowing how to rewrite the middle term. In general, to factor a trinomial ax2 bx c by grouping, choose factors of the product ac that add up to b and use these factors to rewrite the middle term. This technique is illustrated in Example 10. Example 10 Factoring a Trinomial by Grouping Use factoring by grouping to factor 2x 2 5x 3. Solution In the trinomial 2x 2 5x 3, a 2 and c 3, which implies that the product ac is 6. Now, 6 factors as 61 and 6 1 5 b. So, you can rewrite the middle term as 5x 6x x. This produces the following. 2x 2 5x 3 2x 2 6x x 3 2x 2 6x x 3 Rewrite middle term. Group terms. 2xx 3 x 3 Factor groups. x 32x 1 Distributive Property So, the trinomial factors as 2x 2 5x 3 x 32x 1. Now try Exercise 79. Guidelines for Factoring Polynomials 1. Factor out any common factors using the Distributive Property. 2. Factor according to one of the special polynomial forms. 3. Factor as ax2 bx c mx rnx s. 4. Factor by grouping. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 42 Chapter P P.4 Prerequisites EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 13, fill in the blanks. 1. The process of writing a polynomial as a product is called ________. 2. A polynomial is ________ ________ when each of its factors is prime. 3. If a polynomial has more than three terms, a method of factoring called ________ ________ ________ may be used. 4. Match the factored form of the polynomial with its name. a u2 v2 u vu v i Perfect square trinomial 3 3 2 2 b u v u vu uv v ii Difference of two squares c u2 2uv v2 u v2 iii Difference of two cubes SKILLS AND APPLICATIONS In Exercises 58, find the greatest common factor of the expressions. 5. 80, 280 7. 12x 2y 3, 18x 2y, 24x 3y 2 6. 24, 96, 256 8. 15x 23, 42x x 2 2 In Exercises 916, factor out the common factor. 9. 11. 13. 15. 4x 16 2x 3 6x 3xx 5 8x 5 x 32 4x 3 10. 12. 14. 16. 5y 30 3z3 6z2 9z 3xx 2 4x 2 5x 42 5x 4 In Exercises 1722, find the greatest common factor such that the remaining factors have only integer coefficients. 17. 12 x 4 19. 12 x 3 2x 2 5x 21. 23 xx 3 4x 3 18. 13 y 5 20. 13 y 4 5y 2 2y 22. 45 y y 1 2 y 1 In Exercises 2332, completely factor the difference of two squares. 23. x2 81 25. 48y2 27 27. 16x 2 19 29. x 1 2 4 31. 9u2 4v 2 x 2 4x 4 4t 2 4t 1 25y 2 10y 1 9u2 24uv 16v 2 x 2 43x 49 4x2 43 x 19 34. 36. 38. 40. 42. 44. 45. x 3 8 47. y 3 64 8 49. x3 27 51. 8t 3 1 53. u3 27v 3 55. x 23 y3 46. 27 x 3 48. z 3 216 8 50. y3 125 52. 27x 3 8 54. 64x 3 y 3 56. x 3y3 8z3 In Exercises 5770, factor the trinomial. 57. 59. 61. 63. 65. 67. 69. x2 x 2 s 2 5s 6 20 y y 2 x 2 30x 200 3x 2 5x 2 5x 2 26x 5 9z 2 3z 2 58. 60. 62. 64. 66. 68. 70. x 2 5x 6 t2 t 6 24 5z z 2 x 2 13x 42 2x 2 x 1 12x 2 7x 1 5u 2 13u 6 In Exercises 7178, factor by grouping. 71. x 3 x 2 2x 2 73. 2x 3 x 2 6x 3 75. 6 2x 3x3 x4 77. 6x 3 2x 3x 2 1 24. x 2 64 26. 50 98z2 4 2 28. 25 y 64 30. 25 z 5 2 32. 25x 2 16y 2 72. x 3 5x 2 5x 25 74. 5x 3 10x 2 3x 6 76. x 5 2x 3 x 2 2 78. 8x 5 6x 2 12x 3 9 In Exercises 7984, factor the trinomial by grouping. In Exercises 3344, factor the perfect square trinomial. 33. 35. 37. 39. 41. 43. In Exercises 4556, factor the sum or difference of cubes. x 2 10x 25 9x 2 12x 4 36y 2 108y 81 4x 2 4xy y 2 z 2 z 14 3 1 9y2 2 y 16 79. 3x 2 10x 8 81. 6x 2 x 2 83. 15x 2 11x 2 80. 2x 2 9x 9 82. 6x 2 x 15 84. 12x2 13x 1 In Exercises 85120, completely factor the expression. 85. 6x 2 54 87. x3 x2 www.elsolucionario.net 86. 12x 2 48 88. x 3 4x 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.4 x3 16x 90. x 3 9x 2 x 2x 1 92. 16 6x x 2 1 4x 4x 2 94. 9x 2 6x 1 2 3 2x 4x 2x 96. 13x 6 5x 2 2 1 1 1 2 98. 18 x 2 96 x 16 81 x 9 x 8 3x 3 x 2 15x 5 100. 5 x 5x 2 x 3 x 4 4x 3 x 2 4x 102. 3u 2u2 6 u3 2x3 x2 8x 4 104. 3x3 x2 27x 9 1 3 3 1 2 106. 5 x3 x2 x 5 4 x 3x 4 x 9 t 1 2 49 108. x 2 1 2 4x 2 2 2 2 x 8 36x 110. 2t 3 16 5x 3 40 112. 4x2x 1 2x 1 2 2 53 4x 83 4x5x 1 2x 1x 3 2 3x 1 2x 3 73x 2 21 x 2 3x 21 x3 7x2x 2 12x x2 1 27 3x 22x 14 x 2 34x 1 3 2xx 5 4 x 24x 5 3 5x6 146x53x 23 33x 223x6 15 x2 120. x2 14 x 2 15 2 89. 91. 93. 95. 97. 99. 101. 103. 105. 107. 109. 111. 113. 114. 115. 116. 117. 118. 119. b a is shown in the following figure. x x x x x 1 1 1 1 1 x a x x 1 b b c a a b a a a b b b b b d 1 a a b b 1 a 1 1 1 1 b 121. 122. 123. 124. a 2 b 2 a ba b a 2 2ab b 2 a b 2 a 2 2a 1 a 1 2 ab a b 1 a 1b 1 GEOMETRIC MODELING In Exercises 125128, draw a geometric factoring model to represent the factorization. x 125. 126. 127. 128. a a ab x 1 a b a a GEOMETRIC MODELING In Exercises 121124, match the factoring formula with the correct geometric factoring model. The models are labeled a , b , c , and d . For instance, a factoring model for 2 x 2 3x 1 2 x 1x 1 43 Factoring Polynomials 3x 2 7x 2 3x 1x 2 x 2 4x 3 x 3x 1 2x 2 7x 3 2x 1x 3 x 2 3x 2 x 2x 1 1 GEOMETRY In Exercises 129132, write an expression in factored form for the area of the shaded portion of the figure. 129. 130. r r 1 1 1 r+2 1 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 44 131. Chapter P x 8 x x x Prerequisites 132. x 3 x x x x+3 18 4 5 5 x 4 + 3 In Exercises 133138, completely factor the expression. 133. 134. 135. 136. x442x 132x 2x 144x3 x33x2 122x x2 133x2 2x 5435x 425 5x 4342x 532 x2 5324x 34 4x 323x2 52x2 137. 5x 13 3x 15 5x 12 138. 2x 34 4x 12 2x 32 In Exercises 139142, find all values of b for which the trinomial can be factored. 139. x 2 bx 15 141. x 2 bx 50 140. x 2 bx 12 142. x 2 bx 24 In Exercises 143146, find two integer values of c such that the trinomial can be factored. There are many correct answers. 143. 2x 2 5x c 145. 3x 2 x c c An 80 pound bag of concrete mix yields 5 cubic foot of concrete. Find the number of bags required to construct a concrete storage tank having the following dimensions. Outside radius, R 4 feet 2 Inside radius, r 33 feet Height, h feet d Use the table feature of a graphing utility to create a table showing the number of bags of concrete required to construct the storage tank in part c 1 3 with heights of h 2, h 1, h 2, h 2, . . . , h 6 feet. 148. CHEMISTRY The rate of change of an autocatalytic chemical reaction is kQx kx 2, where Q is the amount of the original substance, x is the amount of substance formed, and k is a constant of proportionality. Factor the expression. EXPLORATION TRUE OR FALSE In Exercises 149 and 150, determine whether the statement is true or false. Justify your answer. 149. The difference of two perfect squares can be factored as the product of conjugate pairs. 150. The sum of two perfect squares can be factored as the binomial sum squared. 151. ERROR ANALYSIS 144. 3x 2 10x c 146. 2x 2 9x c Describe the error. 9x 9x 54 3x 63x 9 2 147. GEOMETRY The volume V of concrete used to make the cylindrical concrete storage tank shown in the figure is V R 2h r 2h, where R is the outside radius, r is the inside radius, and h is the height of the storage tank. R 3x 2x 3 152. THINK ABOUT IT Is 3x 6x 1 completely factored Explain. 153. Factor x 2n y 2n as completely as possible. 154. Factor x 3n y 3n as completely as possible. 155. Give an example of a polynomial that is prime with respect to the integers. 156. CAPSTONE Explain what is meant when it is said that a polynomial is in factored form. h r a Factor the expression for the volume. b From the result of part a , show that the volume of concrete is 157. Rewrite u6 v6 as the difference of two squares. Then find a formula for completely factoring u 6 v 6. Use your formula to factor x 6 1 and x 6 64 completely. 2average radiusthickness of the tankh. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.5 Rational Expressions 45 P.5 RATIONAL EXPRESSIONS What you should learn Find domains of algebraic expressions. Simplify rational expressions. Add, subtract, multiply, and divide rational expressions. Simplify complex fractions and rewrite difference quotients. Domain of an Algebraic Expression The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, x 1 x 2 and 2x 3 are equivalent because x 1 x 2 x 1 x 2 xx12 Why you should learn it Rational expressions can be used to solve real life problems. For instance, in Exercise 102 on page 54, a rational expression is used to model the projected numbers of U.S. households banking and paying bills online from 2002 through 2007. 2x 3. Example 1 Finding the Domain of an Algebraic Expression a. The domain of the polynomial 2x 3 3x 4 Dex Images, Inc.Corbis is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted. b. The domain of the radical expression x 2 is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number. c. The domain of the expression x2 x3 is the set of all real numbers except x 3, which would result in division by zero, which is undefined. Now try Exercise 7. The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as 1 , x 2x 1 , x1 or x2 1 x2 1 is a rational expression. Simplifying Rational Expressions Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from 1. To write a fraction in simplest form, divide out common factors. a b c a, c b c 0 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 46 Chapter P Prerequisites The key to success in simplifying rational expressions lies in your ability to factor polynomials. When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. Example 2 WARNING CAUTION In Example 2, do not make the mistake of trying to simplify further by dividing out terms. x6 x6 x2 3 3 Remember that to simplify fractions, divide out common factors, not terms. To learn about other common errors, see Appendix A. Write Simplifying a Rational Expression x 2 4x 12 in simplest form. 3x 6 Solution x2 4x 12 x 6x 2 3x 6 3x 2 x6 , 3 x Factor completely. 2 Divide out common factors. Note that the original expression is undefined when x 2 because division by zero is undefined . To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x 2. Now try Exercise 33. Sometimes it may be necessary to change the sign of a factor by factoring out 1 to simplify a rational expression, as shown in Example 3. Example 3 Write Simplifying Rational Expressions 12 x x2 in simplest form. 2x2 9x 4 Solution 12 x x2 4 x3 x 2x2 9x 4 2x 1x 4 Factor completely. x 43 x 2x 1x 4 3x , x 2x 1 4 4 x x 4 Divide out common factors. Now try Exercise 39. In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of x that must be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. In Example 3, for instance, the restriction x 4 is listed with the simplified expression 1 to make the two domains agree. Note that the value x 2 is excluded from both domains, so it is not necessary to list this value. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.5 Rational Expressions 47 Operations with Rational Expressions To multiply or divide rational expressions, use the properties of fractions discussed in Section P.1. Recall that to divide fractions, you invert the divisor and multiply. Example 4 Multiplying Rational Expressions 2x2 x 6 x2 4x 5 x3 3x2 2x 2x 3x 2 4x2 6x x 5x 1 xx 2x 1 2x2x 3 x 2x 2 , x 2x 5 0, x 1, x 3 2 Now try Exercise 53. In Example 4, the restrictions x 0, x 1, and x 32 are listed with the simplified expression in order to make the two domains agree. Note that the value x 5 is excluded from both domains, so it is not necessary to list this value. Example 5 Dividing Rational Expressions x 3 8 x 2 2x 4 x 3 8 2 x2 4 x3 8 x 4 x3 8 x 2 2x 4 Invert and multiply. x 2x2 2x 4 x 2x2 2x 4 x2 2x 4 x 2x 2 x 2 2x 4, x 2 Divide out common factors. Now try Exercise 55. To add or subtract rational expressions, you can use the LCD least common denominator method or the basic definition a c ad bc , b d bd b 0, d 0. Basic definition This definition provides an efficient way of adding or subtracting two fractions that have no common factors in their denominators. Example 6 WARNING CAUTION When subtracting rational expressions, remember to distribute the negative sign to all the terms in the quantity that is being subtracted. Subtracting Rational Expressions x 2 x3x 4 2x 3 x 3 3x 4 x 33x 4 Basic definition 3x 2 4x 2x 6 x 33x 4 Distributive Property 3x 2 2x 6 x 33x 4 Combine like terms. Now try Exercise 65. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 48 Chapter P Prerequisites For three or more fractions, or for fractions with a repeated factor in the denominators, the LCD method works well. Recall that the least common denominator of several fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. Here is a numerical example. 1 3 2 12 33 24 6 4 3 62 43 34 2 9 8 12 12 12 3 12 1 4 The LCD is 12. Sometimes the numerator of the answer has a factor in common with the denominator. In such cases the answer should be simplified. For instance, in the 3 example above, 12 was simplified to 14. Example 7 Combining Rational Expressions: The LCD Method Perform the operations and simplify. 3 2 x3 2 x1 x x 1 Solution Using the factored denominators x 1, x, and x 1x 1, you can see that the LCD is xx 1x 1. 3 2 x3 x1 x x 1x 1 3xx 1 2x 1x 1 x 3x xx 1x 1 xx 1x 1 xx 1x 1 3xx 1 2x 1x 1 x 3x xx 1x 1 3x 2 3x 2x 2 2 x 2 3x xx 1x 1 Distributive Property 3x 2 2x 2 x 2 3x 3x 2 xx 1x 1 Group like terms. 2x2 6x 2 xx 1x 1 Combine like terms. 2x 2 3x 1 xx 1x 1 Factor. Now try Exercise 67. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.5 Rational Expressions 49 Complex Fractions and the Difference Quotient Fractional expressions with separate fractions in the numerator, denominator, or both are called complex fractions. Here are two examples. x x 1 x2 1 1 and x 2 1 1 To simplify a complex fraction, combine the fractions in the numerator into a single fraction and then combine the fractions in the denominator into a single fraction. Then invert the denominator and multiply. Example 8 Simplifying a Complex Fraction 2 3x x 1 1x 1 1 1 x1 x1 x 3 2 Combine fractions. 2 3x x x2 x 1 Simplify. 2 3x x x1 2 3xx 1 , x xx 2 x2 Invert and multiply. 1 Now try Exercise 73. Another way to simplify a complex fraction is to multiply its numerator and denominator by the LCD of all fractions in its numerator and denominator. This method is applied to the fraction in Example 8 as follows. x 3 x 3 2 1 1 x1 2 1 1 x1 xx 1 xx 1 LCD is xx 1. 2 x 3x xx 1 xx 21 xx 1 2 3xx 1 , x xx 2 www.elsolucionario.net 1 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 50 Chapter P Prerequisites The next three examples illustrate some methods for simplifying rational expressions involving negative exponents and radicals. These types of expressions occur frequently in calculus. To simplify an expression with negative exponents, one method is to begin by factoring out the common factor with the smaller exponent. Remember that when factoring, you subtract exponents. For instance, in 3x52 2x32 the smaller exponent is 52 and the common factor is x52. 3x52 2x32 x5231 2x32 52 x523 2x1 Example 9 3 2x x 52 Simplifying an Expression Simplify the following expression containing negative exponents. x1 2x32 1 2x12 Solution Begin by factoring out the common factor with the smaller exponent. x1 2x32 1 2x12 1 2x32 x 1 2x12 32 1 2x32x 1 2x1 1x 1 2x 32 Now try Exercise 81. A second method for simplifying an expression with negative exponents is shown in the next example. Example 10 Simplifying an Expression with Negative Exponents 4 x 212 x 24 x 212 4 x2 4 x 212 x 24 x 212 4 x 212 4 x 212 4 x2 4 x 21 x 24 x 2 0 4 x 2 32 4 x2 x2 4 x 2 32 4 4 x 2 32 Now try Exercise 83. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.5 Example 11 Rational Expressions 51 Rewriting a Difference Quotient The following expression from calculus is an example of a difference quotient. x h x h Rewrite this expression by rationalizing its numerator. Solution x h x h You can review the techniques for rationalizing a numerator in Section P.2. x h x h x h x x h x x h 2 x 2 hx h x h hx h x 1 x h x , h 0 Notice that the original expression is undefined when h 0. So, you must exclude h 0 from the domain of the simplified expression so that the expressions are equivalent. Now try Exercise 89. Difference quotients, such as that in Example 11, occur frequently in calculus. Often, they need to be rewritten in an equivalent form that can be evaluated when h 0. Note that the equivalent form is not simpler than the original form, but it has the advantage that it is defined when h 0. P.5 EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The set of real numbers for which an algebraic expression is defined is the ________ of the expression. 2. The quotient of two algebraic expressions is a fractional expression and the quotient of two polynomials is a ________ ________. 3. Fractional expressions with separate fractions in the numerator, denominator, or both are called ________ fractions. 4. To simplify an expression with negative exponents, it is possible to begin by factoring out the common factor with the ________ exponent. 5. Two algebraic expressions that have the same domain and yield the same values for all numbers in their domains are called ________. 6. An important rational expression, such as x h2 x2 , that occurs in calculus is called h a ________ ________. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 52 Chapter P Prerequisites SKILLS AND APPLICATIONS 44. ERROR ANALYSIS In Exercises 722, find the domain of the expression. 7. 3x 2 4x 7 9. 4x 3 3, x 0 1 11. 3x 8. 2x 2 5x 2 10. 6x 2 9, x 0 x2 1 13. 2 x 2x 1 x2 5x 6 14. x2 4 x2 2x 3 15. 2 x 6x 9 x2 x 12 16. 2 x 8x 16 17. x 7 19. 2x 5 18. 4 x 20. 4x 5 1 21. x 3 1 22. x 2 12. x2 5 5 2x 6x 2 24. 45. 15x 2 10x 3xy 27. xy x 29. 4y 8y 2 10y 5 x5 31. 10 2x 33. 35. 37. 39. 41. y 2 16 y4 x 3 5x 2 6x x2 4 2 y 7y 12 y 2 3y 18 2 x 2x 2 x 3 x2 4 z3 8 2 z 2z 4 43. ERROR ANALYSIS 18y 2 60y 5 28. 2x 2y xy y 30. 9x 2 9x 2x 2 1 2 46. 0 x 1 2 36. 38. 40. 42. Describe the error. 5 6 3 4 5 6 1 x2 GEOMETRY In Exercises 47 and 48, find the ratio of the area of the shaded portion of the figure to the total area of the figure. 47. 48. x+5 2 r 2x + 3 x+5 2 x 2 25 5x x 2 8x 20 x 2 11x 10 x 2 7x 6 x 2 11x 10 x2 9 3 x x 2 9x 9 y 3 2y 2 3y y3 1 5x3 5x3 5 5 3 3 2x 4 2x 4 2 4 6 4 x3 x2 x 6 12 4x 32. x3 34. 3 x1 3 3 4 4x 1 26. 0 x x2 2x 3 x3 In Exercises 2542, write the rational expression in simplest form. 25. xx 5x 5 xx 5 x 5x 3 x3 In Exercises 45 and 46, complete the table. What can you conclude In Exercises 23 and 24, find the missing factor in the numerator such that the two fractions are equivalent. 23. 25x xx 2 25 2x 15 x 5x 3 x6 3x 2 Describe the error. x3 x+5 In Exercises 4956, perform the multiplication or division and simplify. 5 x1 x 13 xx 3 50. 3 5 x 1 25x 2 x 3 x r r2 4y 16 4y 51. 52. 2 r1 r 1 5y 15 2y 6 t2 t 6 t3 53. 2 t 6t 9 t 2 4 x 2 xy 2y 2 x 54. x 2 3xy 2y 2 x 3 x 2y 49. 55. x 2 36 x 3 6x 2 2 x x x 56. x 2 14x 49 3x 21 x 2 49 x7 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.5 In Exercises 5768, perform the addition or subtraction and simplify. 57. 6 5 x3 5 x x1 x1 3 5 61. x2 2x 59. 63. 4 x 2x 1 x 2 3 5 x1 2x 1 1 x 60. x3 x3 2x 5 62. x5 5x 64. 3x13 x23 3x23 x 31 x 212 2x1 x 212 84. x4 83. 1 x x 2 x 2 x 2 5x 6 2 10 66. 2 x x 2 x 2 2x 8 1 2 1 67. 2 x x 1 x3 x 2 2 1 2 68. x1 x1 x 1 In Exercises 85 88, simplify the difference quotient. 85. ERROR ANALYSIS In Exercises 69 and 70, describe the error. x 4 3x 8 x 4 3x 8 x2 x2 x2 2x 4 2x 2 2 x2 x2 6x x2 8 2 70. 2 xx 2 x x x 2 x6 x x 2 2 8 x 2x 2 2 6x x x 2 4 8 x 2x 2 6x 2 6 2 x x 2 x 2 69. In Exercises 7176, simplify the complex fraction. x 71. x2 x 1 73. x x 1 2 3 x 2x 1 75. x 4 x 4 4 x 2 x 1 x 74. x 12 x t2 t 2 1 t 2 1 76. t2 72. x 2 x x 5 2x2 78. x5 5x3 2 2 5 2 x x 1 x 14 2xx 53 4x 2x 54 2x 2x 112 5x 112 4x 32x 132 2x2x 112 In Exercises 83 and 84, simplify the expression. 2 5x x 3 3x 4 65. 2 1 In Exercises 7782, factor the expression by removing the common factor with the smaller exponent. 77. 79. 80. 81. 82. 58. 53 Rational Expressions 87. x 1 h 1x x h 1 h 1 1 xh4 x4 h 86. 88. 2 1 x2 h xh x xh1 x1 h In Exercises 8994, simplify the difference quotient by rationalizing the numerator. 89. 91. 93. 94. x 2 x 90. 2 t 3 3 92. t z 3 z 3 x 5 5 x x h 1 x 1 h x h 2 x 2 h PROBABILITY In Exercises 95 and 96, consider an experiment in which a marble is tossed into a box whose base is shown in the figure. The probability that the marble will come to rest in the shaded portion of the box is equal to the ratio of the shaded area to the total area of the figure. Find the probability. 95. 96. x 2 x 2x + 1 x+4 x x x+2 4 x x + 2 97. RATE A digital copier copies in color at a rate of 50 pages per minute. a Find the time required to copy one page. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 54 Chapter P Prerequisites b Find the time required to copy x pages. c Find the time required to copy 120 pages. 98. RATE After working together for t hours on a common task, two workers have done fractional parts of the job equal to t3 and t5, respectively. What fractional part of the task has been completed 102. INTERACTIVE MONEY MANAGEMENT The table shows the projected numbers of U.S. households in millions banking online and paying bills online from 2002 through 2007. Source: eMarketer; Forrester Research FINANCE In Exercises 99 and 100, the formula that approximates the annual interest rate r of a monthly installment loan is given by 24NM P N r P NM 12 Year Banking Paying Bills 2002 2003 2004 2005 2006 2007 21.9 26.8 31.5 35.0 40.0 45.0 13.7 17.4 20.9 23.9 26.7 29.1 where N is the total number of payments, M is the monthly payment, and P is the amount financed. Mathematical models for these data are 99. a Approximate the annual interest rate for a four year car loan of 20,000 that has monthly payments of 475. b Simplify the expression for the annual interest rate r, and then rework part a . 100. a Approximate the annual interest rate for a fiveyear car loan of 28,000 that has monthly payments of 525. b Simplify the expression for the annual interest rate r, and then rework part a . Number banking online and Number paying bills online 4t 2 16t 75 2 4t 10 t EXPLORATION where T is the temperature in degrees Fahrenheit and t is the time in hours . a Complete the table. t 0 2 4 6 8 10 14 16 18 20 TRUE OR FALSE In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer. 103. x 2n 12n x n 1n x n 1n 104. x 2 3x 2 x 2, for all values of x x1 12 T t 4.39t 5.5 0.002t2 0.01t 1.0 where t represents the year, with t 2 corresponding to 2002. a Using the models, create a table to estimate the projected numbers of households banking online and the projected numbers of households paying bills online for the given years. b Compare the values given by the models with the actual data. c Determine a model for the ratio of the projected number of households paying bills online to the projected number of households banking online. d Use the model from part c to find the ratios for the given years. Interpret your results. 101. REFRIGERATION When food at room temperature is placed in a refrigerator, the time required for the food to cool depends on the amount of food, the air circulation in the refrigerator, the original temperature of the food, and the temperature of the refrigerator. The model that gives the temperature of food that has an original temperature of 75F and is placed in a 40F refrigerator is T 10 0.728t2 23.81t 0.3 0.049t2 0.61t 1.0 105. THINK ABOUT IT How do you determine whether a rational expression is in simplest form 22 T b What value of T does the mathematical model appear to be approaching 106. CAPSTONE In your own words, explain how to divide rational expressions. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.6 55 The Rectangular Coordinate System and Graphs P.6 THE RECTANGULAR COORDINATE SYSTEM AND GRAPHS What you should learn The Cartesian Plane Plot points in the Cartesian plane. Use the Distance Formula to find the distance between two points. Use the Midpoint Formula to find the midpoint of a line segment. Use a coordinate plane to model and solve real life problems. Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician Rene Descartes 15961650 . The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure P.14. The horizontal real number line is usually called the x axis, and the vertical real number line is usually called the y axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. Why you should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Exercise 70 on page 64, a graph represents the minimum wage in the United States from 1950 through 2009. y axis Quadrant II 3 2 1 Origin 3 2 1 Quadrant I Directed distance x Vertical number line x axis 1 2 Quadrant III 3 FIGURE y axis 1 2 x, y 3 Horizontal number line Directed y distance Quadrant IV P.14 FIGURE x axis P.15 Ariel SkellyCorbis Each point in the plane corresponds to an ordered pair x, y of real numbers x and y, called coordinates of the point. The x coordinate represents the directed distance from the y axis to the point, and the y coordinate represents the directed distance from the x axis to the point, as shown in Figure P.15. Directed distance from y axis 4 3, 4 3 Example 1 1, 2 4 3 1 1 2 2, 3 FIGURE P.16 4 Directed distance from x axis The notation x, y denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended. y 1 x, y 0, 0 1 3, 0 2 3 4 x Plotting Points in the Cartesian Plane Plot the points 1, 2, 3, 4, 0, 0, 3, 0, and 2, 3. Solution To plot the point 1, 2, imagine a vertical line through 1 on the x axis and a horizontal line through 2 on the y axis. The intersection of these two lines is the point 1, 2. The other four points can be plotted in a similar way, as shown in Figure P.16. Now try Exercise 7. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter P Prerequisites The beauty of a rectangular coordinate system is that it allows you to see relationships between two variables. It would be difficult to overestimate the importance of Descartess introduction of coordinates in the plane. Today, his ideas are in common use in virtually every scientific and business related field. Example 2 Year, t Subscribers, N 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 24.1 33.8 44.0 55.3 69.2 86.0 109.5 128.4 140.8 158.7 182.1 207.9 233.0 255.4 Sketching a Scatter Plot From 1994 through 2007, the numbers N in millions of subscribers to a cellular telecommunication service in the United States are shown in the table, where t represents the year. Sketch a scatter plot of the data. Source: CTIA The Wireless Association Solution To sketch a scatter plot of the data shown in the table, you simply represent each pair of values by an ordered pair t, N and plot the resulting points, as shown in Figure P.17. For instance, the first pair of values is represented by the ordered pair 1994, 24.1. Note that the break in the t axis indicates that the numbers between 0 and 1994 have been omitted. N Number of subscribers in millions 56 Subscribers to a Cellular Telecommunication Service 300 250 200 150 100 50 t 1994 1996 1998 2000 2002 2004 2006 Year FIGURE P.17 Now try Exercise 25. In Example 2, you could have let t 1 represent the year 1994. In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 14 instead of 1994 through 2007 . T E C H N O LO G Y The scatter plot in Example 2 is only one way to represent the data graphically. You could also represent the data using a bar graph or a line graph. If you have access to a graphing utility, try using it to represent graphically the data given in Example 2. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.6 The Rectangular Coordinate System and Graphs 57 The Distance Formula a2 + b2 = c2 Recall from the Pythagorean Theorem that, for a right triangle with hypotenuse of length c and sides of lengths a and b, you have c a a 2 b2 c 2 as shown in Figure P.18. The converse is also true. That is, if a 2 b2 c 2, then the triangle is a right triangle. Suppose you want to determine the distance d between two points x1, y1 and x2, y2 in the plane. With these two points, a right triangle can be formed, as shown in Figure P.19. The length of the vertical side of the triangle is y2 y1, and the length of the horizontal side is x2 x1. By the Pythagorean Theorem, you can write b FIGURE P.18 y y d 2 x2 x12 y2 y12 x1, y1 1 d x2 x12 y2 y12 x2 x12 y2 y12. d y 2 y1 Pythagorean Theorem This result is the Distance Formula. y 2 x1, y2 x2, y2 x1 x2 x x 2 x1 FIGURE The Distance Formula The distance d between the points x1, y1 and x2, y2 in the plane is d x2 x12 y2 y12. P.19 Example 3 Finding a Distance Find the distance between the points 2, 1 and 3, 4. Algebraic Solution Let x1, y1 2, 1 and x2, y2 3, 4. Then apply the Distance Formula. d x2 x12 y2 y12 3 2 4 1 Distance Formula Substitute for x1, y1, x2, and y2. 5 2 32 Simplify. 34 Simplify. 5.83 Use a calculator. 2 2 Graphical Solution Use centimeter graph paper to plot the points A2, 1 and B3, 4. Carefully sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment. cm 1 2 3 4 5 6 So, the distance between the points is about 5.83 units. You can use the Pythagorean Theorem to check that the distance is correct. d 2 32 52 Pythagorean Theorem 2 Substitute for d. 34 32 52 Distance checks. 7 34 34 FIGURE P.20 The line segment measures about 5.8 centimeters, as shown in Figure P.20. So, the distance between the points is about 5.8 units. Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 58 Chapter P Prerequisites y Example 4 Show that the points 2, 1, 4, 0, and 5, 7 are vertices of a right triangle. 6 5 Solution d1 = 45 4 The three points are plotted in Figure P.21. Using the Distance Formula, you can find the lengths of the three sides as follows. d3 = 50 3 2 1 Verifying a Right Triangle 5, 7 7 d2 4 2 2 0 1 2 4 1 5 4, 0 1 FIGURE d1 5 2 2 7 1 2 9 36 45 d2 = 5 2, 1 2 3 4 5 x 6 7 d3 5 4 2 7 0 2 1 49 50 Because P.21 d12 d22 45 5 50 d32 you can conclude by the Pythagorean Theorem that the triangle must be a right triangle. Now try Exercise 43. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points x1, y1 and x 2, y 2 is given by the Midpoint Formula Midpoint x1 x 2 y1 y2 , . 2 2 For a proof of the Midpoint Formula, see Proofs in Mathematics on page 72. Example 5 Finding a Line Segments Midpoint Find the midpoint of the line segment joining the points 5, 3 and 9, 3. Solution Let x1, y1 5, 3 and x 2, y 2 9, 3. y 6 9, 3 3 Midpoint x1 x2 y1 y2 , 2 2 5 9 3 3 , 2 2 2, 0 6 x 3 5, 3 3 3 6 FIGURE P.22 Midpoint 6 9 Midpoint Formula 2, 0 Substitute for x1, y1, x2, and y2. Simplify. The midpoint of the line segment is 2, 0, as shown in Figure P.22. Now try Exercise 47 c . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.6 The Rectangular Coordinate System and Graphs 59 Applications Example 6 Finding the Length of a Pass A football quarterback throws a pass from the 28 yard line, 40 yards from the sideline. The pass is caught by a wide receiver on the 5 yard line, 20 yards from the same sideline, as shown in Figure P.23. How long is the pass Solution You can find the length of the pass by finding the distance between the points 40, 28 and 20, 5. Football Pass Distance in yards 35 d x2 x12 y2 y12 40, 28 30 25 20 15 10 20, 5 5 Distance Formula 40 20 2 28 5 2 Substitute for x1, y1, x2, and y2. 400 529 Simplify. 929 Simplify. 30 Use a calculator. 5 10 15 20 25 30 35 40 So, the pass is about 30 yards long. Distance in yards FIGURE Now try Exercise 57. P.23 In Example 6, the scale along the goal line does not normally appear on a football field. However, when you use coordinate geometry to solve real life problems, you are free to place the coordinate system in any way that is convenient for the solution of the problem. Example 7 Estimating Annual Revenue Barnes Noble had annual sales of approximately 5.1 billion in 2005, and 5.4 billion in 2007. Without knowing any additional information, what would you estimate the 2006 sales to have been Source: Barnes Noble, Inc. Solution Sales in billions of dollars y One solution to the problem is to assume that sales followed a linear pattern. With this assumption, you can estimate the 2006 sales by finding the midpoint of the line segment connecting the points 2005, 5.1 and 2007, 5.4. Barnes Noble Sales 5.5 2007, 5.4 5.4 5.3 x1 x2 y1 y2 , 2 2 2005 2007 5.1 5.4 , 2 2 2006, 5.25 Midpoint 5.2 5.1 2005, 5.1 5.0 2006 Year P.24 2006, 5.25 x 2005 FIGURE Midpoint 2007 Midpoint Formula Substitute for x1, x2, y1 and y2. Simplify. So, you would estimate the 2006 sales to have been about 5.25 billion, as shown in Figure P.24. The actual 2006 sales were about 5.26 billion. Now try Exercise 59. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 60 Chapter P Prerequisites Example 8 Translating Points in the Plane The triangle in Figure P.25 has vertices at the points 1, 2, 1, 4, and 2, 3. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure P.26. y y 5 5 4 4 2, 3 Paul Morrell 1, 2 3 2 1 Much of computer graphics, including this computer generated goldfish tessellation, consists of transformations of points in a coordinate plane. One type of transformation, a translation, is illustrated in Example 8. Other types include reflections, rotations, and stretches. x 2 1 1 2 3 4 5 6 7 1 2 3 5 6 7 2 2 3 3 1, 4 4 FIGURE x 2 1 4 P.25 FIGURE P.26 Solution To shift the vertices three units to the right, add 3 to each of the x coordinates. To shift the vertices two units upward, add 2 to each of the y coordinates. Original Point 1, 2 Translated Point 1 3, 2 2 2, 4 1, 4 1 3, 4 2 4, 2 2, 3 2 3, 3 2 5, 5 Now try Exercise 61. The figures provided with Example 8 were not really essential to the solution. Nevertheless, it is strongly recommended that you develop the habit of including sketches with your solutionseven if they are not required. CLASSROOM DISCUSSION Extending the Example Example 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point x, y Transformed Point x, y x, y x, y x, y x, y www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.6 P.6 EXERCISES The Rectangular Coordinate System and Graphs 61 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY 1. Match each term with its definition. a x axis i point of intersection of vertical axis and horizontal axis b y axis ii directed distance from the x axis c origin iii directed distance from the y axis d quadrants iv four regions of the coordinate plane e x coordinate v horizontal real number line f y coordinate vi vertical real number line In Exercises 2 4, fill in the blanks. 2. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ________ plane. 3. The ________ ________ is a result derived from the Pythagorean Theorem. 4. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the ________ ________. SKILLS AND APPLICATIONS In Exercises 5 and 6, approximate the coordinates of the points. y 5. A 6 D y 6. C 4 2 D 2 6 4 2 2 B 4 4 x 2 4 6 4 2 C x 2 B 2 A 4 In Exercises 710, plot the points in the Cartesian plane. 7. 4, 2, 3, 6, 0, 5, 1, 4 8. 0, 0, 3, 1, 2, 4, 1, 1 9. 3, 8, 0.5, 1, 5, 6, 2, 2.5 10. 1, 13 , 34, 3, 3, 4, 43, 32 In Exercises 1524, determine the quadrant s in which x, y is located so that the condition s is are satisfied. 15. 17. 19. 21. 23. x 0 and y 0 x 4 and y 0 y 5 x 0 and y 0 xy 0 16. 18. 20. 22. 24. x 0 and y 0 x 2 and y 3 x 4 x 0 and y 0 xy 0 In Exercises 25 and 26, sketch a scatter plot of the data shown in the table. 25. NUMBER OF STORES The table shows the number y of Wal Mart stores for each year x from 2000 through 2007. Source: Wal Mart Stores, Inc. In Exercises 1114, find the coordinates of the point. 11. The point is located three units to the left of the y axis and four units above the x axis. 12. The point is located eight units below the x axis and four units to the right of the y axis. 13. The point is located five units below the x axis and the coordinates of the point are equal. 14. The point is on the x axis and 12 units to the left of the y axis. www.elsolucionario.net Year, x Number of stores, y 2000 2001 2002 2003 2004 2005 2006 2007 4189 4414 4688 4906 5289 6141 6779 7262 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter P Prerequisites 26. METEOROLOGY The table shows the lowest temperature on record y in degrees Fahrenheit in Duluth, Minnesota for each month x, where x 1 represents January. Source: NOAA Month, x Temperature, y 1 2 3 4 5 6 7 8 9 10 11 12 39 39 29 5 17 27 35 32 22 8 23 34 6, 3, 6, 5 3, 1, 2, 1 2, 6, 3, 6 1, 4, 5, 1 12, 43 , 2, 1 4.2, 3.1, 12.5, 4.8 9.5, 2.6, 3.9, 8.2 28. 30. 32. 34. 36. 43. 44. 45. 46. Right triangle: 4, 0, 2, 1, 1, 5 Right triangle: 1, 3, 3, 5, 5, 1 Isosceles triangle: 1, 3, 3, 2, 2, 4 Isosceles triangle: 2, 3, 4, 9, 2, 7 In Exercises 4756, a plot the points, b find the distance between the points, and c find the midpoint of the line segment joining the points. 47. 49. 51. 53. 55. In Exercises 2738, find the distance between the points. 27. 29. 31. 33. 35. 37. 38. In Exercises 4346, show that the points form the vertices of the indicated polygon. 1, 4, 8, 4 3, 4, 3, 6 8, 5, 0, 20 1, 3, 3, 2 23, 3, 1, 54 1, 1, 9, 7 4, 10, 4, 5 1, 2, 5, 4 12, 1, 52, 43 6.2, 5.4, 3.7, 1.8 In Exercises 39 42, a find the length of each side of the right triangle, and b show that these lengths satisfy the Pythagorean Theorem. y 39. 4 8 13, 5 3 1, 0 4 2 0, 2 1 4, 2 x 4 x 1 2 3 4 8 13, 0 5 y 41. 50 50, 42 40 30 20 10 12, 18 Distance in yards 4, 5 5 1, 12, 6, 0 7, 4, 2, 8 2, 10, 10, 2 13, 13 , 16, 12 16.8, 12.3, 5.6, 4.9 10 20 30 40 50 60 y 40. 48. 50. 52. 54. 56. 57. FLYING DISTANCE An airplane flies from Naples, Italy in a straight line to Rome, Italy, which is 120 kilometers north and 150 kilometers west of Naples. How far does the plane fly 58. SPORTS A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a teammate who is 42 yards from the same endline and 50 yards from the same sideline, as shown in the figure. How long is the pass Distance in yards 62 59. Big Lots y 42. SALES In Exercises 59 and 60, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in 2005, given the sales in 2003 and 2007. Assume that the sales followed a linear pattern. Source: Big Lots, Inc.; Dollar Tree Stores, Inc. 1, 5 6 4 9, 4 Year Sales in millions 2003 2007 4174 4656 4 2 9, 1 2 5, 2 x 1, 1 6 x 8 2 1, 2 6 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 60. Dollar Tree Sales in millions 2003 2007 2800 4243 In Exercises 6164, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. y 3, 6 3 units 4 1, 1 7 5 1, 3 6 units x 2 2, 4 3, 0 5, 3 2 units 2, 3 x 1 3 63. Original coordinates of vertices: 7, 2, 2, 2, 2, 4, 7, 4 Shift: eight units upward, four units to the right 64. Original coordinates of vertices: 5, 8, 3, 6, 7, 6, 5, 2 Shift: 6 units downward, 10 units to the left RETAIL PRICE In Exercises 65 and 66, use the graph, which shows the average retail prices of 1 gallon of whole milk from 1996 through 2007. Source: U.S. Bureau of Labor Statistics Average price in dollars per gallon Year 67 a Estimate the percent increase in the average cost of a 30 second spot from Super Bowl XXXIV in 2000 to Super Bowl XXXVIII in 2004. b Estimate the percent increase in the average cost of a 30 second spot from Super Bowl XXXIV in 2000 to Super Bowl XLII in 2008. 68. ADVERTISING The graph shows the average costs of a 30 second television spot in thousands of dollars during the Academy Awards from 1995 through 2007. Source: Nielson Monitor Plus 1800 1600 1400 1200 1000 800 600 1995 4.00 3.80 3.60 3.40 3.20 3.00 2.80 2.60 1997 1999 2001 2003 2005 2007 Year 1996 1998 2000 2002 2004 2006 Year 65. Approximate the highest price of a gallon of whole milk shown in the graph. When did this occur 66. Approximate the percent change in the price of milk from the price in 1996 to the highest price shown in the graph. 67. ADVERTISING The graph shows the average costs of a 30 second television spot in thousands of dollars during the Super Bowl from 2000 through 2008. Source: Nielson Media and TNS Media Intelligence a Estimate the percent increase in the average cost of a 30 second spot in 1996 to the cost in 2002. b Estimate the percent increase in the average cost of a 30 second spot in 1996 to the cost in 2007. 69. MUSIC The graph shows the numbers of performers who were elected to the Rock and Roll Hall of Fame from 1991 through 2008. Describe any trends in the data. From these trends, predict the number of performers elected in 2010. Source: rockhall.com 10 Number elected 4 2 2000 2001 2002 2003 2004 2005 2006 2007 2008 FIGURE FOR y 62. 5 units 61. 2800 2700 2600 2500 2400 2300 2200 2100 2000 Cost of 30 second TV spot in thousands of dollars Year 63 The Rectangular Coordinate System and Graphs Cost of 30 second TV spot in thousands of dollars Section P.6 8 6 4 2 1991 1993 1995 1997 1999 2001 2003 2005 2007 Year www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 64 Chapter P Prerequisites Minimum wage in dollars 70. LABOR FORCE Use the graph below, which shows the minimum wage in the United States in dollars from 1950 through 2009. Source: U.S. Department of Labor Year, x Pieces of mail, y 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 183 191 197 202 208 207 203 202 206 212 213 212 203 8 7 6 5 4 3 2 1 1950 1960 1970 1980 1990 2000 2010 Year a Which decade shows the greatest increase in minimum wage b Approximate the percent increases in the minimum wage from 1990 to 1995 and from 1995 to 2009. c Use the percent increase from 1995 to 2009 to predict the minimum wage in 2013. d Do you believe that your prediction in part c is reasonable Explain. 71. SALES The Coca Cola Company had sales of 19,805 million in 1999 and 28,857 million in 2007. Use the Midpoint Formula to estimate the sales in 2003. Assume that the sales followed a linear pattern. Source: The Coca Cola Company 72. DATA ANALYSIS: EXAM SCORES The table shows the mathematics entrance test scores x and the final examination scores y in an algebra course for a sample of 10 students. x 22 29 35 40 44 48 53 58 65 76 y 53 74 57 66 79 90 76 93 83 99 a Sketch a scatter plot of the data. b Find the entrance test score of any student with a final exam score in the 80s. c Does a higher entrance test score imply a higher final exam score Explain. 73. DATA ANALYSIS: MAIL The table shows the number y of pieces of mail handled in billions by the U.S. Postal Service for each year x from 1996 through 2008. Source: U.S. Postal Service TABLE FOR 73 a Sketch a scatter plot of the data. b Approximate the year in which there was the greatest decrease in the number of pieces of mail handled. c Why do you think the number of pieces of mail handled decreased 74. DATA ANALYSIS: ATHLETICS The table shows the numbers of mens M and womens W college basketball teams for each year x from 1994 through 2007. Source: National Collegiate Athletic Association Year, x Mens teams, M Womens teams, W 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 858 868 866 865 895 926 932 937 936 967 981 983 984 982 859 864 874 879 911 940 956 958 975 1009 1008 1036 1018 1003 a Sketch scatter plots of these two sets of data on the same set of coordinate axes. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section P.6 b Find the year in which the numbers of mens and womens teams were nearly equal. c Find the year in which the difference between the numbers of mens and womens teams was the greatest. What was this difference EXPLORATION 75. A line segment has x1, y1 as one endpoint and xm, ym as its midpoint. Find the other endpoint x2, y2 of the line segment in terms of x1, y1, xm, and ym. 76. Use the result of Exercise 75 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectively, a 1, 2, 4, 1 and b 5, 11, 2, 4. 77. Use the Midpoint Formula three times to find the three points that divide the line segment joining x1, y1 and x2, y2 into four parts. 78. Use the result of Exercise 77 to find the points that divide the line segment joining the given points into four equal parts. a 1, 2, 4, 1 b 2, 3, 0, 0 79. MAKE A CONJECTURE Plot the points 2, 1, 3, 5, and 7, 3 on a rectangular coordinate system. Then change the sign of the x coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. a The sign of the x coordinate is changed. b The sign of the y coordinate is changed. c The signs of both the x and y coordinates are changed. 80. COLLINEAR POINTS Three or more points are collinear if they all lie on the same line. Use the steps below to determine if the set of points A2, 3, B2, 6, C6, 3 and the set of points A8, 3, B5, 2, C2, 1 are collinear. a For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship exists among these distances for each set of points b Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line c Compare your conclusions from part a with the conclusions you made from the graphs in part b . Make a general statement about how to use the Distance Formula to determine collinearity. The Rectangular Coordinate System and Graphs 65 TRUE OR FALSE In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. 81. In order to divide a line segment into 16 equal parts, you would have to use the Midpoint Formula 16 times. 82. The points 8, 4, 2, 11, and 5, 1 represent the vertices of an isosceles triangle. 83. THINK ABOUT IT When plotting points on the rectangular coordinate system, is it true that the scales on the x and y axes must be the same Explain. 84. CAPSTONE Use the plot of the point x0 , y0 in the figure. Match the transformation of the point with the correct plot. Explain your reasoning. The plots are labeled i , ii , iii , and iv . y x0 , y0 x i y y ii x iii y x y iv x a x0, y0 c x0, 12 y0 x b 2x0, y0 d x0, y0 85. PROOF Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. www.elsolucionario.net y b , c a + b , c 0, 0 a, 0 x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 66 Chapter P Prerequisites Section P.3 Section P.2 Section P.1 P CHAPTER SUMMARY What Did You Learn ExplanationExamples Represent and classify real numbers p. 2 . Real numbers: set of all rational and irrational numbers Rational numbers: real numbers that can be written as the ratio of two integers Irrational numbers: real numbers that cannot be written as the ratio of two integers Real numbers can be represented on the real number line. 1, 2 Order real numbers and use inequalities p. 4 . a b: a is less than b. a b: a is greater than b. a b: a is less than or equal to b. a b: a is greater than or equal to b. 36 Find the absolute values of real numbers and find the distance between two real numbers p. 6 . Absolute value of a: a if a 0 if a 0 Distance between a and b: da, b b a a b 712 Evaluate algebraic expressions p. 8 . To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. 1316 Use the basic rules and properties of algebra p. 9 . The basic rules of algebra, the properties of negation and equality, the properties of zero, and the properties and operations of fractions can be used to perform operations. 1730 Use properties of exponents p. 15 . 1. aman amn 4. a 0 1, a 0 7. ab m a mb m Use scientific notation to represent real numbers p. 17 . A number written in scientific notation has the form c 10n, where 1 c 10 and n is an integer. Use properties of radicals p. 19 to simplify and combine radicals p. 21 . Review Exercises a,a, 2. aman amn 5. abm ambm 8. a2 a2 3. an 1an 6. a m n a mn 3138 3942 n n m n a n ab a a 1. 2. n b mn n n n m n 3. ab ab, b 0 4. a a n a n a n n n an a 5. 6. n even: a a, n odd: A radical expression is in simplest form when 1 all possible factors have been removed from the radical, 2 all fractions have radical free denominators, and 3 the index of the radical is reduced. Radical expressions can be combined if they are like radicals. m 4350 Rationalize denominators and numerators p. 22 . To rationalize a denominator or numerator of the form a bm or a bm, multiply both numerator and denominator by a conjugate. 5156 Use properties of rational exponents p. 23 . If a is a real number and n is a positive integer such that the principal nth root of a exists, then a1n is defined as n a, a1n where 1n is the rational exponent of a. 5760 Write polynomials in standard form p. 28 , and add, subtract, and multiply polynomials p. 29 . In standard form, a polynomial is written with descending powers of x. To add and subtract polynomials, add or subtract the like terms. To find the product of two polynomials, use the FOIL method. 6172 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 67 Chapter Summary ExplanationExamples Use special products to multiply polynomials p. 30 . Sum and difference of same terms: u vu v u2 v2 2 Square of a binomial: u v u2 2uv v2 u v2 u2 2uv v2 Cube of a binomial: u v3 u3 3u 2 v 3uv2 v 3 u v3 u3 3u 2 v 3uv2 v 3 7376 Use polynomials to solve real life problems p. 32 . Polynomials can be used to find the volume of a box. See Example 9. 7780 Remove common factors from polynomials p. 37 . The process of writing a polynomial as a product is called factoring. Removing factoring out any common factors is the first step in completely factoring a polynomial. 81, 82 Factor special polynomial forms p. 38 . Difference of two squares: u2 v2 u vu v Perfect square trinomial: u2 2uv v2 u v2 u2 2uv v2 u v2 3 3 Sum or difference u v u vu2 uv v2 u3 v3 u vu2 uv v2 of two cubes: 83 86 Factor trinomials as the product of two binomials p. 40 . ax2 bx c x x 87, 88 Section P.4 Section P.3 What Did You Learn Section P.6 Section P.5 Factors of a Review Exercises Factors of c Factor polynomials by grouping p. 41 . Polynomials with more than three terms can sometimes be factored by a method called factoring by grouping. See Examples 9 and 10. Find domains of algebraic expressions p. 45 . The set of real numbers for which an algebraic expression is defined is the domain of the expression. 91, 92 Simplify rational expressions p. 45 . When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. 93, 94 Add, subtract, multiply, and divide rational expressions p. 47 . To add or subtract, use the LCD method or the basic a c ad bc , b 0, d 0. To multiply or definition b d bd divide, use the properties of fractions. 9598 Simplify complex fractions and rewrite difference quotients p. 49 . To simplify a complex fraction, combine the fractions in the numerator into a single fraction and then combine the fractions in the denominator into a single fraction. Then invert the denominator and multiply. 99102 Plot points in the Cartesian plane p. 55 . For an ordered pair x, y, the x coordinate is the directed distance from the y axis to the point, and the y coordinate is the directed distance from the x axis to the point. 103106 Use the Distance Formula p. 57 and the Midpoint Formula p. 58 . Distance Formula: d x2 x12 y2 y12 107110 Midpoint Formula: Midpoint Use a coordinate plane to model and solve real life problems p. 59 . x 1 x2 y1 y2 , 2 2 The coordinate plane can be used to find the length of a football pass See Example 6 . www.elsolucionario.net 89, 90 111114 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 68 Chapter P Prerequisites P REVIEW EXERCISES P.1 In Exercises 1 and 2, determine which numbers in the set are a natural numbers, b whole numbers, c integers, d rational numbers, and e irrational numbers. 1. 11, 14, 89, 52, 6, 0.4 3 2. 15, 22, 10 3 , 0, 5.2, 7 3. a b 7 8 4. a 9 25 b 5 7 In Exercises 5 and 6, give a verbal description of the subset of real numbers represented by the inequality, and sketch the subset on the real number line. 5. x 7 6. x 1 In Exercises 7 and 8, find the distance between a and b. 7. a 74, b 48 8. a 112, b 6 In Exercises 912, use absolute value notation to describe the situation. 9. 10. 11. 12. The distance between x and 7 is at least 4. The distance between x and 25 is no more than 10. The distance between y and 30 is less than 5. The distance between z and 16 is greater than 8. In Exercises 1316, evaluate the expression for each value of x. If not possible, state the reason. Expression 13. 12x 7 14. x 2 6x 5 15. x 2 x 1 x 16. x3 21. t2 1 3 3 t2 1 22. 1 3x 4 3x 4 In Exercises 2330, perform the operation s . Write fractional answers in simplest form. In Exercises 3 and 4, use a calculator to find the decimal form of each rational number. If it is a nonterminating decimal, write the repeating pattern. Then plot the numbers on the real number line and place the appropriate inequality sign or between them. 5 6 See www.CalcChat.com for worked out solutions to odd numbered exercises. Values a x 0 b x 1 a x 2 b x 2 a x 1 b x 1 a x 3 b x 3 In Exercises 1722, identify the rule of algebra illustrated by the statement. 17. 2x 3x 10 2x 3x 10 18. 4t 2 4 t 4 2 19. 0 a 5 a 5 2 y4 20. 2 1, y 4 y4 23. 3 42 6 5 18 10 3 25. 27. 64 26 8 29. x 7x 5 12 24. 10 10 26. 16 8 4 28. 416 37 10 30. 9 1 x 6 P.2 In Exercises 3134, simplify each expression. 31. a 3x24x33 b 5y6 10y 32. a 3a26a3 b 36x5 9x10 33. a 2z3 b 8y0 y2 34. a x 2 23 b 40b 35 75b 32 In Exercises 3538, rewrite each expression with positive exponents and simplify. 35. a a2 b2 b a2b43ab2 36. a 62u3v3 12u2v b 34m1n3 92mn3 37. a 5a2 5a2 b 4x13 42x11 b y y 38. a x y11 x3 x 1 In Exercises 39 and 40, write the number in scientific notation. 39. Sales for Nautilus, Inc. in 2007: 501,500,000 Source: Nautilus, Inc. 40. Number of meters in 1 foot: 0.3048 In Exercises 41 and 42, write the number in decimal notation. 41. Distance between the sun and Jupiter: 4.84 42. Ratio of day to year: 2.74 103 www.elsolucionario.net 108 miles http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Review Exercises In Exercises 4346, simplify each expression. 3 272 493 43. a 3 64 44. a 125 3 216 3 45. a 2x3 46. a 3 27 b 81 b 100 4 324 b In Exercises 6568, perform the operation and write the result in standard form. 65. 66. 67. 68. 5 64x6 b 69 3x 2 2x 1 5x 8y 2y 2 3y 8 2xx2 5x 6 3x3 1.5x2 43x In Exercises 47 and 48, simplify each expression. In Exercises 69 and 70, perform the operation. 47. a 50 18 48. a 8x3 2x 69. Add 2x3 5x2 10x 7 and 4x2 7x 2. 70. Subtract 9x4 11x2 16 from 6x4 20x2 x 3. b 232 372 b 18x 5 8x 3 49. WRITING Explain why 5u 3u 22u. 50. ENGINEERING The rectangular cross section of a wooden beam cut from a log of diameter 24 inches see figure will have a maximum strength if its width w and height h are w 83 and h 242 83 . 2 Find the area of the rectangular cross section and write the answer in simplest form. h 24 w In Exercises 5154, rationalize the denominator of the expression. Then, simplify your answer. 51. 3 43 1 53. 2 3 52. 54. 12 In Exercises 7176, find the product. 72. 73. 2x 32 75. 35 235 2 74. 6x 56x 5 76. x 43 77. COMPOUND INTEREST After 2 years, an investment of 2500 compounded annually at an interest rate r will yield an amount of 25001 r2. Write this polynomial in standard form. 78. SURFACE AREA The surface area S of a right circular cylinder is S 2 r 2 2 rh. a Draw a right circular cylinder of radius r and height h. Use the figure to explain how the surface area formula was obtained. b Find the surface area when the radius is 6 inches and the height is 8 inches. 79. GEOMETRY Find a polynomial that represents the total number of square feet for the floor plan shown in the figure. 3 4 1 12 ft 5 1 In Exercises 55 and 56, rationalize the numerator of the expression. Then, simplify your answer. 55. 7 1 2 56. x 2 11 x 3 In Exercises 5760, simplify the expression. 57. 1632 59. 3x252x12 x 1x x 2 71. 3x 65x 1 58. 6423 60. x 113x 114 16 ft 80. GEOMETRY Use the area model to write two different expressions for the area. Then equate the two expressions and name the algebraic property that is illustrated. x 5 x P.3 In Exercises 6164, write the polynomial in standard form. Identify the degree and leading coefficient. 61. 3 11x 2 63. 4 12x 2 3 62. 3x 3 5x 5 x 4 64. 12x 7x 2 6 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 70 Chapter P Prerequisites P.4 In Exercises 8190, completely factor the expression. 81. 83. 85. 87. 89. x x 25x 2 49 x 3 64 2x 2 21x 10 x3 x 2 2x 2 3 82. 84. 86. 88. 90. xx 3 4x 3 x 2 12x 36 8x 3 27 3x 2 14x 8 x 3 4x 2 2x 8 P.5 In Exercises 91 and 92, find the domain of the expression. 91. 1 x6 x 2 64 53x 24 94. x 3 27 x6 3x 4x 2 5 x 2 2x 2 3x 2 In Exercises 99 and 100, simplify the complex fraction. 2x 3 2x 3 100. 1 1 2x 2x 3 3a 1 101. 1 1 xh3 x3 102. h P.6 In Exercises 103 and 104, plot the points in the Cartesian plane. 103. 5, 5, 2, 0, 3, 6, 1, 7 104. 0, 6, 8, 1, 4, 2, 3, 3 x 70 75 80 85 90 95 100 y 70 77 85 95 109 130 150 a Sketch a scatter plot of the data shown in the table. b Find the change in the apparent temperature when the actual temperature changes from 70F to 100F. EXPLORATION TRUE OR FALSE In Exercises 115 and 116, determine whether the statement is true or false. Justify your answer. 115. A binomial sum squared is equal to the sum of the terms squared. 116. x n y n factors as conjugates for all values of n. 117. THINK ABOUT IT Is the following statement true for all nonzero real numbers a and b Explain. In Exercises 105 and 106, determine the quadrant s in which x, y is located so that the condition s is are satisfied. 105. x 0 and y 2 113. SALES Starbucks had annual sales of 2.17 billion in 2000 and 10.38 billion in 2008. Use the Midpoint Formula to estimate the sales in 2004. Source: Starbucks Corp. 114. METEOROLOGY The apparent temperature is a measure of relative discomfort to a person from heat and high humidity. The table shows the actual temperatures x in degrees Fahrenheit versus the apparent temperatures y in degrees Fahrenheit for a relative humidity of 75. 1 In Exercises 101 and 102, simplify the difference quotient. 1 1 2x h 2x h Shift: eight units downward, four units to the left 112. Original coordinates of vertices: Shift: three units upward, two units to the left x2 4 x2 2 x4 2x 2 8 x2 2 4x 6 2x 3x 96. x 12 x 2 2x 3 1 1x 97. x 1 x2 x 1 2 In Exercises 111 and 112, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. 0, 1, 3, 3, 0, 5, 3, 3 x2 95. a x 1 99. a x 1 108. 2, 6, 4, 3 110. 1.8, 7.4, 0.6, 14.5 4, 8, 6, 8, 4, 3, 6, 3 In Exercises 9598, perform the indicated operation and simplify. 98. 107. 3, 8, 1, 5 109. 5.6, 0, 0, 8.2 111. Original coordinates of vertices: 92. x 4 In Exercises 93 and 94, write the rational expression in simplest form. 93. In Exercises 107110, a plot the points, b find the distance between the points, and c find the midpoint of the line segment joining the points. ax b 1 b ax 106. xy 4 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Test P CHAPTER TEST 71 See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Place or between the real numbers 10 3 and 4. 2. Find the distance between the real numbers 5.4 and 334. 3. Identify the rule of algebra illustrated by 5 x 0 5 x. In Exercises 4 and 5, evaluate each expression without using a calculator. 3 4. a 27 5. a 5 2 125 5 5 18 8 27 b 2 b c 5 c 5.4 108 3 103 3 3 d 2 32 d 3 3 1043 In Exercises 6 and 7, simplify each expression. x2y 2 3 6. a 3z 22z3 2 b u 24u 23 c 7. a 9z8z 32z3 b 4x35x13 c 16v 3 1 5 8. Write the polynomial 3 2x5 3x3 x 4 in standard form. Identify the degree and leading coefficient. In Exercises 912, perform the operation and simplify. 9. x 2 3 3x 8 x 2 10. x 5 x 5 x x 1 12. 4 x 1 2 5x 20 11. x4 4x 2 2 13. Factor a 2x 4 3x 3 2x 2 and b x3 2x 2 4x 8 completely. 16 4 14. Rationalize each denominator. a 3 b 16 1 2 15. Find the domain of 16. Multiply: 2 3 3x 3x 2x FIGURE FOR 19 x 6x . 1x y2 8y 16 2y 4 8y 16 y 43. 17. A T shirt company can produce and sell x T shirts per day. The total cost C in dollars for producing x T shirts is C 1480 6x, and the total revenue R in dollars is R 15x. Find the profit obtained by selling 225 T shirts per day. 18. Plot the points 2, 5 and 6, 0. Find the coordinates of the midpoint of the line segment joining the points and the distance between the points. 19. Write an expression for the area of the shaded region in the figure at the left, and simplify the result. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS What does the word proof mean to you In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula p. 58 The midpoint of the line segment joining the points x1, y1 and x2, y2 is given by the Midpoint Formula Midpoint x 1 x2 y1 y2 , . 2 2 Proof The Cartesian Plane The Cartesian plane was named after the French mathematician Rene Descartes 15961650 . While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. Using the figure, you must show that d1 d2 and d1 d2 d3. y x1, y1 d1 x +2 x , y +2 y 1 d3 2 1 2 d2 x 2, y 2 x By the Distance Formula, you obtain d1 x1 x2 x1 2 2 y1 y2 y1 2 2 2 1 x2 x12 y2 y12 2 d2 x 2 x1 x2 2 y 2 2 y1 y2 2 1 x2 x12 y2 y12 2 d3 x2 x12 y2 y12 So, it follows that d1 d2 and d1 d2 d3. 72 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROBLEM SOLVING This collection of thought provoking and challenging exercises further explores and expands upon concepts learned in this chapter. Weight minimum Diameter minimum Diameter maximum Mens Womens 7.26 kg 4.0 kg 110 mm 95 mm 130 mm 110 mm a Find the maximum and minimum volumes of both the mens and womens shots. b The density of an object is an indication of how heavy the object is. To find the density of an object, divide its mass weight by its volume. Find the maximum and minimum densities of both the mens and womens shots. c A shot is usually made out of iron. If a ball of cork has the same volume as an iron shot, do you think they would have the same density Explain your reasoning. 2. Find an example for which a b a b, and an example for which a b a b. Then prove that a b a b for all a, b. 3. A major feature of Epcot Center at Disney World is called Spaceship Earth. The building is shaped as a sphere and weighs 1.6 107 pounds, which is equal in weight to 1.58 108 golf balls. Use these values to find the approximate weight in pounds of one golf ball. Then convert the weight to ounces. Source: Disney.com 4. The average life expectancies at birth in 2005 for men and women were 75.2 years and 80.4 years, respectively. Assuming an average healthy heart rate of 70 beats per minute, find the numbers of beats in a lifetime for a man and for a woman. Source: National Center for Health Statistics 5. The accuracy of an approximation to a number is related to how many significant digits there are in the approximation. Write a definition of significant digits and illustrate the concept with examples. 6. The table shows the census population y in millions of the United States for each census year x from 1950 through 2000. Source: U.S. Census Bureau Year, x Population, y 1950 1960 1970 1980 1990 2000 151.33 179.32 203.30 226.54 248.72 281.42 a Sketch a scatter plot of the data. Describe any trends in the data. b Find the increase in population from each census year to the next. c Over which decade did the population increase the most the least d Find the percent increase in population from each census year to the next. e Over which decade was the percent increase the greatest the least 7. Find the annual depreciation rate r from the bar graph below. To find r by the declining balances method, use the formula r1 C S 1n where n is the useful life of the item in years , S is the salvage value in dollars , and C is the original cost in dollars . Value in thousands of dollars 1. The NCAA states that the mens and womens shots for track and field competition must comply with the following specifications. Source: NCAA 14 12 Cost: 12,000 10 8 Salvage value: 3,225 6 4 2 n 0 1 2 3 4 Year 73 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Planet x Mercury Venus Earth Mars Jupiter 0.387 0.723 1.000 1.524 5.203 x y1 2x1 x 2 y2 0.615 1.000 1.881 11.860 3 y 9. A stained glass window is designed in the shape of a rectangle with a semicircular arch see figure . The width of the window is 2 feet and the perimeter is approximately 13.14 feet. Find the smallest amount of glass required to construct the window. x3 1 x 2 2 3x 2 1 x 2 Change y2 so that y1 y 2. 12. Prove that 2x 3 x , 2y 3 y 1 2 1 2 is one of the points of trisection of the line segment joining x1, y1 and x2, y2. Find the midpoint of the line segment joining 2x 3 x , 2y 3 y 1 0.241 y y2 by letting x 0 and evaluating y1 11. Verify that y1 and y2. 2 1 2 and x2, y2 to find the second point of trisection. 13. Use the results of Exercise 12 to find the points of trisection of the line segment joining each pair of points. a 1, 2, 4, 1 b 2, 3, 0, 0 14. Although graphs can help visualize relationships between two variables, they can also be used to mislead people. The graphs shown below represent the same data points. a Which of the two graphs is misleading, and why Discuss other ways in which graphs can be misleading. b Why would it be beneficial for someone to use a misleading graph 2 ft 10. The volume V in cubic inches of the box shown in the figure is modeled by V 2x3 x2 8x 4 Company profits 8. Johannes Kepler 15711630 , a well known German astronomer, discovered a relationship between the average distance of a planet from the sun and the time or period it takes the planet to orbit the sun. People then knew that planets that are closer to the sun take less time to complete an orbit than planets that are farther from the sun. Kepler discovered that the distance and period are related by an exact mathematical formula. The table shows the average distances x in astronomical units and periods y in years for the five planets that are closest to the sun. By completing the table, can you rediscover Keplers relationship Write a paragraph that summarizes your conclusions. where x is measured in inches. Find an expression for the surface area of the box. Then find the surface area when x 6 inches. 40 30 20 10 0 J M M J S N Month Company profits 2x + 1 50 34.4 34.0 33.6 33.2 32.8 32.4 32.0 J M M J S N Month 74 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Equations, Inequalities, and Mathematical Modeling 1.1 Graphs of Equations 1.2 Linear Equations in One Variable 1.3 Modeling with Linear Equations 1.4 Quadratic Equations and Applications 1.5 Complex Numbers 1.6 Other Types of Equations 1.7 Linear Inequalities in One Variable 1.8 Other Types of Inequalities 1 In Mathematics The methods used for solving equations are similar to the methods used for solving inequalities. In Real Life istockphoto.com Real life data can be modeled by many types of equations. These include linear, quadratic, radical, rational, and higher order polynomial equations. Inequalities can also be used to model and solve real life problems. For instance, inequalities can be used to represent the range of the target heart rates for a 20 year old and a 40 yearold. See Exercises 109 and 110, page 147. IN CAREERS There are many careers that use equations and inequalities. Several are listed below. Electrician Exercise 80, page 86 Physicist Exercises 93 and 94, page 106 Anthropologist Exercise 107, page 94 Physical Chemist Exercise 130, page 149 75 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 76 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.1 GRAPHS OF EQUATIONS What you should learn Sketch graphs of equations. Find x and y intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of circles. Use graphs of equations in solving real life problems. Why you should learn it The graph of an equation can help you see relationships between real life quantities. For example, in Exercise 79 on page 86, a graph can be used to estimate the life expectancies of children who are born in 2015. The Graph of an Equation In Section P.6, you used a coordinate system to represent graphically the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane. Frequently, a relationship between two quantities is expressed as an equation in two variables. For instance, y 7 3x is an equation in x and y. An ordered pair a, b is a solution or solution point of an equation in x and y if the equation is true when a is substituted for x, and b is substituted for y. For instance, 1, 4 is a solution of y 7 3x because 4 7 31 is a true statement. In this section you will review some basic procedures for sketching the graph of an equation in two variables. The graph of an equation is the set of all points that are solutions of the equation. Example 1 Determining Solution Points Determine whether a 2, 13 and b 1, 3 lie on the graph of y 10x 7. Solution a. y 10x 7 13 102 7 Write original equation. 13 13 2, 13 is a solution. Substitute 2 for x and 13 for y. The point 2, 13 does lie on the graph of y 10x 7 because it is a solution point of the equation. John GriffinThe Image Works b. y 10x 7 3 101 7 Write original equation. 3 1, 3 is not a solution. 17 Substitute 1 for x and 3 for y. The point 1, 3 does not lie on the graph of y 10x 7 because it is not a solution point of the equation. Now try Exercise 7. The basic technique used for sketching the graph of an equation is the point plotting method. Sketching the Graph of an Equation by Point Plotting When evaluating an expression or an equation, remember to follow the Basic Rules of Algebra. To review these rules, see Section P.1. 1. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation. 2. Make a table of values showing several solution points. 3. Plot these points on a rectangular coordinate system. 4. Connect the points with a smooth curve or line. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.1 Graphs of Equations 77 When making a table of solution points, be sure to use positive, zero, and negative values of x. Example 2 Sketching the Graph of an Equation Sketch the graph of y 7 3x. Solution Because the equation is already solved for y, construct a table of values that consists of several solution points of the equation. For instance, when x 1, y 7 31 10 which implies that 1, 10 is a solution point of the graph. x y 7 3x x, y 1 10 1, 10 0 7 0, 7 1 4 1, 4 2 1 2, 1 3 2 3, 2 4 5 4, 5 From the table, it follows that 1, 10, 0, 7, 1, 4, 2, 1, 3, 2, and 4, 5 are solution points of the equation. After plotting these points, you can see that they appear to lie on a line, as shown in Figure 1.1. The graph of the equation is the line that passes through the six plotted points. y 1, 10 8 6 4 0, 7 1, 4 2 2, 1 x 4 2 2 4 6 FIGURE 2 4 6 8 10 3, 2 4, 5 1.1 Now try Exercise 15. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 78 Chapter 1 Equations, Inequalities, and Mathematical Modeling Example 3 Sketching the Graph of an Equation Sketch the graph of y x 2 2. Solution Because the equation is already solved for y, begin by constructing a table of values. 2 1 0 1 2 3 2 1 2 1 2 7 2, 2 1, 1 0, 2 1, 1 2, 2 3, 7 x yx 2 2 One of your goals in this course is to learn to classify the basic shape of a graph from its equation. For instance, you will learn that the linear equation in Example 2 has the form x, y Next, plot the points given in the table, as shown in Figure 1.2. Finally, connect the points with a smooth curve, as shown in Figure 1.3. y y y mx b and its graph is a line. Similarly, the quadratic equation in Example 3 has the form y ax 2 bx c and its graph is a parabola. 3, 7 3, 7 6 6 4 4 2 2 y = x2 2 2, 2 4 x 2 1, 1 FIGURE 2, 2 2, 2 2 1, 1 0, 2 4 4 1.2 2 1, 1 FIGURE 2, 2 x 2 1, 1 0, 2 4 1.3 Now try Exercise 17. The point plotting method demonstrated in Examples 2 and 3 is easy to use, but it has some shortcomings. With too few solution points, you can misrepresent the graph of an equation. For instance, if only the four points 2, 2, 1, 1, 1, 1, and 2, 2 in Figure 1.2 were plotted, any one of the three graphs in Figure 1.4 would be reasonable. y y 4 4 4 2 2 2 x 2 FIGURE y 2 2 x 2 2 x 2 1.4 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.1 79 Graphs of Equations T E C H N O LO G Y To graph an equation involving x and y on a graphing utility, use the following procedure. 1. Rewrite the equation so that y is isolated on the left side. 2. Enter the equation in the graphing utility. 3. Determine a viewing window that shows all important features of the graph. 4. Graph the equation. Intercepts of a Graph It is often easy to determine the solution points that have zero as either the x coordinate or the y coordinate. These points are called intercepts because they are the points at which the graph intersects or touches the x or y axis. It is possible for a graph to have no intercepts, one intercept, or several intercepts, as shown in Figure 1.5. y y y x x No x intercepts One y intercept FIGURE 1.5 Three x intercepts One y intercept y x One x intercept Two y intercepts x No intercepts Note that an x intercept can be written as the ordered pair x, 0 and a y intercept can be written as the ordered pair 0, y. Some texts denote the x intercept as the x coordinate of the point a, 0 and the y intercept as the y coordinate of the point 0, b rather than the point itself. Unless it is necessary to make a distinction, we will use the term intercept to mean either the point or the coordinate. y Example 4 5 4 3 2 Identify the x and y intercepts of the graph of y x3 1 y = x3 + 1 x 4 3 2 1 2 3 4 5 2 3 4 5 FIGURE 1.6 Identifying x and y Intercepts shown in Figure 1.6. Solution From the figure, you can see that the graph of the equation y x3 1 has an x intercept where y is zero at 1, 0 and a y intercept where x is zero at 0, 1. Now try Exercise 19. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 80 Chapter 1 Equations, Inequalities, and Mathematical Modeling Symmetry Graphs of equations can have symmetry with respect to one of the coordinate axes or with respect to the origin. Symmetry with respect to the x axis means that if the Cartesian plane were folded along the x axis, the portion of the graph above the x axis would coincide with the portion below the x axis. Symmetry with respect to the y axis or the origin can be described in a similar manner, as shown in Figure 1.7. y y y x, y x, y x, y x, y x x x x, y x, y x axis symmetry FIGURE 1.7 y axis symmetry Origin symmetry Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need only half as many solution points to sketch the graph. There are three basic types of symmetry, described as follows. Graphical Tests for Symmetry 1. A graph is symmetric with respect to the x axis if, whenever x, y is on the graph, x, y is also on the graph. 2. A graph is symmetric with respect to the y axis if, whenever x, y is on the graph, x, y is also on the graph. 3. A graph is symmetric with respect to the origin if, whenever x, y is on the graph, x, y is also on the graph. You can conclude that the graph of y x 2 2 is symmetric with respect to the y axis because the point x, y is also on the graph of y x2 2. See the table below and Figure 1.8. y 7 6 5 4 3 2 1 3, 7 2, 2 3, 7 2, 2 x 4 3 2 1, 1 3 y axis symmetry FIGURE 1.8 2 3 4 5 x 3 2 1 1 2 3 y 7 2 1 1 2 7 3, 7 2, 2 1, 1 1, 1 2, 2 3, 7 x, y 1, 1 y = x2 2 Algebraic Tests for Symmetry 1. The graph of an equation is symmetric with respect to the x axis if replacing y with y yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the y axis if replacing x with x yields an equivalent equation. 3. The graph of an equation is symmetric with respect to the origin if replacing x with x and y with y yields an equivalent equation. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.1 Example 5 2 Solution 1, 2 x axis: y = 2x 3 1 y 2x3 1 1 y axis: 2 1 Replace y with y. Result is not an equivalent equation. y 2x3 Write original equation. y 2x3 Replace x with x. y 2x Simplify. Result is not an equivalent equation. 3 1, 2 2 Origin: 1.9 y 2x3 Write original equation. y 2x 3 y y 5, 2 1 1, 0 Simplify. 4 Equivalent equation Now try Exercise 25. x 3 Replace y with y and x with x. Of the three tests for symmetry, the only one that is satisfied is the test for origin symmetry see Figure 1.9 . 2, 1 2 2x3 y 2x3 x y2 = 1 2 Write original equation. y 2x3 x FIGURE Testing for Symmetry Test y 2x3 for symmetry with respect to both axes and the origin. y 2 81 Graphs of Equations 5 1 Example 6 2 Using Symmetry as a Sketching Aid Use symmetry to sketch the graph of x y 2 1. FIGURE 1.10 Solution Of the three tests for symmetry, the only one that is satisfied is the test for x axis symmetry because x y2 1 is equivalent to x y2 1. So, the graph is symmetric with respect to the x axis. Using symmetry, you only need to find the solution points above the x axis and then reflect them to obtain the graph, as shown in Figure 1.10. Now try Exercise 41. In Example 7, x 1 is an absolute value expression. You can review the techniques for evaluating an absolute value expression in Section P.1. Example 7 Sketching the Graph of an Equation Sketch the graph of y x 1. Solution This equation fails all three tests for symmetry and consequently its graph is not symmetric with respect to either axis or to the origin. The absolute value sign indicates that y is always nonnegative. Create a table of values and plot the points, as shown in Figure 1.11. From the table, you can see that x 0 when y 1. So, the y intercept is 0, 1. Similarly, y 0 when x 1. So, the x intercept is 1, 0. y 6 5 y = x 1 2, 3 4 3 4, 3 3, 2 2, 1 1, 2 2 0, 1 3 2 1 x x 1, 0 2 3 4 5 y x 1 x, y 2 1 0 1 2 3 4 3 2 1 0 1 2 3 2, 3 1, 2 0, 1 1, 0 2, 1 3, 2 4, 3 2 FIGURE 1.11 Now try Exercise 45. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 82 Chapter 1 Equations, Inequalities, and Mathematical Modeling y Throughout this course, you will learn to recognize several types of graphs from their equations. For instance, you will learn to recognize that the graph of a seconddegree equation of the form y ax 2 bx c Center: h, k is a parabola see Example 3 . The graph of a circle is also easy to recognize. Circles Radius: r Point on circle: x, y Consider the circle shown in Figure 1.12. A point x, y is on the circle if and only if its distance from the center h, k is r. By the Distance Formula, x 1.12 FIGURE x h2 y k2 r. By squaring each side of this equation, you obtain the standard form of the equation of a circle. Standard Form of the Equation of a Circle The point x, y lies on the circle of radius r and center h, k if and only if x h 2 y k 2 r 2. WARNING CAUTION Be careful when you are finding h and k from the standard equation of a circle. For instance, to find the correct h and k from the equation of the circle in Example 8, rewrite the quantities x 12 and y 22 using subtraction. From this result, you can see that the standard form of the equation of a circle with its center at the origin, h, k 0, 0, is simply x 2 y 2 r 2. Example 8 x 12 x 12, Circle with center at origin Finding the Equation of a Circle The point 3, 4 lies on a circle whose center is at 1, 2, as shown in Figure 1.13. Write the standard form of the equation of this circle. y 22 y 22 Solution So, h 1 and k 2. The radius of the circle is the distance between 1, 2 and 3, 4. r x h2 y k2 y 6 3, 4 4 1, 2 6 x 2 2 2 4 FIGURE 1.13 4 Distance Formula 3 1 4 2 Substitute for x, y, h, and k. 42 22 Simplify. 16 4 Simplify. 20 Radius 2 2 Using h, k 1, 2 and r 20, the equation of the circle is x h2 y k2 r 2 Equation of circle x 1 2 y 22 20 2 x 1 2 y 2 2 20. Substitute for h, k, and r. Standard form Now try Exercise 65. You will learn more about writing equations of circles in Section 4.4. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.1 Graphs of Equations 83 Application In this course, you will learn that there are many ways to approach a problem. Three common approaches are illustrated in Example 9. You should develop the habit of using at least two approaches to solve every problem. This helps build your intuition and helps you check that your answers are reasonable. A Numerical Approach: Construct and use a table. A Graphical Approach: Draw and use a graph. An Algebraic Approach: Use the rules of algebra. Example 9 Recommended Weight The median recommended weight y in pounds for men of medium frame who are 25 to 59 years old can be approximated by the mathematical model y 0.073x 2 6.99x 289.0, 62 x 76 where x is the mans height in inches . Company Source: Metropolitan Life Insurance a. Construct a table of values that shows the median recommended weights for men with heights of 62, 64, 66, 68, 70, 72, 74, and 76 inches. b. Use the table of values to sketch a graph of the model. Then use the graph to estimate graphically the median recommended weight for a man whose height is 71 inches. c. Use the model to confirm algebraically the estimate you found in part b . Solution Weight, y 62 64 66 68 70 72 74 76 136.2 140.6 145.6 151.2 157.4 164.2 171.5 179.4 a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure 1.14. From the graph, you can estimate that a height of 71 inches corresponds to a weight of about 161 pounds. Recommended Weight y 180 Weight in pounds Height, x 170 160 150 140 130 x 62 64 66 68 70 72 74 76 Height in inches FIGURE 1.14 c. To confirm algebraically the estimate found in part b , you can substitute 71 for x in the model. y 0.073712 6.9971 289.0 160.70 So, the graphical estimate of 161 pounds is fairly good. Now try Exercise 79. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 84 Chapter 1 1.1 Equations, Inequalities, and Mathematical Modeling EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. An ordered pair a, b is a ________ of an equation in x and y if the equation is true when a is substituted for x, and b is substituted for y. The set of all solution points of an equation is the ________ of the equation. The points at which a graph intersects or touches an axis are called the ________ of the graph. A graph is symmetric with respect to the ________ if, whenever x, y is on the graph, x, y is also on the graph. The equation x h2 y k2 r 2 is the standard form of the equation of a ________ with center ________ and radius ________. 6. When you construct and use a table to solve a problem, you are using a ________ approach. 2. 3. 4. 5. SKILLS AND APPLICATIONS In Exercises 714, determine whether each point lies on the graph of the equation. Equation 7. 8. 9. 10. 11. 12. 13. 14. y x 4 y 5 x y x 2 3x 2 y 4 x 2 y x 1 2 2x y 3 0 x2 y2 20 y 13x3 2x 2 a a a a a a a a Points b 0, 2 b 1, 2 b 2, 0 b 1, 5 b 2, 3 b 1, 2 3, 2 b 16 2, 3 b 5, 3 5, 0 2, 8 6, 0 1, 0 1, 1 4, 2 3, 9 x 1 1 2 y x, y In Exercises 1924, graphically estimate the x and y intercepts of the graph. 19. y x 32 20. y 16 4x 2 y y 20 10 8 6 4 2 8 4 0 1 2 5 2 4 2 21. y x 2 2 0 1 4 3 3 y 3 5 4 3 2 2 1 22. y2 4 x y 3 16. y 4 x 1 x 1 2 4 6 8 x, y 1 x 1 1 2 4 5 x y 4 3 2 1 x, y 1 3 1 23. y 2 2x3 17. y x 2 3x x 0 x y x 1 2 x In Exercises 1518, complete the table. Use the resulting solution points to sketch the graph of the equation. 15. y 2x 5 18. y 5 x 2 24. y x3 4x y 0 1 2 y 5 4 3 3 y 1 x, y 1 1 2 1 www.elsolucionario.net x 2 3 x 1 3 2 3 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.1 In Exercises 25 32, use the algebraic tests to check for symmetry with respect to both axes and the origin. 25. x 2 y 0 27. y x 3 x 29. y 2 x 1 31. xy 2 10 0 26. x y 2 0 28. y x 4 x 2 3 30. y 1 x2 1 32. xy 4 y y 34. 58. y 6 xx 60. y 2 x 57. y xx 6 59. y x 3 In Exercises 61 68, write the standard form of the equation of the circle with the given characteristics. In Exercises 33 36, assume that the graph has the indicated type of symmetry. Sketch the complete graph of the equation. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 33. 85 Graphs of Equations 4 61. 62. 63. 64. 65. 66. 67. 68. Center: 0, 0; Radius: 4 Center: 0, 0; Radius: 5 Center: 2, 1; Radius: 4 Center: 7, 4; Radius: 7 Center: 1, 2; Solution point: 0, 0 Center: 3, 2; Solution point: 1, 1 Endpoints of a diameter: 0, 0, 6, 8 Endpoints of a diameter: 4, 1, 4, 1 4 2 2 x 4 2 x 4 2 2 4 6 8 69. 71. 73. 74. 4 y axis symmetry x axis symmetry y 35. 4 2 y 36. 4 4 2 2 x 2 2 4 4 2 4 x 2 4 2 4 Origin symmetry y axis symmetry In Exercises 3748, identify any intercepts and test for symmetry. Then sketch the graph of the equation. 37. y 3x 1 39. y x 2 2x 41. y x 3 3 43. y x 3 45. y x 6 47. x y 2 1 38. y 2x 3 40. y x 2 2x 42. y x 3 1 44. y 1 x 46. y 1 x 48. x y 2 5 In Exercises 4960, use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. 1 49. y 5 2x 51. y x 2 4x 3 2x 53. y x1 3 x 2 55. y The symbol 2 50. y 3x 1 52. y x 2 x 2 4 54. y 2 x 1 3 x 1 56. y In Exercises 6974, find the center and radius of the circle, and sketch its graph. x 2 y 2 25 x 12 y 32 9 x 12 2 y 12 2 94 x 22 y 32 169 70. x 2 y 2 36 72. x 2 y 1 2 1 75. DEPRECIATION A hospital purchases a new magnetic resonance imaging MRI machine for 500,000. The depreciated value y reduced value after t years is given by y 500,000 40,000t, 0 t 8. Sketch the graph of the equation. 76. CONSUMERISM You purchase an all terrain vehicle ATV for 8000. The depreciated value y after t years is given by y 8000 900t, 0 t 6. Sketch the graph of the equation. 77. GEOMETRY A regulation NFL playing field including the end zones of length x and width y has a perimeter 2 1040 of 3463 or 3 yards. a Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. b Show that the width of the rectangle is 520 520 y x and its area is A x x . 3 3 c Use a graphing utility to graph the area equation. Be sure to adjust your window settings. d From the graph in part c , estimate the dimensions of the rectangle that yield a maximum area. e Use your schools library, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL playing field and compare your findings with the results of part d . indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 86 Chapter 1 Equations, Inequalities, and Mathematical Modeling 78. GEOMETRY A soccer playing field of length x and width y has a perimeter of 360 meters. a Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. b Show that the width of the rectangle is y 180 x and its area is A x180 x. c Use a graphing utility to graph the area equation. Be sure to adjust your window settings. d From the graph in part c , estimate the dimensions of the rectangle that yield a maximum area. e Use your schools library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part d . 79. POPULATION STATISTICS The table shows the life expectancies of a child at birth in the United States for selected years from 1920 to 2000. Source: U.S. National Center for Health Statistics Year Life Expectancy, y 1920 1930 1940 1950 1960 1970 1980 1990 2000 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4 77.0 e Do you think this model can be used to predict the life expectancy of a child 50 years from now Explain. 80. ELECTRONICS The resistance y in ohms of 1000 feet of solid copper wire at 68 degrees Fahrenheit can be approximated by the model y 10,770 0.37, 5 x 100 x2 where x is the diameter of the wire in mils 0.001 inch . Source: American Wire Gage a Complete the table. x 5 10 20 30 40 50 y x 60 70 80 90 100 y b Use the table of values in part a to sketch a graph of the model. Then use your graph to estimate the resistance when x 85.5. c Use the model to confirm algebraically the estimate you found in part b . d What can you conclude in general about the relationship between the diameter of the copper wire and the resistance EXPLORATION A model for the life expectancy during this period is y 0.0025t 2 0.574t 44.25, 20 t 100 where y represents the life expectancy and t is the time in years, with t 20 corresponding to 1920. a Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data Explain. b Determine the life expectancy in 1990 both graphically and algebraically. c Use the graph to determine the year when life expectancy was approximately 76.0. Verify your answer algebraically. d One projection for the life expectancy of a child born in 2015 is 78.9. How does this compare with the projection given by the model 81. THINK ABOUT IT Find a and b if the graph of y ax 2 bx 3 is symmetric with respect to a the y axis and b the origin. There are many correct answers. 82. CAPSTONE Match the equation or equations with the given characteristic. y 3x3 3x ii y x 32 3 x iii y 3x 3 iv y v y 3x2 3 vi y x 3 i a b c d Symmetric with respect to the y axis Three x intercepts Symmetric with respect to the x axis 2, 1 is a point on the graph e Symmetric with respect to the origin f Graph passes through the origin www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.2 Linear Equations in One Variable 87 1.2 LINEAR EQUATIONS IN ONE VARIABLE What you should learn Identify different types of equations. Solve linear equations in one variable. Solve equations that lead to linear equations. Find x and y intercepts of graphs of equations algebraically. Use linear equations to model and solve real life problems. Why you should learn it Linear equations are used in many real life applications. For example, in Exercise 110 on page 95, linear equations can be used to model the number of women in the civilian work force over time. Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. For example 3x 5 7, x 2 x 6 0, and 2x 4 are equations. To solve an equation in x means to find all values of x for which the equation is true. Such values are solutions. For instance, x 4 is a solution of the equation 3x 5 7 because 34 5 7 is a true statement. The solutions of an equation depend on the kinds of numbers being considered. For instance, in the set of rational numbers, x 2 10 has no solution because there is no rational number whose square is 10. However, in the set of real numbers, the equation has the two solutions x 10 and x 10. An equation that is true for every real number in the domain of the variable is called an identity. For example x2 9 x 3x 3 Identity is an identity because it is a true statement for any real value of x. The equation Andrew DouglasMasterfile x 1 3x2 3x Identity where x 0, is an identity because it is true for any nonzero real value of x. An equation that is true for just some or even none of the real numbers in the domain of the variable is called a conditional equation. For example, the equation x2 9 0 Conditional equation is conditional because x 3 and x 3 are the only values in the domain that satisfy the equation. The equation 2x 4 2x 1 is conditional because there are no real values of x for which the equation is true. Learning to solve conditional equations is the primary focus of this chapter. Linear Equations in One Variable Definition of Linear Equation A linear equation in one variable x is an equation that can be written in the standard form ax b 0 where a and b are real numbers with a www.elsolucionario.net 0. http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 88 Chapter 1 Equations, Inequalities, and Mathematical Modeling HISTORICAL NOTE A linear equation has exactly one solution. To see this, consider the following steps. Remember that a 0. ax b 0 Write original equation. ax b British Museum x This ancient Egyptian papyrus, discovered in 1858, contains one of the earliest examples of mathematical writing in existence. The papyrus itself dates back to around 1650 B.C., but it is actually a copy of writings from two centuries earlier. The algebraic equations on the papyrus were written in words. Diophantus, a Greek who lived around A.D. 250, is often called the Father of Algebra. He was the first to use abbreviated word forms in equations. b a Subtract b from each side. Divide each side by a. To solve a conditional equation in x, isolate x on one side of the equation by a sequence of equivalent and usually simpler equations, each having the same solution s as the original equation. The operations that yield equivalent equations come from the Substitution Principle and the Properties of Equality studied in Chapter P. Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following steps. Given Equation 2x x 4 Equivalent Equation x4 2. Add or subtract the same quantity to from each side of the equation. x16 x5 3. Multiply or divide each side of the equation by the same nonzero quantity. 2x 6 x3 4. Interchange the two sides of the equation. 2x x2 1. Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation. Example 1 Solving a Linear Equation a. 3x 6 0 Original equation 3x 6 Add 6 to each side. x2 Divide each side by 3. b. 5x 4 3x 8 2x 4 8 2x 12 x 6 Original equation Subtract 3x from each side. Subtract 4 from each side. Divide each side by 2. Now try Exercise 33. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.2 Linear Equations in One Variable 89 After solving an equation, you should check each solution in the original equation. For instance, you can check the solution of Example 1 a as follows. 3x 6 0 32 6 0 Write original equation. Substitute 2 for x. 00 Solution checks. Try checking the solution of Example 1 b . Some equations have no solutions because all the x terms sum to zero and a contradictory false statement such as 0 5 or 12 7 is obtained. For instance, the equation xx1 has no solution. Watch for this type of equation in the exercises. Example 2 T E C H N O LO G Y Solve You can use a graphing utility to check that a solution is reasonable. One way to do this is to graph the left side of the equation, then graph the right side of the equation, and determine the point of intersection. For instance, in Example 2, if you graph the equations y1 6x 1 4 The left side y2 37x 1 The right side in the same viewing window, they should intersect at x 13, as shown in the graph below. 1 13, 4 6 6x 1 4 37x 1. Solution 6x 1 4 37x 1 6x 6 4 21x 3 6x 2 21x 3 15x 2 3 Distributive Property Simplify. Subtract 21x from each side. 15x 5 x Write original equation. Add 2 to each side. 1 3 Divide each side by 15. Check Check this solution by substituting 13 for x in the original equation. 0 2 Solving a Linear Equation 6x 1 4 37x 1 6 13 1 4 37 13 1 6 43 4 3 73 1 6 43 4 3 43 12 24 3 43 8 4 4 4 4 Write original equation. 1 Substitute 3 for x. Simplify. Simplify. Multiply. Simplify. Solution checks. 13. So, the solution is x Note that if you subtracted 6x from each side of the equation and then subtracted 3 from each side of the equation, you would still obtain the solution x 13. Now try Exercise 39. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 90 Chapter 1 Equations, Inequalities, and Mathematical Modeling Equations That Lead to Linear Equations An equation with a single fraction on each side can be cleared of denominators by cross multiplying. To do this, multiply the left numerator by the right denominator and the right numerator by the left denominator as follows. a c b d ad cb To solve an equation involving fractional expressions, find the least common denominator LCD of all terms and multiply every term by the LCD. This process will clear the original equation of fractions and produce a simpler equation to work with. Example 3 Solve 3x x 2. 3 4 Solution x 3x 2 3 4 Original equation Cross multiply. An Equation Involving Fractional Expressions Write original equation. x 3x 12 12 122 3 4 Multiply each term by the LCD of 12. 4x 9x 24 Divide out and multiply. 13x 24 x Combine like terms. 24 13 Divide each side by 13. The solution is x 24 13 . Check this in the original equation. Now try Exercise 43. When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous solution is one that does not satisfy the original equation. Therefore, it is essential that you check your solutions. Example 4 Solve An Equation with an Extraneous Solution 1 3 6x . x 2 x 2 x2 4 Solution The LCD is x 2 4, or x 2x 2. Multiply each term by this LCD. Recall that the least common denominator of two or more fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. For instance, in Example 4, by factoring each denominator you can determine that the LCD is x 2x 2. 1 3 6x x 2x 2 x 2x 2 2 x 2x 2 x2 x2 x 4 x 2 3x 2 6x, x 2 x 2 3x 6 6x x 2 3x 6 4x 8 x 2 Extraneous solution In the original equation, x 2 yields a denominator of zero. So, x 2 is an extraneous solution, and the original equation has no solution. Now try Exercise 63. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.2 Linear Equations in One Variable 91 Finding Intercepts Algebraically In Section 1.1, you learned to find x and y intercepts using a graphical approach. Because all the points on the x axis have a y coordinate equal to zero, and all the points on the y axis have an x coordinate equal to zero, you can use an algebraic approach to find x and y intercepts, as follows. Finding Intercepts Algebraically 1. To find x intercepts, set y equal to zero and solve the equation for x. 2. To find y intercepts, set x equal to zero and solve the equation for y. Here is an example. 1 y 4x 1 0 4x 1 1 4x 4 x y 4x 1 y 40 1 y 1 1 So, the x intercept of y 4x 1 is 4, 0 and the y intercept is 0, 1. Female Participants in High School Athletics Number of female participants in millions y 3.5 Example 5 3.0 2.5 2.0 y 0.042t 2.73, 1.0 0.5 t 0 1 2 3 4 5 6 7 8 Year 0 2000 FIGURE 1.15 Female Participants in Athletic Programs The number y in millions of female participants in high school athletic programs in the United States from 1999 through 2008 can be approximated by the linear model y = 0.042t + 2.73 1.5 1 Application 1 t 8 where t 0 represents 2000. a Find algebraically the y intercept of the graph of the linear model shown in Figure 1.15. b Assuming that this linear pattern continues, find the year in which there will be 3.36 million female participants. Source: National Federation of State High School Associations Solution a. To find the y intercept, let t 0 and solve for y, as follows. y 0.042t 2.73 Write original equation. 0.0420 2.73 Substitute 0 for t. 2.73 Simplify. So, the y intercept is 0, 2.73. b. Let y 3.36 and solve the equation 3.36 0.042t 2.73 for t. 3.36 0.042t 2.73 Write original equation. 0.63 0.042t Subtract 2.73 from each side. 15 t Divide each side by 0.042. Because t 0 represents 2000, t 15 must represent 2015. So, from this model, there will be 3.36 million female participants in 2015. Now try Exercise 109. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 92 Chapter 1 1.2 Equations, Inequalities, and Mathematical Modeling EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. 5. An ________ is a statement that equates two algebraic expressions. To find all values that satisfy an equation is to ________ the equation. There are two types of equations, ________ and ________ equations. A linear equation in one variable is an equation that can be written in the standard form ________. When solving an equation, it is possible to introduce an ________ solution, which is a value that does not satisfy the original equation. 6. To solve a conditional equation, isolate the variable on one side using transformations that produce ________ ________. SKILLS AND APPLICATIONS In Exercises 718, determine whether each value of x is a solution of the equation. Equation 7. 5x 3 3x 5 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. a c 7 3x 5x 17 a c 3x 2 2x 5 a 2 2x 2 c 5x3 2x 3 a 3 4x 2x 11 c 5 4 a 3 2x x c a 6x 19 x 2 7 14 c a 1 4 3 x2 c x 5x 3 a 24 2 c 3x 2 4 a c 3 x 8 3 a c 2 6x 11x 35 0 a c 2 10x 21x 10 0 a c Values x0 b x4 d x 3 b x8 d x 3 b x4 d x2 b x0 d x 12 b x0 d x 2 b x 12 d x 1 b x0 d x 3 b x7 d x3 b x9 d x2 b x 35 d 5 x 3 b 7 d x2 2 x5 b x 13 d x 5 x 10 x0 x3 x1 x 5 x 2 x 10 x4 x 14 x1 x7 x 2 x5 x 2 x9 x2 x 6 x 5 x8 x 27 5 x3 x 52 x 2 In Exercises 1930, determine whether the equation is an identity or a conditional equation. 19. 2x 1 2x 2 20. 3x 2 5x 4 21. 6x 3 5 2x 10 22. 3x 2 5 3x 1 23. 4x 1 2x 2x 2 24. 7x 3 4x 37 x 25. x 2 8x 5 x 42 11 26. x 2 23x 2 x 2 6x 4 1 4x 5 3 27. 3 28. 24 x1 x1 x x 29. 2x 1 2x 1 30. 14x 4 14 x 4 In Exercises 31 and 32, justify each step of the solution. 4x 32 83 4x 32 32 83 32 4x 51 4x 51 4 4 51 x 4 32. 3x 4 10 7 3x 12 10 7 3x 2 7 3x 2 2 7 2 3x 9 3x 9 3 3 x3 31. In Exercises 3348, solve the equation and check your solution. 33. 35. 37. 39. x 11 15 7 2x 25 3x 5 2x 7 4y 2 5y 7 6y www.elsolucionario.net 34. 36. 38. 40. 7 x 19 7x 2 23 5x 3 6 2x 5y 1 8y 5 6y http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.2 41. x 32x 3 8 5x 42. 9x 10 5x 22x 5 5x 1 1 x x 3x 43. x 44. 3 4 2 2 5 2 10 45. 32z 5 14z 24 0 3x 1 46. x 2 10 2 4 47. 0.25x 0.7510 x 3 48. 0.60x 0.40100 x 50 In Exercises 4952, solve the equation using two different methods. Then explain which method is easier. 49. 3x 4x 4 8 3 50. 3z z 6 8 10 51. 2x 4 5x 5 3 52. 4y 16 2y 3 5 In Exercises 5374, solve the equation and check your solution. If not possible, explain why. 53. x 8 2x 2 x 54. 8x 2 32x 1 2x 5 100 4x 5x 6 55. 6 3 4 17 y 32 y 56. 100 y y 5x 4 2 10x 3 1 57. 58. 5x 4 3 5x 6 2 13 5 15 6 59. 10 4 60. 4 3 x x x x 2 1 2 61. 3 2 62. 0 z2 x x5 x 4 63. 20 x4 x4 7 8x 64. 4 2x 1 2x 1 2 1 2 65. x 4x 2 x 4 x 2 4 6 15 66. x 1 3x 1 3x 1 1 1 10 67. x 3 x 3 x2 9 1 3 4 x 2 x 3 x2 x 6 3 4 1 6 2 3x 5 69. 2 70. 2 x 3x x x3 x x3 x 3x 68. 71. 72. 73. 74. Linear Equations in One Variable 93 x 22 5 x 32 4x 1 3x x 5 x 22 x 2 4x 1 2x 12 4x2 x 6 GRAPHICAL ANALYSIS In Exercises 7580, use a graphing utility to graph the equation and approximate any x intercepts. Set y 0 and solve the resulting equation. Compare the results with the graphs x intercept s . 75. y 2x 1 4 77. y 20 3x 10 79. y 38 59 x 76. y 43x 2 78. y 10 2x 2 80. y 6x 616 11 x In Exercises 8190, find the x and y intercepts of the graph of the equation algebraically. 81. y 12 5x 83. y 32x 1 85. 2x 3y 10 87. 82. y 16 3x 84. y 5 6 x 86. 4x 5y 12 2x 8 3y 0 5 88. 89. 4y 0.75x 1.2 0 8x 5 2y 0 3 90. 3y 2.5x 3.4 0 91. A student states that the solution of the equation 2 5 1 xx 2 x x2 is x 2. Describe and correct the students error. 92. A student states that the equation 3x 2 3x 6 is an identity. Describe and correct the students error. In Exercises 9396, solve the equation for x. Round your solution to three decimal places. 93. 0.275x 0.725500 x 300 94. 2.763 4.52.1x 5.1432 6.32x 5 2 4.405 1 3 6 95. 96. 18 7.398 x x 6.350 x In Exercises 97104, solve for x. 97. 98. 99. 101. 102. 103. 4x 1 ax x 5 4 2x 2b ax 3 100. 5 ax 12 bx 6x ax 2x 5 1 19x 2 ax x 9 53x 6b 12 8 3ax 2ax 6x 3 4x 1 4 104. 5x ax 225 x 1 10 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 94 Chapter 1 Equations, Inequalities, and Mathematical Modeling 105. GEOMETRY The surface area S of the circular cylinder shown in the figure is S 2 25 2 5h. c Complete the table to determine if there is a height of an adult for which an anthropologist would not be able to determine whether the femur belonged to a male or a female. Find the height h of the cylinder if the surface area is 471 square feet. Use 3.14 for . Female femur length, y Height, x 5 ft 60 70 80 90 100 110 h ft 106. GEOMETRY The surface area S of the rectangular solid in the figure is S 224 24x 26x. Find the length x of the box if the surface area is 248 square centimeters. 4 cm x 6 cm 107. ANTHROPOLOGY The relationship between the length of an adults femur thigh bone and the height of the adult can be approximated by the linear equations d Solve part c algebraically by setting the two equations equal to each other and solving for x. Compare your solutions. Do you believe an anthropologist would ever have the problem of not being able to determine whether a femur belonged to a male or a female Why or why not 108. TAX CREDITS Use the following information about a possible tax credit for a family consisting of two adults and two children see figure . Earned income: E Subsidy a grant of money : 0 E 20,000 Female S 10,000 12 E, y 0.449x 12.15 Male Total income: T E S where y is the length of the femur in inches and x is the height of the adult in inches see figure . x in. Thousands of dollars y 0.432x 10.44 y in. Male femur length, y Total income T Subsidy S 18 14 10 6 2 E 0 femur 2 4 6 8 10 12 14 16 18 20 Earned income in thousands of dollars a An anthropologist discovers a femur belonging to an adult human female. The bone is 16 inches long. Estimate the height of the female. b From the foot bones of an adult human male, an anthropologist estimates that the persons height was 69 inches. A few feet away from the site where the foot bones were discovered, the anthropologist discovers a male adult femur that is 19 inches long. Is it likely that both the foot bones and the thigh bone came from the same person a Write the total income T in terms of E. b Find the earned income E if the subsidy is 6600. c Find the earned income E if the total income is 13,800. d Find the subsidy S if the total income is 12,500. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.2 109. NEWSPAPERS The number of newspapers y in the United States from 1996 through 2007 can be approximated by the model y 7.69t 1480.7, 4 t 7, where t represents the year, with t 0 corresponding to 2000. Source: Editor Publisher Co. a Sketch a graph of the model. Graphically estimate the y intercept of the graph. b Find the y intercept of the graph algebraically. c Assuming this linear pattern continues, find the year in which the number of newspapers will be 1373. Does your answer seem reasonable Explain. 110. LABOR STATISTICS The number of women y in millions in the civilian work force in the United States from 2000 through 2007 see figure can be approximated by the model y 0.66t 66.1, 0 t 7, where t represents the year, with t 0 corresponding to 2000. Source: U.S. Bureau of Labor Statistics Number of women in millions y 80 70 60 50 40 30 20 10 95 Linear Equations in One Variable 115. The equation 2x 3 1 2x 5 has no solution. 116. The equation 3x 1 2 3x 6 is an identity and therefore has all real number solutions. 1 3 has no solution x2 x2 because x 2 is an extraneous solution. 117. The equation 2 118. THINK ABOUT IT What is meant by equivalent equations Give an example of two equivalent equations. 119. THINK ABOUT IT a Complete the table. 1 x 0 1 2 3 4 3.2x 5.8 b Use the table in part a to determine the interval in which the solution of the equation 3.2x 5.8 0 is located. Explain your reasoning. c Complete the table. x 1.5 1.6 1.7 1.8 1.9 2.0 3.2x 5.8 t 0 1 2 3 4 5 6 7 Year 0 2000 a According to this model, during which year did the number reach 70 million b Explain how you can solve part a graphically and algebraically. 111. OPERATING COST A delivery company has a fleet of vans. The annual operating cost C per van is C 0.32m 2500, where m is the number of miles traveled by a van in a year. What number of miles will yield an annual operating cost of 10,000 112. FLOOD CONTROL A river has risen 8 feet above its flood stage. The water begins to recede at a rate of 3 inches per hour. Write a mathematical model that shows the number of feet above flood stage after t hours. If the water continually recedes at this rate, when will the river be 1 foot above its flood stage EXPLORATION TRUE OR FALSE In Exercises 113117, determine whether the statement is true or false. Justify your answer. 113. The equation x3 x 10 is a linear equation. 114. The equation x 2 9x 5 4 x 3 has no real solution. d Use the table in part c to determine the interval in which the solution of the equation 3.2x 5.8 0 is located. Explain how this process can be used to approximate the solution to any desired degree of accuracy. 120. Use the procedure in Exercise 119 to approximate the solution of the equation 0.3x 1.5 2 0, accurate to two decimal places. 121. GRAPHICAL REASONING a Use a graphing utility to graph the equation y 3x 6. b Use the result of part a to estimate the x intercept of the graph. c Explain how the x intercept is related to the solution of the equation 3x 6 0, as shown in Example 1 a . 122. CAPSTONE a Explain the difference between a conditional equation and an identity. b Describe the steps used to transform an equation into an equivalent equation. c What is meant by an equation having an extraneous solution www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 96 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.3 MODELING WITH LINEAR EQUATIONS What you should learn Use a verbal model in a problemsolving plan. Write and use mathematical models to solve real life problems. Solve mixture problems. Use common formulas to solve real life problems. Why you should learn it You can use linear equations to find the percent changes in the prices of various items or services. See Exercises 5356 on page 104. Introduction to Problem Solving In this section, you will learn how algebra can be used to solve problems that occur in real life situations. The process of translating phrases or sentences into algebraic expressions or equations is called mathematical modeling. A good approach to mathematical modeling is to use two stages. Begin by using the verbal description of the problem to form a verbal model. Then, after assigning labels to the quantities in the verbal model, form a mathematical model or algebraic equation. Verbal Description Verbal Model Algebraic Equation When you are constructing a verbal model, it is helpful to look for a hidden equality. For instance, in the following example the hidden equality equates your annual income to 24 paychecks and one bonus check. Example 1 Using a Verbal Model You have accepted a job for which your annual salary will be 32,300. This salary includes a year end bonus of 500. You will be paid twice a month. What will your gross pay pay before taxes be for each paycheck Solution Tony Freeman PhotoEdit Because there are 12 months in a year and you will be paid twice a month, it follows that you will receive 24 paychecks during the year. Verbal Model: Labels: Income for year 24 paychecks Bonus Income for year 32,300 Amount of each paycheck x Bonus 500 dollars dollars dollars Equation: 32,300 24x 500 The algebraic equation for this problem is a linear equation in the variable x, which you can solve as follows. 32,300 24x 500 32,300 500 24x 500 500 Write original equation. Subtract 500 from each side. 31,800 24x Simplify. 31,800 24x 24 24 Divide each side by 24. 1325 x Simplify. So, your gross pay for each paycheck will be 1325. Now try Exercise 37. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.3 Modeling with Linear Equations 97 A fundamental step in writing a mathematical model to represent a real life problem is translating key words and phrases into algebraic expressions and equations. The following list gives several examples. Translating Key Words and Phrases Key Words and Phrases Verbal Description Algebraic Expression or Equation Equality: Equals, equal to, is, are, was, will be, represents The sale price S is 10 less than the list price L. S L 10 Addition: Sum, plus, greater than, increased by, more than, exceeds, total of The sum of 5 and x Seven more than y Subtraction: Difference, minus, less than, decreased by, subtracted from, reduced by, the remainder The difference of 4 and b Three less than z 4b z3 Multiplication: Product, multiplied by, twice, times, percent of Two times x Three percent of t 2x 0.03t Division: Quotient, divided by, ratio, per The ratio of x to 8 x 8 5 x or x 5 7 y or y 7 Using Mathematical Models Example 2 Finding the Percent of a Raise You have accepted a job that pays 10 an hour. You are told that after a two month probationary period, your hourly wage will be increased to 11 an hour. What percent raise will you receive after the two month period Solution Verbal Model: Labels: Equation: Raise Percent Old wage Old wage 10 New wage 11 Raise 11 10 1 Percent r dollars per hour dollars per hour dollars per hour percent in decimal form 1 r 10 1 10 r 0.1 r Divide each side by 10. Rewrite fraction as a decimal. You will receive a raise of 0.1 or 10. Now try Exercise 49. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 98 Chapter 1 Equations, Inequalities, and Mathematical Modeling Example 3 Writing the unit for each label in a real life problem helps you determine the unit for the answer. This is called unit analysis. When the same unit of measure occurs in the numerator and denominator of an expression, you can divide out the unit. For instance, unit analysis verifies that the unit for time in the formula below is hours. Time distance rate miles miles hour miles hours Finding the Percent of Monthly Expenses Your family has an annual income of 57,000 and the following monthly expenses: mortgage 1100 , car payment 375 , food 295 , utilities 240 , and credit cards 220 . The total value of the monthly expenses represents what percent of your familys annual income Solution The total amount of your familys monthly expenses is 2230. The total monthly expenses for 1 year are 26,760. Verbal Model: Monthly expenses Percent Labels: Income 57,000 Monthly expenses 26,760 Percent r Income dollars dollars in decimal form Equation: 26,760 r 57,000 26,760 r 57,000 hours miles 0.469 r Divide each side by 57,000. Use a calculator. Your familys monthly expenses are approximately 0.469 or 46.9 of your familys annual income. Now try Exercise 51. Example 4 Finding the Dimensions of a Room A rectangular kitchen is twice as long as it is wide, and its perimeter is 84 feet. Find the dimensions of the kitchen. Solution For this problem, it helps to sketch a diagram, as shown in Figure 1.16. w l FIGURE 1.16 Verbal Model: 2 Length 2 Width Perimeter Labels: Perimeter 84 Width w Length l 2w feet feet feet Equation: 22w 2w 84 6w 84 w 14 Group like terms. Divide each side by 6. Because the length is twice the width, you have l 2w Length is twice width. 214 28. Substitute and simplify. So, the dimensions of the room are 14 feet by 28 feet. Now try Exercise 57. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.3 Example 5 Modeling with Linear Equations 99 A Distance Problem A plane is flying nonstop from Atlanta to Portland, a distance of about 2700 miles, as shown in Figure 1.17. After 1.5 hours in the air, the plane flies over Kansas City a distance of 820 miles from Atlanta . Estimate the time it will take the plane to fly from Atlanta to Portland. Solution Portland Kansas City Atlanta Verbal Model: Labels: Distance Rate 1.17 Time Distance 2700 Time t distance to Kansas City 820 Rate time to Kansas City 1.5 Equation: 2700 FIGURE miles hours miles per hour 820 t 1.5 4050 820t 4050 t 820 4.94 t The trip will take about 4.94 hours, or about 4 hours and 56 minutes. Now try Exercise 61. Example 6 An Application Involving Similar Triangles To determine the height of the Aon Center Building in Chicago , you measure the shadow cast by the building and find it to be 142 feet long, as shown in Figure 1.18. Then you measure the shadow cast by a four foot post and find it to be 6 inches long. Estimate the buildings height. Solution To solve this problem, you use a result from geometry that states that the ratios of corresponding sides of similar triangles are equal. Verbal Model: Labels: x ft 48 in. Equation: 142 ft 6 in. Not drawn to scale FIGURE 1.18 Height of building Length of buildings shadow Height of post Length of posts shadow Height of building x Length of buildings shadow 142 Height of post 4 feet 48 inches Length of posts shadow 6 feet feet inches inches x 48 142 6 x 1136 So, the Aon Center Building is about 1136 feet high. Now try Exercise 67. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 100 Chapter 1 Equations, Inequalities, and Mathematical Modeling Mixture Problems Problems that involve two or more rates are called mixture problems. They are not limited to mixtures of chemical solutions, as shown in Examples 7 and 8. Example 7 A Simple Interest Problem You invested a total of 10,000 at 412 and 512 simple interest. During 1 year, the two accounts earned 508.75. How much did you invest in each account Example 7 uses the simple interest formula I Prt, where I is the interest, P is the principal original deposit , r is the annual interest rate in decimal form , and t is the time in years. Notice that in this example the amount invested, 10,000, is separated into two parts, x and 10,000 x. Solution Verbal Model: Labels: 1 1 Interest from 42 Interest from 52 Total interest 1 Amount invested at 42 x dollars Amount invested at 512 10,000 x dollars Interest from Interest from 412 Prt x0.0451 1 52 Prt 10,000 x0.0551 dollars dollars Total interest 508.75 dollars Equation: 0.045x 0.05510,000 x 508.75 0.01x 41.25 So, 4125 was invested at 412 x 4125 and 10,000 x or 5875 was invested at 512. Now try Exercise 71. Example 8 An Inventory Problem A store has 30,000 of inventory in single disc DVD players and multi disc DVD players. The profit on a single disc player is 22 and the profit on a multi disc player is 40. The profit for the entire stock is 35. How much was invested in each type of DVD player Solution Verbal Model: Labels: Profit from Profit from Total single disc players multi disc players profit Inventory of single disc players x Inventory of multi disc players 30,000 x Profit from single disc players 0.22x Profit from multi disc players 0.4030,000 x Total profit 0.3530,000 10,500 dollars dollars dollars dollars dollars Equation: 0.22x 0.4030,000 x 10,500 0.18x 1500 x 8333.33 So, 8333.33 was invested in single disc DVD players and 30,000 x or 21,666.67 was invested in multi disc DVD players. Now try Exercise 73. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.3 Modeling with Linear Equations 101 Common Formulas A literal equation is an equation that contains more than one variable. A formula is an example of a literal equation. Many common types of geometric, scientific, and investment problems use ready made equations called formulas. Knowing these formulas will help you translate and solve a wide variety of real life applications. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Square Rectangle Circle A s2 A lw A r2 P 4s P 2l 2w C 2 r Triangle 1 A bh 2 Pabc w r s a c h l s b Cube Rectangular Solid V s3 Circular Cylinder 4 V r3 3 V r 2h V lwh h Sphere . r s l w h r s s Miscellaneous Common Formulas Temperature: 9 F C 32 5 F degrees Fahrenheit, C degrees Celsius 5 C F 32 9 Simple Interest: I Prt I interest, P principal original deposit , r annual interest rate in decimal form , t time in years Compound Interest: AP 1 r n nt n compoundings number of times interest is calculated per year, t time in years, A balance, P principal original deposit , r annual interest rate in decimal form Distance: d rt d distance traveled, r rate, t time www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 102 Chapter 1 Equations, Inequalities, and Mathematical Modeling When working with applied problems, you often need to rewrite a literal equation in terms of another variable. You can use the methods for solving linear equations to solve some literal equations for a specified variable. For instance, the formula for the perimeter of a rectangle, P 2l 2w, can be rewritten or solved for w as 1 w 2P 2l . Example 9 A cylindrical can has a volume of 200 cubic centimeters cm3 and a radius of 4 centimeters cm , as shown in Figure 1.19. Find the height of the can. 4 cm h FIGURE 1.19 Using a Formula Solution The formula for the volume of a cylinder is V r 2h. To find the height of the can, solve for h. h V r2 Then, using V 200 and r 4, find the height. h 200 4 2 Substitute 200 for V and 4 for r. 200 16 Simplify denominator. 3.98 Use a calculator. You can use unit analysis to check that your answer is reasonable. 200 cm3 3.98 cm 16 cm2 Now try Exercise 95. CLASSROOM DISCUSSION Translating Algebraic Formulas Most people use algebraic formulas every day sometimes without realizing it because they use a verbal form or think of an oftenrepeated calculation in steps. Translate each of the following verbal descriptions into an algebraic formula, and demonstrate the use of each formula. a. Designing Billboards The letters on a sign or billboard are designed to be readable at a certain distance. Take half the letter height in inches and multiply by 100 to find the readable distance in feet.Thos. Hodgson, Hodgson Signs Source: Rules of Thumb by Tom Parker b. Percent of Calories from Fat To calculate percent of calories from fat, multiply grams of total fat per serving by 9, divide by the number of calories per serving, and then multiply by 100. Source: Good Housekeeping c. Building Stairs A set of steps will be comfortable to use if two times the height of one riser plus the width of one tread is equal to 26 inches. Alice Lukens Bachelder, gardener Source: Rules of Thumb by Tom Parker www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.3 1.3 EXERCISES 103 Modeling with Linear Equations See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 1 and 2, fill in the blanks. 1. The process of translating phrases or sentences into algebraic expressions or equations is called ________ ________. 2. A good approach to mathematical modeling is a two stage approach, using a verbal description to form a ________ ________, and then, after assigning labels to the quantities, forming an ________ ________. In Exercises 38, write the formula for the given quantity. 3. 5. 7. 8. Area of a circle: ________ 4. Perimeter of a rectangle: ________ Volume of a cube: ________ 6. Volume of a circular cylinder: ________ Balance if P dollars is invested at r compounded monthly for t years: ________ Simple interest if P dollars is invested at r for t years: ________ SKILLS AND APPLICATIONS In Exercises 918, write a verbal description of the algebraic expression without using the variable. 9. x 4 u 11. 5 y4 13. 5 15. 3b 2 17. 4 p 1 p 10. t 10 2 12. x 3 z 10 14. 7 16. 12xx 5 q 43 q 18. 2q 30. The total revenue obtained by selling x units at 12.99 per unit In Exercises 3134, translate the statement into an algebraic expression or equation. 31. Thirty percent of the list price L 32. The amount of water in q quarts of a liquid that is 28 water 33. The percent of 672 that is represented by the number N 34. The percent change in sales from one month to the next if the monthly sales are S1 and S2, respectively In Exercises 1930, write an algebraic expression for the verbal description. In Exercises 35 and 36, write an expression for the area of the region in the figure. 19. The sum of two consecutive natural numbers 20. The product of two consecutive natural numbers 35. 21. The product of two consecutive odd integers, the first of which is 2n 1 22. The sum of the squares of two consecutive even integers, the first of which is 2n 23. The distance traveled in t hours by a car traveling at 55 miles per hour 24. The travel time for a plane traveling at a rate of r kilometers per hour for 900 kilometers 25. The amount of acid in x liters of a 20 acid solution 26. The sale price of an item that is discounted 33 of its list price L 27. The perimeter of a rectangle with a width x and a length that is twice the width 28. The area of a triangle with base 16 inches and height h inches 29. The total cost of producing x units for which the fixed costs are 2500 and the cost per unit is 40 36. 4 2 b 3 x 2x 4 +1 b x 8 NUMBER PROBLEMS In Exercises 3742, write a mathematical model for the problem and solve. 37. The sum of two consecutive natural numbers is 525. Find the numbers. 38. The sum of three consecutive natural numbers is 804. Find the numbers. 39. One positive number is 5 times another number. The difference between the two numbers is 148. Find the numbers. 40. One positive number is 15 of another number. The difference between the two numbers is 76. Find the numbers. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 104 Chapter 1 Equations, Inequalities, and Mathematical Modeling 41. Find two consecutive integers whose product is 5 less than the square of the smaller number. 42. Find two consecutive natural numbers such that the difference of their reciprocals is 14 the reciprocal of the smaller number. In Exercises 4348, solve the percent equation. 43. 45. 46. 47. 48. What is 30 of 45 44. What is 175 of 360 432 is what percent of 1600 459 is what percent of 340 12 is 12 of what number 70 is 40 of what number 49. FINANCE A salespersons weekly paycheck is 15 less than a second salespersons paycheck. The two paychecks total 1125. Find the amount of each paycheck. 50. DISCOUNT The price of a swimming pool has been discounted 16.5. The sale price is 1210.75. Find the original list price of the pool. 51. FINANCE A family has annual loan payments equaling 32 of their annual income. During the year, their loan payments total 15,125.50. What is their annual income 52. FINANCE A family has a monthly mortgage payment of 500, which is 16 of their monthly income. What is their monthly income In Exercises 5356, the prices of various items are given for 2000 and 2007. Find the percent change for each item. Sources: U.S. Energy Information Association, SNL Kagan, U.S. Bureau of Labor Statistics, CTIA The Wireless Association Item 53. Gallon of regular unleaded gasoline 2000 1.51 2007 2.80 54. Monthly cable rate 55. Pound of 100 ground beef 56. Monthly bill for cellular phone service 30.37 1.63 45.27 42.72 2.23 49.79 57. DIMENSIONS OF A ROOM A room is 1.5 times as long as it is wide, and its perimeter is 25 meters. a Draw a diagram that represents the problem. Identify the length as l and the width as w. b Write l in terms of w and write an equation for the perimeter in terms of w. c Find the dimensions of the room. 58. DIMENSIONS OF A PICTURE FRAME A picture frame has a total perimeter of 3 meters. The height of 2 the frame is 3 times its width. a Draw a diagram that represents the problem. Identify the width as w and the height as h. b Write h in terms of w and write an equation for the perimeter in terms of w. c Find the dimensions of the picture frame. 59. COURSE GRADE To get an A in a course, you must have an average of at least 90 on four tests of 100 points each. The scores on your first three tests were 87, 92, and 84. What must you score on the fourth test to get an A for the course 60. COURSE GRADE You are taking a course that has four tests. The first three tests are 100 points each and the fourth test is 200 points. To get an A in the course, you must have an average of at least 90 on the four tests. Your scores on the first three tests were 87, 92, and 84. What must you score on the fourth test to get an A for the course 61. TRAVEL TIME You are driving on a Canadian freeway to a town that is 500 kilometers from your home. After 30 minutes you pass a freeway exit that you know is 50 kilometers from your home. Assuming that you continue at the same constant speed, how long will it take for the entire trip 62. TRAVEL TIME Students are traveling in two cars to a football game 135 miles away. The first car leaves on time and travels at an average speed of 45 miles per hour. 1 The second car starts 2 hour later and travels at an average speed of 55 miles per hour. How long will it take the second car to catch up to the first car Will the second car catch up to the first car before the first car arrives at the game 63. AVERAGE SPEED A truck driver traveled at an average speed of 55 miles per hour on a 200 mile trip to pick up a load of freight. On the return trip with the truck fully loaded , the average speed was 40 miles per hour. What was the average speed for the round trip 64. WIND SPEED An executive flew in the corporate jet to a meeting in a city 1500 kilometers away. After traveling the same amount of time on the return flight, the pilot mentioned that they still had 300 kilometers to go. The air speed of the plane was 600 kilometers per hour. How fast was the wind blowing Assume that the wind direction was parallel to the flight path and constant all day. 65. PHYSICS Light travels at the speed of approximately 3.0 108 meters per second. Find the time in minutes required for light to travel from the sun to Earth an approximate distance of 1.5 1011 meters . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.3 66. RADIO WAVES Radio waves travel at the same speed as light, approximately 3.0 108 meters per second. Find the time required for a radio wave to travel from Mission Control in Houston to NASA astronauts on the surface of the moon 3.84 108 meters away. 67. HEIGHT OF A BUILDING To obtain the height of the Chrysler Building in New York, you measure the buildings shadow and find that it is 87 feet long. You also measure the shadow of a four foot stake and find that it is 4 inches long. How tall is the Chrysler Building 68. HEIGHT OF A TREE To obtain the height of a tree see figure , you measure the trees shadow and find that it is 8 meters long. You also measure the shadow of a two meter lamppost and find that it is 75 centimeters long. How tall is the tree Modeling with Linear Equations 73. INVENTORY A nursery has 40,000 of inventory in dogwood trees and red maple trees. The profit on a dogwood tree is 25 and the profit on a red maple tree is 17. The profit for the entire stock is 20. How much was invested in each type of tree 74. INVENTORY An automobile dealer has 600,000 of inventory in minivans and alternative fueled vehicles. The profit on a minivan is 24 and the profit on an alternative fueled vehicle is 28. The profit for the entire stock is 25. How much was invested in each type of vehicle 75. MIXTURE PROBLEM Using the values in the table, determine the amounts of solutions 1 and 2 needed to obtain the specified amount and concentration of the final mixture. Concentration 2m 8m 75 cm 105 a b c d Solution 1 Solution 2 Final solution Amount of final solution 10 25 15 70 30 50 45 90 25 30 30 75 100 gal 5L 10 qt 25 gal Not drawn to scale 69. FLAGPOLE HEIGHT A person who is 6 feet tall walks away from a flagpole toward the tip of the shadow of the flagpole. When the person is 30 feet from the flagpole, the tips of the persons shadow and the shadow cast by the flagpole coincide at a point 5 feet in front of the person. a Draw a diagram that gives a visual representation of the problem. Let h represent the height of the flagpole. b Find the height of the flagpole. 70. SHADOW LENGTH A person who is 6 feet tall walks away from a 50 foot tower toward the tip of the towers shadow. At a distance of 32 feet from the tower, the persons shadow begins to emerge beyond the towers shadow. How much farther must the person walk to be completely out of the towers shadow 71. INVESTMENT You plan to invest 12,000 in two 1 funds paying 42 and 5 simple interest. There is more risk in the 5 fund. Your goal is to obtain a total annual interest income of 580 from the investments. What is the smallest amount you can invest in the 5 fund and still meet your objective 72. INVESTMENT You plan to invest 25,000 in two 1 funds paying 3 and 42 simple interest. There is 1 more risk in the 42 fund. Your goal is to obtain a total annual interest income of 1000 from the investments. 1 What is the smallest amount you can invest in the 42 fund and still meet your objective 76. MIXTURE PROBLEM A 100 concentrate is to be mixed with a mixture having a concentration of 40 to obtain 55 gallons of a mixture with a concentration of 75. How much of the 100 concentrate will be needed 77. MIXTURE PROBLEM A forester mixes gasoline and oil to make 2 gallons of mixture for his two cycle chainsaw engine. This mixture is 32 parts gasoline and 1 part two cycle oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil 78. MIXTURE PROBLEM A grocer mixes peanuts that cost 1.49 per pound and walnuts that cost 2.69 per pound to make 100 pounds of a mixture that costs 2.21 per pound. How much of each kind of nut is put into the mixture 79. COMPANY COSTS An outdoor furniture manufacturer has fixed costs of 14,000 per month and average variable costs of 12.75 per unit manufactured. The company has 110,000 available to cover the monthly costs. How many units can the company manufacture Fixed costs are those that occur regardless of the level of production. Variable costs depend on the level of production. 80. COMPANY COSTS A plumbing supply company has fixed costs of 10,000 per month and average variable costs of 9.30 per unit manufactured. The company has 85,000 available to cover the monthly costs. How many units can the company manufacture Fixed costs are those that occur regardless of the level of production. Variable costs depend on the level of production. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 106 Chapter 1 Equations, Inequalities, and Mathematical Modeling In Exercises 8192, solve for the indicated variable. 81. AREA OF A TRIANGLE Solve for h: A 12 bh 82. AREA OF A TRAPEZOID Solve for b: A 12 a bh 83. MARKUP Solve for C: S C RC 84. INVESTMENT AT SIMPLE INTEREST Solve for r: A P Prt 85. VOLUME OF AN OBLATE SPHEROID Solve for b: V 43 a2b 86. VOLUME OF A SPHERICAL SEGMENT Solve for r: V 13 h 2 3r h 87. FREE FALLING BODY Solve for a: h v0 t 12at 2 88. LENSMAKERS EQUATION 1 1 1 Solve for R1: n 1 f R1 R2 89. CAPACITANCE IN SERIES CIRCUITS 1 Solve for C1: C 1 1 C1 C2 90. ARITHMETIC PROGRESSION n Solve for a: S 2a n 1d 2 EXPLORATION TRUE OR FALSE In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer. 101. 8 less than z cubed divided by the difference of z squared and 9 can be written as z3 8z 92. 102. The volume of a cube with a side of length 9.5 inches is greater than the volume of a sphere with a radius of 5.9 inches. 91. ARITHMETIC PROGRESSION Solve for n: L a n 1 d 92. GEOMETRIC PROGRESSION Solve for r: S rL a r1 PHYSICS In Exercises 93 and 94, you have a uniform beam of length L with a fulcrum x feet from one end see figure . Objects with weights W1 and W2 are placed at opposite ends of the beam. The beam will balance when W1 x W2L x. Find x such that the beam will balance. W2 W1 x 95. VOLUME OF A BILLIARD BALL A billiard ball has a volume of 5.96 cubic inches. Find the radius of a billiard ball. 96. LENGTH OF A TANK The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank. 97. TEMPERATURE The average daily temperature in San Diego, California is 64.4F. What is San Diegos average daily temperature in degrees Celsius Source: NOAA 98. TEMPERATURE The average daily temperature in Duluth, Minnesota is 39.1F. What is Duluths average daily temperature in degrees Celsius Source: NOAA 99. TEMPERATURE The highest temperature ever recorded in Phoenix, Arizona was 50C. What is this temperature in degrees Fahrenheit Source: NOAA 100. TEMPERATURE The lowest temperature ever recorded in Louisville, Kentucky was 30C. What is this temperature in degrees Fahrenheit Source: NOAA Lx 93. Two children weighing 50 pounds and 75 pounds are playing on a seesaw that is 10 feet long. 94. A person weighing 200 pounds is attempting to move a 550 pound rock with a bar that is 5 feet long. 103. Consider the linear equation ax b 0. a What is the sign of the solution if ab 0 b What is the sign of the solution if ab 0 In each case, explain your reasoning. 104. CAPSTONE Arrange the following statements in the proper order to obtain a strategy for modeling and solving a real life problem. Assign labels to each part of the verbal model numbers to the known quantities and letters or expressions to the variable quantities. Answer the original question and check that your answer satisfies the original problem as stated. Solve the algebraic equation. Ask yourself what you need to know to solve the problem and then write a verbal model that includes arithmetic operations to describe the problem. Write an algebraic equation based on the verbal model. 105. Write a linear equation that has the solution x 3. There are many correct answers. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 Quadratic Equations and Applications 107 1.4 QUADRATIC EQUATIONS AND APPLICATIONS What you should learn Solve quadratic equations by factoring. Solve quadratic equations by extracting square roots. Solve quadratic equations by completing the square. Use the Quadratic Formula to solve quadratic equations. Use quadratic equations to model and solve real life problems. Why you should learn it Quadratic equations can be used to model and solve real life problems. For instance, in Exercise 123 on page 119, you will use a quadratic equation to model average admission prices for movie theaters from 2001 through 2008. Factoring A quadratic equation in x is an equation that can be written in the general form ax 2 bx c 0 where a, b, and c are real numbers with a 0. A quadratic equation in x is also called a second degree polynomial equation in x. In this section, you will study four methods for solving quadratic equations: factoring, extracting square roots, completing the square, and the Quadratic Formula. The first method is based on the Zero Factor Property from Section P.1. If ab 0, then a 0 or b 0. Zero Factor Property To use this property, write the left side of the general form of a quadratic equation as the product of two linear factors. Then find the solutions of the quadratic equation by setting each linear factor equal to zero. Example 1 a. Solving a Quadratic Equation by Factoring 2x 2 9x 7 3 2x2 Original equation 9x 4 0 Write in general form. 2x 1x 4 0 Factor. x IndiapictureAlamy 2x 1 0 x40 The solutions are x b. x 4 12 Set 2nd factor equal to 0. Original equation 3x2x 1 0 2x 1 0 Set 1st factor equal to 0. and x 4. Check these in the original equation. 6x 2 3x 0 3x 0 1 2 Factor. x0 x 1 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. The solutions are x 0 and x 12. Check these in the original equation. Now try Exercise 15. Be sure you see that the Zero Factor Property works only for equations written in general form in which the right side of the equation is zero . So, all terms must be collected on one side before factoring. For instance, in the equation x 5x 2 8, it is incorrect to set each factor equal to 8. To solve this equation, you must multiply the binomials on the left side of the equation, and then subtract 8 from each side. After simplifying the left side of the equation, you can use the Zero Factor Property to solve the equation. Try to solve this equation correctly. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 108 Chapter 1 Equations, Inequalities, and Mathematical Modeling Extracting Square Roots T E C H N O LO G Y You can use a graphing utility to check graphically the real solutions of a quadratic equation. Begin by writing the equation in general form. Then set y equal to the left side and graph the resulting equation. The x intercepts of the equation represent the real solutions of the original equation. You can use the zero or root feature of a graphing utility to approximate the x intercepts of the graph. For example, to check the solutions of 6x 2 3x 0, graph y 6x 2 3x, and use the zero or root feature to approximate the x intercepts 1 to be 0, 0 and 2, 0, as shown below. These x intercepts represent the solutions x 0 and x 12, as found in Example 1 b . 3 There is a nice shortcut for solving quadratic equations of the form u 2 d, where d 0 and u is an algebraic expression. By factoring, you can see that this equation has two solutions. u2 d Write original equation. u2 d 0 Write in general form. u d u d 0 Factor. u d 0 u d Set 1st factor equal to 0. u d 0 u d Set 2nd factor equal to 0. Because the two solutions differ only in sign, you can write the solutions together, using a plus or minus sign, as u d. This form of the solution is read as u is equal to plus or minus the square root of d. Solving an equation of the form u 2 d without going through the steps of factoring is called extracting square roots. Extracting Square Roots The equation u 2 d, where d 0, has exactly two solutions: u d and u d. These solutions can also be written as u d. 3 , 0 1 2 0, 0 1 3 Example 2 Extracting Square Roots Solve each equation by extracting square roots. b. x 32 7 a. 4x 2 12 Solution a. 4x 2 12 Write original equation. x2 3 Divide each side by 4. x 3 Extract square roots. When you take the square root of a variable expression, you must account for both positive and negative solutions. So, the solutions are x 3 and x 3. Check these in the original equation. b. x 32 7 x 3 7 x 3 7 Write original equation. Extract square roots. Add 3 to each side. The solutions are x 3 7. Check these in the original equation. Now try Exercise 33. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 Quadratic Equations and Applications 109 Completing the Square The equation in Example 2 b was given in the form x 32 7 so that you could find the solution by extracting square roots. Suppose, however, that the equation had been given in the general form x 2 6x 2 0. Because this equation is equivalent to the original, it has the same two solutions, x 3 7. However, the left side of the equation is not factorable, and you cannot find its solutions unless you rewrite the equation by completing the square. Note that when you complete the square to solve a quadratic equation, you are just rewriting the equation so it can be solved by extracting square roots. Completing the Square To complete the square for the expression x 2 bx, add b2 2, which is the square of half the coefficient of x. Consequently, 2 x 2 . b x 2 bx Example 3 2 b 2 Completing the Square: Leading Coefficient Is 1 Solve x 2 2x 6 0 by completing the square. Solution x 2 2x 6 0 Write original equation. x 2x 6 2 Add 6 to each side. x 2x 1 6 1 2 2 2 Add 12 to each side. 2 half of 2 x 1 7 2 Simplify. x 1 7 Take square root of each side. x 1 7 Subtract 1 from each side. The solutions are x 1 7. Check these in the original equation as follows. Check x2 2x 6 0 1 72 21 7 6 0 8 27 2 27 6 0 8260 Write original equation. Substitute 1 7 for x. Multiply. Solution checks. Check the second solution in the original equation. Now try Exercise 41. When solving quadratic equations by completing the square, you must add b2 2 to each side in order to maintain equality. If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square, as shown in Example 4. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 110 Chapter 1 Equations, Inequalities, and Mathematical Modeling Example 4 Completing the Square: Leading Coefficient Is Not 1 Solve 2x 2 8x 3 0 by completing the square. Solution 2x 2 8x 3 0 Write original equation. 2x 2 8x 3 x 2 4x Subtract 3 from each side. 3 2 Divide each side by 2. 3 x2 4x 22 22 2 Add 22 to each side. 2 half of 4 5 2 x 22 x2 You can review rationalizing denominators in Section P.2. x2 Simplify. 52 Take square root of each side. 10 Rationalize denominator. 2 x 2 The solutions are x 2 10 Subtract 2 from each side. 2 10 2 . Check these in the original equation. Now try Exercise 43. Example 5 Completing the Square: Leading Coefficient Is Not 1 3x2 4x 5 0 Original equation 3x 4x 5 2 Add 5 to each side. 4 5 x2 x 3 3 4 2 x2 x 3 3 2 Divide each side by 3. 5 2 3 3 2 4 4 19 x2 x 3 9 9 x 32 x 2 19 9 19 2 3 3 x 19 2 3 3 Add 3 to each side. 2 2 Simplify. Perfect square trinomial Extract square roots. Solutions Now try Exercise 47. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 111 Quadratic Equations and Applications The Quadratic Formula Often in mathematics you are taught the long way of solving a problem first. Then, the longer method is used to develop shorter techniques. The long way stresses understanding and the short way stresses efficiency. For instance, you can think of completing the square as a long way of solving a quadratic equation. When you use completing the square to solve quadratic equations, you must complete the square for each equation separately. In the following derivation, you complete the square once in a general setting to obtain the Quadratic Formula a shortcut for solving quadratic equations. ax 2 bx c 0 ax2 Write in general form, a bx c Subtract c from each side. b c x2 x a a b b x2 x a 2a half of ba 2 0. Divide each side by a. c b a 2a 2 Complete the square. 2 x 2a b x 2 b2 4ac 4a2 b 2a Simplify. b 4a 4ac x 2 2 b2 4ac b 2a 2a Extract square roots. Solutions Note that because 2a represents the same numbers as 2a, you can omit the absolute value sign. So, the formula simplifies to x b b2 4ac . 2a The Quadratic Formula You can solve every quadratic equation by completing the square or using the Quadratic Formula. The solutions of a quadratic equation in the general form ax 2 bx c 0, a 0 are given by the Quadratic Formula x b b2 4ac . 2a The Quadratic Formula is one of the most important formulas in algebra. You should learn the verbal statement of the Quadratic Formula: Negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 112 Chapter 1 Equations, Inequalities, and Mathematical Modeling In the Quadratic Formula, the quantity under the radical sign, b2 4ac, is called the discriminant of the quadratic expression ax 2 bx c. It can be used to determine the nature of the solutions of a quadratic equation. Solutions of a Quadratic Equation The solutions of a quadratic equation ax2 bx c 0, a as follows. If the discriminant b2 4ac is 0, can be classified 1. positive, then the quadratic equation has two distinct real solutions and its graph has two x intercepts. 2. zero, then the quadratic equation has one repeated real solution and its graph has one x intercept. 3. negative, then the quadratic equation has no real solutions and its graph has no x intercepts. If the discriminant of a quadratic equation is negative, as in case 3 above, then its square root is imaginary not a real number and the Quadratic Formula yields two complex solutions. You will study complex solutions in Section 1.5. When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form. Example 6 The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve x 2 3x 9. Solution The general form of the equation is x2 3x 9 0. The discriminant is b2 4ac 9 36 45, which is positive. So, the equation has two real solutions. You can solve the equation as follows. x 2 3x 9 0 Write in general form. x b b2 4ac 2a Quadratic Formula x 3 32 419 21 Substitute a 1, b 3, and c 9. x 3 45 2 Simplify. x 3 35 2 Simplify. The two solutions are: x 3 35 2 and x 3 35 . 2 Check these in the original equation. Now try Exercise 81. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 Quadratic Equations and Applications 113 Applications Quadratic equations often occur in problems dealing with area. Here is a simple example. A square room has an area of 144 square feet. Find the dimensions of the room. To solve this problem, let x represent the length of each side of the room. Then, by solving the equation x 2 144 you can conclude that each side of the room is 12 feet long. Note that although the equation x 2 144 has two solutions, x 12 and x 12, the negative solution does not make sense in the context of the problem, so you choose the positive solution. Example 7 Finding the Dimensions of a Room A bedroom is 3 feet longer than it is wide see Figure 1.20 and has an area of 154 square feet. Find the dimensions of the room. w w+3 FIGURE 1.20 Solution Verbal Model: Width of room Labels: Width of room w Length of room w 3 Area of room 154 Length Area of room of room feet feet square feet ww 3 154 Equation: w2 3w 154 0 w 11w 14 0 w 11 0 w 11 w 14 0 w 14 Choosing the positive value, you find that the width is 11 feet and the length is w 3, or 14 feet. You can check this solution by observing that the length is 3 feet longer than the width and that the product of the length and width is 154 square feet. Now try Exercise 113. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 114 Chapter 1 Equations, Inequalities, and Mathematical Modeling Another common application of quadratic equations involves an object that is falling or projected into the air . The general equation that gives the height of such an object is called a position equation, and on Earths surface it has the form s 16t 2 v0 t s0. In this equation, s represents the height of the object in feet , v0 represents the initial velocity of the object in feet per second , s0 represents the initial height of the object in feet , and t represents the time in seconds . Example 8 Falling Time A construction worker on the 24th floor of a building project see Figure 1.21 accidentally drops a wrench and yells Look out below Could a person at ground level hear this warning in time to get out of the way Note: The speed of sound is about 1100 feet per second. Solution 235 ft Assume that each floor of the building is 10 feet high, so that the wrench is dropped from a height of 235 feet the construction workers hand is 5 feet below the ceiling of the 24th floor . Because sound travels at about 1100 feet per second, it follows that a person at ground level hears the warning within 1 second of the time the wrench is dropped. To set up a mathematical model for the height of the wrench, use the position equation s 16t 2 v0 t s0. Because the object is dropped rather than thrown, the initial velocity is v0 0 feet per second. Moreover, because the initial height is s0 235 feet, you have the following model. s 16t 2 0t 235 16t2 235 FIGURE 1.21 After the wrench has fallen for 1 second, its height is 1612 235 219 feet. After the wrench has fallen for 2 seconds, its height is 1622 235 171 feet. To find the number of seconds it takes the wrench to hit the ground, let the height s be zero and solve the equation for t. s 16t 2 235 0 16t 2 16t 2 235 t2 The position equation used in Example 8 ignores air resistance. This implies that it is appropriate to use the position equation only to model falling objects that have little air resistance and that fall over short distances. t 235 16 235 4 t 3.83 235 Write position equation. Substitute 0 for height. Add 16t 2 to each side. Divide each side by 16. Extract positive square root. Use a calculator. The wrench will take about 3.83 seconds to hit the ground. If the person hears the warning 1 second after the wrench is dropped, the person still has almost 3 seconds to get out of the way. Now try Exercise 119. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 Quadratic Equations and Applications 115 A third type of application of a quadratic equation is one in which a quantity is changing over time t according to a quadratic model. Example 9 Quadratic Modeling: Internet Users From 2000 through 2008, the estimated numbers of Internet users I in millions in the United States can be modeled by the quadratic equation I 1.446t 2 23.45t 122.9, 0 t 8 where t represents the year, with t 0 corresponding to 2000. According to this model, in which year did the number of Internet users reach or surpass 200 million Source: International Telecommunication UnionThe Nielsen Company Algebraic Solution Numerical Solution To find the year in which the number of Internet users reached 200 million, you can solve the equation You can estimate the year in which the number of Internet users reached or surpassed 200 million by constructing a table of values. The table below shows the number of Internet users for each year from 2000 through 2008. 1.446t2 23.45t 122.9 200. To begin, write the equation in general form. 1.446t 2 23.45t 77.1 0 Year t I Then apply the Quadratic Formula. 2000 0 122.9 b b2 4ac 2a 2001 1 144.9 2002 2 164.0 2003 3 180.2 2004 4 193.6 2005 5 204.0 2006 6 211.5 2007 7 216.2 2008 8 218.0 t 23.45 23.452 41.44677.1 t 21.446 23.45 103.96 2.892 4.6 or 11.6 Choose the smaller value t 4.6. Because t 0 corresponds to 2000, it follows that t 4.6 must correspond to 2004. So, the number of Internet users should have reached 200 million during the year 2004. From the table, you can see that sometime during 2004 the number of Internet users reached 200 million. Now try Exercise 123. T E C H N O LO G Y You can also use a graphical approach to solve Example 9. Use a graphing utility to graph y1 1.446t2 23.45t 122.9 and y2 200 in the same viewing window. Then use the intersect feature to find the point s of intersection of the two graphs. You should obtain t y 4.6, which verifies the answer obtained algebraically. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 116 Chapter 1 Equations, Inequalities, and Mathematical Modeling A fourth type of application that often involves a quadratic equation is one dealing with the hypotenuse of a right triangle. In these types of applications, the Pythagorean Theorem is often used. The Pythagorean Theorem states that a2 b2 c2 Pythagorean Theorem where a and b are the legs of a right triangle and c is the hypotenuse. Example 10 2x An Application Involving the Pythagorean Theorem Athletic Center An L shaped sidewalk from the athletic center to the library on a college campus is shown in Figure 1.22. The sidewalk was constructed so that the length of one sidewalk forming the L was twice as long as the other. The length of the diagonal sidewalk that cuts across the grounds between the two buildings is 32 feet. How many feet does a person save by walking on the diagonal sidewalk 32 ft Solution Library Using the Pythagorean Theorem, you have the following. x 2 2x2 322 5x 2 1024 x FIGURE x2 1.22 204.8 Pythagorean Theorem Combine like terms. Divide each side by 5. x 204.8 Take the square root of each side. x 204.8 Extract positive square root. The total distance covered by walking on the L shaped sidewalk is x 2x 3x 3204.8 42.9 feet. Walking on the diagonal sidewalk saves a person about 42.9 32 10.9 feet. Now try Exercise 125. CLASSROOM DISCUSSION Comparing Solution Methods In this section, you studied four algebraic methods for solving quadratic equations. Solve each of the quadratic equations below in several different ways. Write a short paragraph explaining which method s you prefer. Does your preferred method depend on the equation a. b. c. d. x 2 4x 5 0 x 2 4x 0 x 2 4x 3 0 x 2 4x 6 0 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 1.4 EXERCISES Quadratic Equations and Applications 117 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. A ________ ________ in x is an equation that can be written in the general form ax2 bx c 0, where a, b, and c are real numbers with a 0. 2. A quadratic equation in x is also called a ________ ________ equation in x. 3. Four methods that can be used to solve a quadratic equation are ________, extracting ________ ________ , ________ the ________, and the ________ ________. 4. The part of the Quadratic Formula, b2 4ac, known as the ________, determines the type of solutions of a quadratic equation. 5. The general equation that gives the height of an object that is falling is called a ________ ________. 6. An important theorem that is sometimes used in applications that require solving quadratic equations is the ________ ________. SKILLS AND APPLICATIONS In Exercises 712, write the quadratic equation in general form. 7. 2x 2 3 5x 9. x 32 3 1 11. 53x 2 10 12x 8. x 2 16x 10. 13 3x 72 0 12. xx 2 5x 2 1 In Exercises 1324, solve the quadratic equation by factoring. 3x 0 13. 2 15. x 2x 8 0 17. x 2 10x 25 0 19. 3 5x 2x 2 0 21. x 2 4x 12 3 23. 4 x 2 8x 20 0 6x 2 10 14. 2 16. x 10x 9 0 18. 4x 2 12x 9 0 20. 2x 2 19x 33 22. x 2 8x 12 1 24. 8 x 2 x 16 0 9x 2 In Exercises 2538, solve the equation by extracting square roots. 25. x 2 49 27. x 2 11 29. 3x 2 81 31. x 122 16 33. x 2 2 14 35. 2x 12 18 37. x 72 x 3 2 26. x 2 144 28. x 2 32 30. 9x 2 36 32. x 52 25 34. x 92 24 36. 4x 72 44 38. x 52 x 4 2 45. 7 2x x2 0 47. 2x 2 5x 8 0 46. x 2 x 1 0 48. 3x 2 4x 7 0 In Exercises 4956, rewrite the quadratic portion of the algebraic expression as the sum or difference of two squares by completing the square. 1 x 2 2x 5 4 51. 2 x 4x 3 49. 53. 55. 1 4x2 4x 9 1 50. 1 x 2 12x 19 52. 5 x 2 25x 11 54. 1 4x2 4x 25 56. 6x x2 1 16 6x x2 GRAPHICAL ANALYSIS In Exercises 57 64, a use a graphing utility to graph the equation, b use the graph to approximate any x intercepts of the graph, c set y 0 and solve the resulting equation, and d compare the result of part c with the x intercepts of the graph. 57. y x 3 2 4 59. y 1 x 22 61. y 4x 2 4x 3 63. y x 2 3x 4 58. y x 42 1 60. y 9 x 82 62. y 4x 2 1 64. y x 2 5x 24 In Exercises 39 48, solve the quadratic equation by completing the square. In Exercises 6572, use the discriminant to determine the number of real solutions of the quadratic equation. 39. x 2 4x 32 0 41. x 2 6x 2 0 43. 9x 2 18x 3 65. 2x 2 5x 5 0 67. 2x 2 x 1 0 69. 13x 2 5x 25 0 71. 0.2x 2 1.2x 8 0 40. x 2 2x 3 0 42. x 2 8x 14 0 44. 4x2 4x 1 www.elsolucionario.net 66. 5x 2 4x 1 0 68. x 2 4x 4 0 70. 47 x 2 8x 28 0 72. 9 2.4x 8.3x 2 0 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 118 Chapter 1 Equations, Inequalities, and Mathematical Modeling In Exercises 7396, use the Quadratic Formula to solve the equation. 73. 75. 77. 79. 81. 83. 85. 87. 89. 91. 93. 95. 2x 2 x 1 0 16x 2 8x 3 0 2 2x x 2 0 x 2 12x 16 0 x 2 8x 4 0 12x 9x 2 3 9x2 30x 25 0 4x 2 4x 7 28x 49x 2 4 8t 5 2t 2 y 52 2y 1 2 3 2x 8x 2 74. 76. 78. 80. 82. 84. 86. 88. 90. 92. 94. 96. 2x 2 x 1 0 25x 2 20x 3 0 x 2 10x 22 0 4x 8 x 2 2x 2 3x 4 0 9x 2 37 6x 36x 2 24x 7 0 16x 2 40x 5 0 3x x 2 1 0 25h2 80h 61 0 z 62 2z 57x 142 8x In Exercises 97104, use the Quadratic Formula to solve the equation. Round your answer to three decimal places. 97. 98. 99. 100. 101. 102. 103. 104. 5.1x 2 1.7x 3.2 0 2x 2 2.50x 0.42 0 0.067x 2 0.852x 1.277 0 0.005x 2 0.101x 0.193 0 422x 2 506x 347 0 1100x 2 326x 715 0 12.67x 2 31.55x 8.09 0 3.22x 2 0.08x 28.651 0 114. DIMENSIONS OF A GARDEN A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens see figure . The gardener wants the enclosed area to be 350 square meters. What dimensions should the gardener use to obtain this area y x x 4x + 3y = 100 115. PACKAGING An open box with a square base see figure is to be constructed from 108 square inches of material. The height of the box is 3 inches. What are the dimensions of the box Hint: The surface area is S x 2 4xh. 3 in. x x 116. PACKAGING An open gift box is to be made from a square piece of material by cutting four centimeter squares from the corners and turning up the sides see figure . The volume of the finished box is to be 576 cubic centimeters. Find the size of the original piece of material. In Exercises 105112, solve the equation using any convenient method. 105. x 2 2x 1 0 107. x 32 81 11 109. x2 x 4 0 106. 11x 2 33x 0 108. x2 14x 49 0 3 110. x2 3x 4 0 111. x 12 x 2 112. 3x 4 2x2 7 113. FLOOR SPACE The floor of a one story building is 14 feet longer than it is wide see figure . The building has 1632 square feet of floor space. w w + 14 a Write a quadratic equation for the area of the floor in terms of w. b Find the length and width of the floor. 4 cm 4 cm x 4 cm 4 cm x x x 4 cm 117. MOWING THE LAWN Two landscapers must mow a rectangular lawn that measures 100 feet by 200 feet. Each wants to mow no more than half of the lawn. The first starts by mowing around the outside of the lawn. The mower has a 24 inch cut. How wide a strip must the first landscaper mow on each of the four sides in order to mow no more than half of the lawn Approximate the required number of trips around the lawn the first landscaper must take. 118. SEATING A rectangular classroom seats 72 students. If the seats were rearranged with three more seats in each row, the classroom would have two fewer rows. Find the original number of seats in each row. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 In Exercises 119122, use the position equation given in Example 8 as the model for the problem. 119. MILITARY A C 141 Starlifter flying at 25,000 feet over level terrain drops a 500 pound supply package. a How long will it take until the supply package strikes the ground b The plane is flying at 500 miles per hour. How far will the supply package travel horizontally during its descent 120. EIFFEL TOWER You drop a coin from the top of the Eiffel Tower in Paris. The building has a height of 984 feet. a Use the position equation to write a mathematical model for the height of the coin. b Find the height of the coin after 4 seconds. c How long will it take before the coin strikes the ground 121. SPORTS Some Major League Baseball pitchers can throw a fastball at speeds of up to and over 100 miles per hour. Assume a Major League Baseball pitcher throws a baseball straight up into the air at 100 miles per hour from a height of 6 feet 3 inches. a Use the position equation to write a mathematical model for the height of the baseball. b Find the height of the baseball after 3 seconds, 4 seconds, and 5 seconds. What must have occurred sometime in the interval 3 t 5 Explain. c How many seconds is the baseball in the air 122. CN TOWER At 1815 feet tall, the CN Tower in Toronto, Ontario is the worlds tallest self supporting structure. An object is dropped from the top of the tower. a Use the position equation to write a mathematical model for the height of the object. b Complete the table. Time, t 0 2 4 6 8 10 12 119 Quadratic Equations and Applications 123. DATA ANALYSIS: MOVIE TICKETS The average admission prices P for movie theaters from 2001 through 2008 can be approximated by the model P 0.0103t2 0.119t 5.55, 1 t 8 where t represents the year, with t 1 corresponding to 2001. Source: Motion Picture Association of America, Inc. a Use the model to complete the table to determine when the average admission price reached or surpassed 6.50. t 1 2 3 4 5 6 7 8 P b Verify your result from part a algebraically. c Use the model to predict the average admission price for movie theaters in 2014. Is this prediction reasonable How does this value compare with the admission price where you live 124. DATA ANALYSIS: MEDIAN INCOME The median incomes I in dollars of U.S. households from 2000 through 2007 can be approximated by the model I 187.65t2 119.1t 42,013, 0 t 7 where t represents the year, with t 0 corresponding to 2000. Source: U.S. Census Bureau a Use a graphing utility to graph the model. Then use the graph to determine in which year the median income reached or surpassed 45,000. b Verify your result from part a algebraically. c Use the model to predict the median incomes of U.S. households in 2014 and 2018. Can this model be used to predict the median income of U.S. households after 2007 Before 2000 Explain. 125. BOATING A winch is used to tow a boat to a dock. The rope is attached to the boat at a point 15 feet below the level of the winch see figure . Height, s c From the table in part b , determine the time interval during which the object reaches the ground. Numerically approximate the time it takes the object to reach the ground. d Find the time it takes the object to reach the ground algebraically. How close was your numerical approximation e Use a graphing utility with the appropriate viewing window to verify your answer s to parts c and d . 15 ft l x Not drawn to scale a Use the Pythagorean Theorem to write an equation giving the relationship between l and x. b Find the distance from the boat to the dock when there is 75 feet of rope out. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 120 Chapter 1 Equations, Inequalities, and Mathematical Modeling 126. FLYING SPEED Two planes leave simultaneously from Chicagos OHare Airport, one flying due north and the other due east see figure . The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane. N D 0.032t2 0.21t 5.6, 0 t 8 where t represents the year, with t 0 corresponding to 2000. Source: U.S. Department of the Treasury a Use the model to complete the table to determine when the total public debt reached or surpassed 7 trillion. t 2440 mi W 0 1 2 3 4 5 6 7 8 D b Verify your result from part a algebraically and graphically. E S 127. GEOMETRY The hypotenuse of an isosceles right triangle is 9 centimeters long. How long are its sides 128. GEOMETRY An equilateral triangle has a height of 16 inches. How long is one of its sides Hint: Use the height of the triangle to partition the triangle into two congruent right triangles. 129. REVENUE The demand equation for a product is p 20 0.0002x, where p is the price per unit and x is the number of units sold. The total revenue for selling x units is Revenue xp x20 0.0002x. How many units must be sold to produce a revenue of 500,000 130. REVENUE The demand equation for a product is p 60 0.0004x, where p is the price per unit and x is the number of units sold. The total revenue for selling x units is c Use the model to predict the public debt in 2014. Is this prediction reasonable Explain. 136. BIOLOGY The metabolic rate of an ectothermic organism increases with increasing temperature within a certain range. Experimental data for the oxygen consumption C in microliters per gram per hour of a beetle at certain temperatures can be approximated by the model C 0.45x 2 1.65x 50.75, 10 x 25 where x is the air temperature in degrees Celsius. a The oxygen consumption is 150 microliters per gram per hour. What is the air temperature b The temperature is increased from 10C to 20C. The oxygen consumption is increased by approximately what factor 137. GEOMETRY An above ground swimming pool with the dimensions shown in the figure is to be constructed such that the volume of water in the pool is 1024 cubic feet. The height of the pool is to be 4 feet. Revenue xp x60 0.0004x. How many units must be sold to produce a revenue of 220,000 COST In Exercises 131134, use the cost equation to find the number of units x that a manufacturer can produce for the given cost C. Round your answer to the nearest positive integer. 131. C 0.125x 2 20x 500 132. C 0.5x 2 15x 5000 133. C 800 0.04x 0.002x 2 C 14,000 C 11,500 C 1680 x2 134. C 800 10x 4 C 896 135. PUBLIC DEBT The total public debt D in trillions of dollars in the United States at the beginning of each year from 2000 through 2008 can be approximated by the model 4 ft x x+1 Not drawn to scale a What are the possible dimensions of the base b One cubic foot of water weighs approximately 62.4 pounds. Find the total weight of the water in the pool. c A water pump is filling the pool at a rate of 5 gallons per minute. Find the time that will be required for the pump to fill the pool. Hint: One gallon of water is approximately 0.13368 cubic foot. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.4 138. FLYING DISTANCE A commercial jet flies to three cities whose locations form the vertices of a right triangle see figure . The total flight distance from Oklahoma City to Austin to New Orleans and back to Oklahoma City is approximately 1348 miles. It is 560 miles between Oklahoma City and New Orleans. Approximate the other two distances. Oklahoma City 56 0m i Austin New Orleans TRUE OR FALSE In Exercises 139 and 140, determine whether the statement is true or false. Justify your answer. 139. The quadratic equation 3x 2 x 10 has two real solutions. 140. If 2x 3x 5 8, then either 2x 3 8 or x 5 8. 141. To solve the equation 2x 2 3x 15x, a student divides each side by x and solves the equation 2x 3 15. The resulting solution x 6 satisfies the original equation. Is there an error Explain. 142. The graphs show the solutions of equations plotted on the real number line. In each case, determine whether the solution s is are for a linear equation, a quadratic equation, both, or neither. Explain. x a b 145. 147. 149. 150. 3 and 5 146. 6 and 9 8 and 14 148. 61 and 25 1 2 and 1 2 3 5 and 3 5 151. From each graph, can you tell whether the discriminant is positive, zero, or negative Explain your reasoning. Find each discriminant to verify your answers. a x2 2x 0 b x2 2x 1 0 y x x a b c d 143. Solve 3x 42 x 4 2 0 in two ways. a Let u x 4, and solve the resulting equation for u. Then solve the u solution for x. b Expand and collect like terms in the equation, and solve the resulting equation for x. c Which method is easier Explain. 2 4 2 x 2 4 c x2 2x 2 0 y 2 2 b x 2 c a 6 2 a c y 6 x b d 144. CAPSTONE Match the equation with a method you would use to solve it. Explain your reasoning. Use each method once and do not solve the equations. a 3x2 5x 11 0 i Factoring b x2 10x 3 ii Extracting square roots 2 c x 16x 64 0 iii Completing the square 2 d x 15 0 iv Quadratic Formula THINK ABOUT IT In Exercises 145150, write a quadratic equation that has the given solutions. There are many correct answers. EXPLORATION a 121 Quadratic Equations and Applications x 2 4 How many solutions would part c have if the linear term was 2x If the constant was 2 152. THINK ABOUT IT Is it possible for a quadratic equation to have only one x intercept Explain. 153. PROOF Given that the solutions of a quadratic equation are x b b2 4ac 2a, show that a the sum of the solutions is S ba and b the product of the solutions is P ca. PROJECT: POPULATION To work an extended application analyzing the population of the United States, visit this texts website at academic.cengage.com. Data Source: U.S. Census Bureau www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 122 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.5 COMPLEX NUMBERS What you should learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers in standard form. Find complex solutions of quadratic equations. Why you should learn it You can use complex numbers to model and solve real life problems in electronics. For instance, in Exercise 89 on page 128, you will learn how to use complex numbers to find the impedance of an electrical circuit. The Imaginary Unit i In Section 1.4, you learned that some quadratic equations have no real solutions. For instance, the quadratic equation x 2 1 0 has no real solution because there is no real number x that can be squared to produce 1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i 1 Imaginary unit where i 2 1. By adding real numbers to real multiples of this imaginary unit, the set of complex numbers is obtained. Each complex number can be written in the standard form a bi. For instance, the standard form of the complex number 5 9 is 5 3i because 5 9 5 321 5 31 5 3i. In the standard form a bi, the real number a is called the real part of the complex number a bi, and the number bi where b is a real number is called the imaginary part of the complex number. Definition of a Complex Number Richard MegnaFundamental Photographs If a and b are real numbers, the number a bi is a complex number, and it is said to be written in standard form. If b 0, the number a bi a is a real number. If b 0, the number a bi is called an imaginary number. A number of the form bi, where b 0, is called a pure imaginary number. The set of real numbers is a subset of the set of complex numbers, as shown in Figure 1.23. This is true because every real number a can be written as a complex number using b 0. That is, for every real number a, you can write a a 0i. Real numbers Complex numbers Imaginary numbers FIGURE 1.23 Equality of Complex Numbers Two complex numbers a bi and c di, written in standard form, are equal to each other a bi c di Equality of two complex numbers if and only if a c and b d. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.5 Complex Numbers 123 Operations with Complex Numbers To add or subtract two complex numbers, you add or subtract the real and imaginary parts of the numbers separately. Addition and Subtraction of Complex Numbers If a bi and c di are two complex numbers written in standard form, their sum and difference are defined as follows. Sum: a bi c di a c b d i Difference: a bi c di a c b d i The additive identity in the complex number system is zero the same as in the real number system . Furthermore, the additive inverse of the complex number a bi is a bi a bi. Additive inverse So, you have a bi a bi 0 0i 0. Example 1 Adding and Subtracting Complex Numbers a. 4 7i 1 6i 4 7i 1 6i Remove parentheses. 4 1 7i 6i Group like terms. 5i Write in standard form. b. 1 2i 4 2i 1 2i 4 2i Remove parentheses. 1 4 2i 2i Group like terms. 3 0 Simplify. 3 Write in standard form. c. 3i 2 3i 2 5i 3i 2 3i 2 5i 2 2 3i 3i 5i 0 5i 5i d. 3 2i 4 i 7 i 3 2i 4 i 7 i 3 4 7 2i i i 0 0i 0 Now try Exercise 21. Note in Examples 1 b and 1 d that the sum of two complex numbers can be a real number. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 124 Chapter 1 Equations, Inequalities, and Mathematical Modeling Many of the properties of real numbers are valid for complex numbers as well. Here are some examples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Property of Multiplication Over Addition Notice below how these properties are used when two complex numbers are multiplied. a bic di ac di bi c di Distributive Property ac ad i bci bd i 2 Distributive Property ac ad i bci bd 1 i 2 1 ac bd ad i bci Commutative Property ac bd ad bci Associative Property Rather than trying to memorize this multiplication rule, you should simply remember how the Distributive Property is used to multiply two complex numbers. Example 2 Multiplying Complex Numbers a. 42 3i 42 43i Distributive Property 8 12i The procedure described above is similar to multiplying two polynomials and combining like terms, as in the FOIL Method shown in Section P.3. For instance, you can use the FOIL Method to multiply the two complex numbers from Example 2 b . F O I Simplify. b. 2 i4 3i 24 3i i4 3i 8 6i 4i 3i 2 i 2 1 8 3 6i 4i Group like terms. 11 2i Write in standard form. 9 6i 6i L Distributive Property 8 6i 4i 31 c. 3 2i3 2i 33 2i 2i3 2i 2 i4 3i 8 6i 4i 3i2 Distributive Property 4i 2 Distributive Property Distributive Property 9 6i 6i 41 i 2 1 94 Simplify. 13 Write in standard form. d. 3 2i 3 2i3 2i 2 Square of a binomial 33 2i 2i3 2i Distributive Property 9 6i 6i Distributive Property 4i 2 9 6i 6i 41 i 2 1 9 12i 4 Simplify. 5 12i Write in standard form. Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.5 Complex Numbers 125 Complex Conjugates Notice in Example 2 c that the product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form a bi and a bi, called complex conjugates. a bia bi a 2 abi abi b2i 2 a2 b21 You can compare complex conjugates with the method for rationalizing denominators in Section P.2. a 2 b2 Example 3 Multiplying Conjugates Multiply each complex number by its complex conjugate. a. 1 i b. 4 3i Solution a. The complex conjugate of 1 i is 1 i. 1 i1 i 12 i 2 1 1 2 b. The complex conjugate of 4 3i is 4 3i. 4 3i 4 3i 42 3i 2 16 9i 2 16 91 25 Now try Exercise 41. Note that when you multiply the numerator and denominator of a quotient of complex numbers by c di c di you are actually multiplying the quotient by a form of 1. You are not changing the original expression, you are only creating an expression that is equivalent to the original expression. To write the quotient of a bi and c di in standard form, where c and d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain a bi a bi c di c di c di c di Example 4 ac bd bc ad i . c2 d2 Standard form Writing a Quotient of Complex Numbers in Standard Form 2 3i 2 3i 4 2i 4 2i 4 2i 4 2i Multiply numerator and denominator by complex conjugate of denominator. 8 4i 12i 6i 2 16 4i 2 Expand. 8 6 16i 16 4 i 2 1 2 16i 20 Simplify. 1 4 i 10 5 Write in standard form. Now try Exercise 53. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 126 Chapter 1 Equations, Inequalities, and Mathematical Modeling Complex Solutions of Quadratic Equations You can review the techniques for using the Quadratic Formula in Section 1.4. WARNING CAUTION The definition of principal square root uses the rule ab ab for a 0 and b 0. This rule is not valid if both a and b are negative. For example, 55 5151 When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as 3, which you know is not a real number. By factoring out i 1, you can write this number in standard form. 3 31 31 3i The number 3i is called the principal square root of 3. Principal Square Root of a Negative Number If a is a positive number, the principal square root of the negative number a is defined as a ai. Example 5 Writing Complex Numbers in Standard Form a. 312 3 i12 i 36 i 2 61 6 5i5 i b. 48 27 48i 27 i 43i 33i 3i 25i 2 c. 1 3 2 1 3i2 12 23i 3 2i 2 5i 2 5 whereas 1 23 i 31 55 25 5. 2 23 i To avoid problems with square roots of negative numbers, be sure to convert complex numbers to standard form before multiplying. Now try Exercise 63. Example 6 Complex Solutions of a Quadratic Equation Solve a x 2 4 0 and b 3x 2 2x 5 0. Solution a. x 2 4 0 Write original equation. x 4 2 Subtract 4 from each side. x 2i Extract square roots. b. 3x2 2x 5 0 2 22 435 x 23 Write original equation. Quadratic Formula 2 56 6 Simplify. 2 214i 6 Write 56 in standard form. 1 14 i 3 3 Write in standard form. Now try Exercise 69. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.5 1.5 EXERCISES Complex Numbers 127 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY 1. Match the type of complex number with its definition. a Real number i a bi, a 0, b b Imaginary number ii a bi, a 0, b c Pure imaginary number iii a bi, b 0 0 0 In Exercises 24, fill in the blanks. 2. The imaginary unit i is defined as i ________, where i 2 ________. 3. If a is a positive number, the ________ ________ root of the negative number a is defined as a a i. 4. The numbers a bi and a bi are called ________ ________, and their product is a real number a2 b2. SKILLS AND APPLICATIONS In Exercises 5 8, find real numbers a and b such that the equation is true. 5. a bi 12 7i 6. a bi 13 4i 7. a 1 b 3i 5 8i 8. a 6 2bi 6 5i In Exercises 920, write the complex number in standard form. 9. 11. 13. 15. 17. 19. 8 25 2 27 80 14 10i i 2 0.09 10. 12. 14. 16. 18. 20. 5 36 1 8 4 75 4i 2 2i 0.0049 In Exercises 2130, perform the addition or subtraction and write the result in standard form. 22. 13 2i 5 6i 7 i 3 4i 24. 3 2i 6 13i 9 i 8 i 2 8 5 50 8 18 4 32 i 28. 25 10 11i 15i 13i 14 7i 32 52i 53 11 i 3 30. 1.6 3.2i 5.8 4.3i 21. 23. 25. 26. 27. 29. 39. 2 3i2 2 3i2 In Exercises 41 48, write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. 41. 43. 45. 47. 9 2i 1 5i 20 6 32. 7 2i3 5i 1 i3 2i 34. 8i 9 4i 12i1 9i 14 10 i14 10i 3 15 i3 15i 37. 6 7i2 38. 5 4i2 31. 33. 35. 36. 42. 44. 46. 48. 8 10i 3 2i 15 1 8 In Exercises 4958, write the quotient in standard form. 49. 3 i 14 2i 13 1i 6 7i 1 2i 8 16i 2i 5i 2 3i2 50. 2 4 5i 5i 53. 5i 9 4i 55. i 3i 57. 4 5i 2 51. 52. 54. 56. 58. In Exercises 5962, perform the operation and write the result in standard form. 2 3 1i 1i i 2i 61. 3 2i 3 8i 59. In Exercises 31 40, perform the operation and write the result in standard form. 40. 1 2i2 1 2i2 2i 5 2i 2i 1i 3 62. i 4i 60. In Exercises 6368, write the complex number in standard form. 63. 6 2 65. 15 www.elsolucionario.net 2 64. 5 10 66. 75 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 128 Chapter 1 Equations, Inequalities, and Mathematical Modeling 67. 3 57 10 92. Write each of the powers of i as i, i, 1, or 1. a i 40 b i 25 c i 50 d i 67 68. 2 6 2 In Exercises 6978, use the Quadratic Formula to solve the quadratic equation. 69. 71. 73. 75. 77. x 2 2x 2 0 4x 2 16x 17 0 4x 2 16x 15 0 3 2 2 x 6x 9 0 1.4x 2 2x 10 0 70. 72. 74. 76. 78. x 2 6x 10 0 9x 2 6x 37 0 16t 2 4t 3 0 7 2 3 5 8 x 4 x 16 0 4.5x 2 3x 12 0 In Exercises 7988, simplify the complex number and write it in standard form. 79. 6i 3 i 2 81. 14i 5 3 83. 72 1 85. 3 i 87. 3i4 80. 4i 2 2i 3 82. i 3 6 84. 2 1 86. 2i 3 88. i6 89. IMPEDANCE The opposition to current in an electrical circuit is called its impedance. The impedance z in a parallel circuit with two pathways satisfies the equation 1 1 1 z z1 z 2 where z1 is the impedance in ohms of pathway 1 and z2 is the impedance of pathway 2. a The impedance of each pathway in a parallel circuit is found by adding the impedances of all components in the pathway. Use the table to find z1 and z2. b Find the impedance z. Symbol Impedance Resistor Inductor Capacitor a b c a bi ci 1 16 2 20 9 10 EXPLORATION TRUE OR FALSE In Exercises 9396, determine whether the statement is true or false. Justify your answer. 93. There is no complex number that is equal to its complex conjugate. 94. i6 is a solution of x 4 x 2 14 56. 95. i 44 i 150 i 74 i 109 i 61 1 96. The sum of two complex numbers is always a real number. 97. PATTERN RECOGNITION Complete the following. i1 i i2 1 i3 i i4 1 i5 i6 i7 i8 9 10 11 i i i i12 What pattern do you see Write a brief description of how you would find i raised to any positive integer power. 98. CAPSTONE Consider the binomials x 5 and 2x 1 and the complex numbers 1 5i and 2 i. a Find the sum of the binomials and the sum of the complex numbers. b Find the difference of the binomials and the difference of the complex numbers. c Describe the similarities and differences in your results for parts a and b . d Find the product of the binomials and the product of the complex numbers. e Explain why the products you found in part d are not related in the same way as your results in parts a and b . f Write a brief paragraph that compares operations with binomials and operations with complex numbers. 99. ERROR ANALYSIS Describe the error. 66 66 36 6 90. Cube each complex number. a 2 b 1 3 i c 1 3 i 91. Raise each complex number to the fourth power. a 2 b 2 c 2i d 2i 100. PROOF Prove that the complex conjugate of the product of two complex numbers a1 b1i and a 2 b2i is the product of their complex conjugates. 101. PROOF Prove that the complex conjugate of the sum of two complex numbers a1 b1i and a 2 b2i is the sum of their complex conjugates. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 Other Types of Equations 129 1.6 OTHER TYPES OF EQUATIONS What you should learn Polynomial Equations Solve polynomial equations of degree three or greater. Solve equations involving radicals. Solve equations involving fractions or absolute values. Use polynomial equations and equations involving radicals to model and solve real life problems. In this section you will extend the techniques for solving equations to nonlinear and nonquadratic equations. At this point in the text, you have only four basic methods for solving nonlinear equationsfactoring, extracting square roots, completing the square, and the Quadratic Formula. So the main goal of this section is to learn to rewrite nonlinear equations in a form to which you can apply one of these methods. Example 1 shows how to use factoring to solve a polynomial equation, which is an equation that can be written in the general form Why you should learn it Polynomial equations, radical equations, and absolute value equations can be used to model and solve real life problems. For instance, in Exercise 108 on page 138, a radical equation can be used to model the total monthly cost of airplane flights between Chicago and Denver. a n x n an1x n1 . . . a2x2 a1x a0 0. Example 1 Solving a Polynomial Equation by Factoring Solve 3x 4 48x 2. Solution First write the polynomial equation in general form with zero on one side, factor the other side, and then set each factor equal to zero and solve. 3x 4 48x 2 3x 4 3x 2 3x 2 x2 48x 2 0 Write in general form. 16 0 Factor out common factor. x 4x 4 0 3x 2 0 Austin BrownGetty Images Write original equation. Write in factored form. x0 Set 1st factor equal to 0. x40 x 4 Set 2nd factor equal to 0. x40 x4 Set 3rd factor equal to 0. You can check these solutions by substituting in the original equation, as follows. Check 304 480 2 344 484 2 344 484 2 0 checks. 4 checks. 4 checks. So, you can conclude that the solutions are x 0, x 4, and x 4. Now try Exercise 5. A common mistake that is made in solving an equation like that in Example 1 is to divide each side of the equation by the variable factor x 2. This loses the solution x 0. When solving an equation, always write the equation in general form, then factor the equation and set each factor equal to zero. Do not divide each side of an equation by a variable factor in an attempt to simplify the equation. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 130 Chapter 1 Equations, Inequalities, and Mathematical Modeling For a review of factoring special polynomial forms, see Section P.4. T E C H N O LO G Y Example 2 You can use a graphing utility to check graphically the solutions of the equation in Example 2. To do this, graph the equation y x3 3x 2 2 4 Solve x 3 3x 2 3x 9 0. Solution x3 3x 2 3x 9 0 Write original equation. x 3 3x 3 0 Factor by grouping. 3x 9. Then use the zero or root feature to approximate any x intercepts. As shown below, the x intercept of the graph occurs at 3, 0, confirming the real solution of x 3 found in Example 2. 3, 0 Solving a Polynomial Equation by Factoring x2 x 3x 2 3 0 x30 x 30 2 Distributive Property x3 Set 1st factor equal to 0. x 3i Set 2nd factor equal to 0. The solutions are x 3, x 3i, and x 3i. Now try Exercise 13. 9 Occasionally, mathematical models involve equations that are of quadratic type. In general, an equation is of quadratic type if it can be written in the form 14 Try using a graphing utility to check the solutions found in Example 3. au 2 bu c 0 where a 0 and u is an algebraic expression. Example 3 Solving an Equation of Quadratic Type Solve x4 3x 2 2 0. Solution This equation is of quadratic type with u x 2. x 2 2 3x 2 2 0 To solve this equation, you can factor the left side of the equation as the product of two second degree polynomials. x4 3x 2 2 0 u2 Write original equation. 3u x22 3x2 2 0 x2 1 x2 Quadratic form 2 0 Partially factor. x 1x 1x 2 2 0 Factor completely. x10 x 1 Set 1st factor equal to 0. x10 x1 Set 2nd factor equal to 0. x 2 Set 3rd factor equal to 0. x2 20 The solutions are x 1, x 1, x 2, and x 2. Check these in the original equation. Now try Exercise 17. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 Other Types of Equations 131 Equations Involving Radicals Operations such as squaring each side of an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity all can introduce extraneous solutions. So, when you use any of these operations, checking your solutions is crucial. Example 4 Solving Equations Involving Radicals a. 2x 7 x 2 Original equation 2x 7 x 2 Isolate radical. 2x 7 x 2 4x 4 Square each side. 0 x 2x 3 Write in general form. 0 x 3x 1 Factor. 2 x30 x 3 Set 1st factor equal to 0. x10 x1 Set 2nd factor equal to 0. By checking these values, you can determine that the only solution is x 1. b. 2x 5 x 3 1 Original equation 2x 5 x 3 1 Isolate 2x 5. 2x 5 x 3 2x 3 1 Square each side. 2x 5 x 2 2x 3 Combine like terms. x 3 2x 3 Isolate 2x 3. x 6x 9 4x 3 2 When an equation contains two radicals, it may not be possible to isolate both. In such cases, you may have to raise each side of the equation to a power at two different stages in the solution, as shown in Example 4 b . Square each side. x 10x 21 0 Write in general form. x 3x 7 0 Factor. 2 x30 x3 Set 1st factor equal to 0. x70 x7 Set 2nd factor equal to 0. The solutions are x 3 and x 7. Check these in the original equation. Now try Exercise 37. Example 5 Solving an Equation Involving a Rational Exponent x 423 25 Original equation x 4 25 Rewrite in radical form. x 4 15,625 Cube each side. 3 2 2 x 4 125 x 129, x 121 Take square root of each side. Add 4 to each side. Now try Exercise 51. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 132 Chapter 1 Equations, Inequalities, and Mathematical Modeling Equations with Fractions or Absolute Values To solve an equation involving fractions, multiply each side of the equation by the least common denominator LCD of all terms in the equation. This procedure will clear the equation of fractions. For instance, in the equation 1 2 2 x2 1 x x you can multiply each side of the equation by xx 2 1. Try doing this and solve the resulting equation. You should obtain one solution: x 1. Example 6 Solve Solving an Equation Involving Fractions 3 2 1. x x2 Solution For this equation, the least common denominator of the three terms is x x 2 , so you begin by multiplying each term of the equation by this expression. 2 3 1 x x2 Write original equation. 2 3 x x 2 xx 2 xx 21 x x2 2x 2 3x xx 2 Simplify. 2x 4 x 2 5x x2 Multiply each term by the LCD. Simplify. 3x 4 0 Write in general form. x 4x 1 0 Factor. x40 x4 Set 1st factor equal to 0. x10 x 1 Set 2nd factor equal to 0. Check x 4 Check x 1 3 2 1 x x2 2 3 1 x x2 2 3 1 4 42 2 3 1 1 1 2 2 1 1 1 3 1 2 2 2 2 1 1 2 2 So, the solutions are x 4 and x 1. Now try Exercise 65. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 You can review the definition of absolute value in Section P.1. Other Types of Equations 133 To solve an equation involving an absolute value, remember that the expression inside the absolute value signs can be positive or negative. This results in two separate equations, each of which must be solved. For instance, the equation x 2 3 results in the two equations x 2 3 and x 2 3, which implies that the equation has two solutions: x 5 and x 1. Example 7 Solving an Equation Involving Absolute Value Solve x 2 3x 4x 6. Solution Because the variable expression inside the absolute value signs can be positive or negative, you must solve the following two equations. First Equation x 2 3x 4x 6 Use positive expression. x2 x 6 0 Write in general form. x 3x 2 0 Factor. x30 x 3 Set 1st factor equal to 0. x20 x2 Set 2nd factor equal to 0. Second Equation x 2 3x 4x 6 x2 Use negative expression. 7x 6 0 Write in general form. x 1x 6 0 Factor. x10 x1 Set 1st factor equal to 0. x60 x6 Set 2nd factor equal to 0. Check 32 33 43 6 Substitute 3 for x. 18 18 22 32 42 6 3 checks. 2 1 31 41 6 2 does not check. 22 62 36 46 6 1 checks. 2 2 18 18 Substitute 2 for x. Substitute 1 for x. Substitute 6 for x. 6 does not check. The solutions are x 3 and x 1. Now try Exercise 73. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 134 Chapter 1 Equations, Inequalities, and Mathematical Modeling Applications It would be impossible to categorize the many different types of applications that involve nonlinear and nonquadratic models. However, from the few examples and exercises that are given, you will gain some appreciation for the variety of applications that can occur. Example 8 Reduced Rates A ski club chartered a bus for a ski trip at a cost of 480. In an attempt to lower the bus fare per skier, the club invited nonmembers to go along. After five nonmembers joined the trip, the fare per skier decreased by 4.80. How many club members are going on the trip Solution Begin the solution by creating a verbal model and assigning labels. Verbal Model: Labels: Equation: Cost per skier Number of skiers Cost of trip Cost of trip 480 Number of ski club members x Number of skiers x 5 480 Original cost per member x 480 Cost per skier 4.80 x x 480 dollars people people dollars per person dollars per person 4.80 x 5 480 480 4.8x x 5 480 x Write 480 4.8xx 5 480x Multiply each side by x. 480x 2400 24x 480x 4.8x2 4.8x2 24x 2400 0 x2 5x 500 0 x 25x 20 0 480x 4.80 as a fraction. Multiply. Subtract 480x from each side. Divide each side by 4.8. Factor. x 25 0 x 25 x 20 0 x 20 Choosing the positive value of x, you can conclude that 20 ski club members are going on the trip. Check this in the original statement of the problem, as follows. 4.8020 5 480 480 20 24 4.8025 480 480 480 Substitute 20 for x. Simplify. 20 checks. Now try Exercise 99. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 Other Types of Equations 135 Interest in a savings account is calculated by one of three basic methods: simple interest, interest compounded n times per year, and interest compounded continuously. The next example uses the formula for interest that is compounded n times per year. AP 1 r n nt In this formula, A is the balance in the account, P is the principal or original deposit , r is the annual interest rate in decimal form , n is the number of compoundings per year, and t is the time in years. In Chapter 5, you will study a derivation of the formula above for interest compounded continuously. Example 9 Compound Interest When you were born, your grandparents deposited 5000 in a long term investment in which the interest was compounded quarterly. Today, on your 25th birthday, the value of the investment is 25,062.59. What is the annual interest rate for this investment Solution r n nt Formula: AP 1 Labels: Balance A 25,062.59 Principal P 5000 Time t 25 Compoundings per year n 4 Annual interest rate r Equation: 25,062.59 5000 1 25,062.59 r 1 5000 4 5.0125 1 r 4 r 4 100 100 dollars dollars years compoundings per year percent in decimal form 425 Divide each side by 5000. Use a calculator. 5.01251100 1 r 4 Raise each side to reciprocal power. 1.01625 1 r 4 Use a calculator. 0.01625 r 4 0.065 r Subtract 1 from each side. Multiply each side by 4. The annual interest rate is about 0.065, or 6.5. Check this in the original statement of the problem. Now try Exercise 103. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 136 Chapter 1 1.6 Equations, Inequalities, and Mathematical Modeling EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The equation an x n an1 x n1 . . . a2 x 2 a1x a0 0 is a ________ equation in x written in general form. 2. Squaring each side of an equation, multiplying each side of an equation by a variable quantity, and raising each side of an equation to a rational power are all operations that can introduce ________ solutions to a given equation. 3. The equation 2x 4 x 2 1 0 is of ________ ________. 4 6 5 of fractions, multiply each side of the equation by the least common x x3 denominator ________. 4. To clear the equation SKILLS AND APPLICATIONS In Exercises 530, find all solutions of the equation. Check your solutions in the original equation. In Exercises 3558, find all solutions of the equation. Check your solutions in the original equation. 6x4 14x 2 0 x 4 81 0 x 3 512 0 5x3 30x 2 45x 0 x3 3x 2 x 3 0 x3 2x 2 3x 6 0 x4 x3 x 1 0 x4 2x 3 8x 16 0 x4 4x2 3 0 19. 4x4 65x 2 16 0 21. x6 7x3 8 0 1 8 23. 2 15 0 x x 35. 37. 39. 41. 43. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 5. 7. 9. 11. 13. 14. 15. 16. 17. 6. 8. 10. 12. 36x3 100x 0 x6 64 0 27x 3 343 0 9x4 24x3 16x 2 0 18. x4 5x 2 36 0 20. 36t 4 29t 2 7 0 22. x6 3x3 2 0 24. 1 3 2 2 x x x x 2 3x x 2 2 0 x x 5 60 26. 6 x1 x 1 2 25. 2 2 27. 2x 9x 5 28. 6x 7x 3 0 13 23 29. 3x 2x 5 30. 9t 23 24t 13 16 0 GRAPHICAL ANALYSIS In Exercises 3134, a use a graphing utility to graph the equation, b use the graph to approximate any x intercepts of the graph, c set y 0 and solve the resulting equation, and d compare the result of part c with the x intercepts of the graph. 31. 32. 33. 34. y x 3 2x 2 3x y 2x 4 15x 3 18x 2 y x 4 10x 2 9 y x 4 29x 2 100 3x 12 0 x 10 4 0 3 2x 5 3 0 36. 38. 40. 42. 44. 7x 4 0 5 x 3 0 3 3x 1 5 0 26 11x 4 x x 31 9x 5 x 1 3x 1 x 5 x 5 x x 5 1 x x 20 10 x 5 x 5 10 2x 1 2x 3 1 x 2 2x 3 1 4x 3 6x 17 3 x 532 8 x 332 8 x 323 8 x 223 9 x 2 532 27 x2 x 2232 27 3xx 112 2x 132 0 4x2x 113 6xx 143 0 GRAPHICAL ANALYSIS In Exercises 5962, a use a graphing utility to graph the equation, b use the graph to approximate any x intercepts of the graph, c set y 0 and solve the resulting equation, and d compare the result of part c with the x intercepts of the graph. 59. y 11x 30 x 60. y 2x 15 4x 61. y 7x 36 5x 16 2 4 4 62. y 3x x www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 In Exercises 6376, find all solutions of the equation. Check your solutions in the original equation. 3 1 x 2 4 5 x x 3 6 1 1 3 x x1 4 3 1 x1 x2 30 x x x 3 4x 1 x x 1 3 x2 4 x 2 x1 x1 0 3 x2 2x 5 11 3x 2 7 x x 2 x 24 x 2 6x 3x 18 x 1 x 2 5 x 15 x 2 15x 63. x 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 86. 2.4x 12.4x 0.28 0 87. 4x 23 8x13 3.6 0 88. 8.4x23 1.2x13 24 0 89. 91. 93. 95. 97. 1 4 1 x x1 9 5 78. y x x1 79. y x 1 2 80. y x 2 3 77. y In Exercises 8188, find the real solutions of the equation algebraically. Round your answers to three decimal places. 3.2x 4 1.5x 2 2.1 0 0.1x4 2.4x2 3.6 0 7.08x 6 4.15x 3 9.6 0 5.25x6 0.2x3 1.55 0 1.8x 6x 5.6 0 137 THINK ABOUT IT In Exercises 8998, find an equation that has the given solutions. There are many correct answers. GRAPHICAL ANALYSIS In Exercises 77 80, a use a graphing utility to graph the equation, b use the graph to approximate any x intercepts of the graph, c set y 0 and solve the resulting equation, and d compare the result of part c with the x intercepts of the graph. 81. 82. 83. 84. 85. Other Types of Equations 4, 7 73, 67 3, 3, 4 i, i 1, 1, i, i 90. 92. 94. 96. 98. 0, 2, 9 18, 45 27, 7 2i, 2i 4i, 4i, 6, 6 99. CHARTERING A BUS A college charters a bus for 1700 to take a group to a museum. When six more students join the trip, the cost per student drops by 7.50. How many students were in the original group 100. RENTING AN APARTMENT Three students are planning to rent an apartment for a year and share equally in the cost. By adding a fourth person, each person could save 75 a month. How much is the monthly rent 101. AIRSPEED An airline runs a commuter flight between Portland, Oregon and Seattle, Washington, which are 145 miles apart. If the average speed of the plane could be increased by 40 miles per hour, the travel time would be decreased by 12 minutes. What airspeed is required to obtain this decrease in travel time 102. AVERAGE SPEED A family drove 1080 miles to their vacation lodge. Because of increased traffic density, their average speed on the return trip was decreased by 6 miles per hour and the trip took 1 22 hours longer. Determine their average speed on the way to the lodge. 103. MUTUAL FUNDS A deposit of 2500 in a mutual fund reaches a balance of 3052.49 after 5 years. What annual interest rate on a certificate of deposit compounded monthly would yield an equivalent return 104. MUTUAL FUNDS A sales representative for a mutual funds company describes a guaranteed investment fund that the company is offering to new investors. You are told that if you deposit 10,000 in the fund you will be guaranteed a return of at least 25,000 after 20 years. Assume the interest is compounded quarterly. a What is the annual interest rate if the investment only meets the minimum guaranteed amount b After 20 years, you receive 32,000. What is the annual interest rate www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 138 Chapter 1 Equations, Inequalities, and Mathematical Modeling 105. NUMBER OF DOCTORS The number of medical doctors D in thousands in the United States from 1998 through 2006 can be modeled by D 431.61 121.8t, C 0.2x 1 8 t 16 where t represents the year, with t 8 corresponding to 1998. Source: American Medical Association a In which year did the number of medical doctors reach 875,000 b Use the model to predict when the number of medical doctors will reach 1,000,000. Is this prediction reasonable Explain. 106. VOTING POPULATION The total voting age population P in millions in the United States from 1990 through 2006 can be modeled by P 182.17 1.552t , 0 t 16 1.00 0.018t where t represents the year, with t 0 corresponding to 1990. Source: U.S. Census Bureau a In which year did the total voting age population reach 210 million b Use the model to predict when the total voting age population will reach 245 million. Is this prediction reasonable Explain. 107. SATURATED STEAM The temperature T in degrees Fahrenheit of saturated steam increases as pressure increases. This relationship is approximated by the model T 75.82 2.11x 43.51x, 5 x 40 where x is the absolute pressure in pounds per square inch . a Use the model to complete the table. Absolute pressure, x 5 10 15 20 Temperature, T Absolute pressure, x 108. AIRLINE PASSENGERS An airline offers daily flights between Chicago and Denver. The total monthly cost C in millions of dollars of these flights is where x is the number of passengers in thousands . The total cost of the flights for June is 2.5 million dollars. How many passengers flew in June 109. DEMAND The demand equation for a video game is modeled by p 40 0.01x 1 where x is the number of units demanded per day and p is the price per unit. Approximate the demand when the price is 37.55. 110. DEMAND The demand equation for a high definition television set is modeled by p 800 0.01x 1 where x is the number of units demanded per month and p is the price per unit. Approximate the demand when the price is 750. 111. BASEBALL A baseball diamond has the shape of a square in which the distance from home plate to 1 second base is approximately 1272 feet. Approximate the distance between the bases. 112. METEOROLOGY A meteorologist is positioned 100 feet from the point where a weather balloon is launched. When the balloon is at height h, the distance d in feet between the meteorologist and the balloon is d 1002 h2. a Use a graphing utility to graph the equation. Use the trace feature to approximate the value of h when d 200. b Complete the table. Use the table to approximate the value of h when d 200. h 25 30 35 40 Temperature, T b The temperature of steam at sea level is 212F. Use the table in part a to approximate the absolute pressure at this temperature. c Solve part b algebraically. d Use a graphing utility to verify your solutions for parts b and c . 160 165 170 175 180 185 d c Find h algebraically when d 200. d Compare the results of the three methods. In each case, what information did you gain that was not apparent in another solution method www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.6 113. GEOMETRY You construct a cone with a base radius of 8 inches. The lateral surface area S of the cone can be represented by the equation S 864 h2 where h is the height of the cone. a Use a graphing utility to graph the equation. Use the trace feature to approximate the value of h when S 350 square inches. b Complete the table. Use the table to approximate the value of h when S 350. h 8 9 10 11 12 13 S c Find h algebraically when S 350. d Compare the results of the three methods. In each case, what information did you gain that was not apparent in another solution method 114. LABOR Working together, two people can complete a task in 8 hours. Working alone, one person takes 2 hours longer than the other to complete the task. How long would it take for each person to complete the task 115. LABOR Working together, two people can complete a task in 12 hours. Working alone, one person takes 3 hours longer than the other to complete the task. How long would it take for each person to complete the task 116. POWER LINE A power station is on one side of 3 a river that is 4 mile wide, and a factory is 8 miles downstream on the other side of the river, as shown in the figure. It costs 24 per foot to run power lines over land and 30 per foot to run them under water. Other Types of Equations 139 In Exercises 117 and 118, solve for the indicated variable. 117. A PERSONS TANGENTIAL SPEED IN A ROTOR gR Solve for g: v s 118. INDUCTANCE 1 Q2 q Solve for Q: i LC EXPLORATION TRUE OR FALSE In Exercises 119121, determine whether the statement is true or false. Justify your answer. 119. An equation can never have more than one extraneous solution. 120. When solving an absolute value equation, you will always have to check more than one solution. 121. The equation x 10 x 10 0 has no solution. 122. CAPSTONE When solving an equation, list three operations that can introduce an extraneous solution. Write an equation that has an extraneous solution. In Exercises 123 and 124, find x such that the distance between the given points is 13. Explain your results. 123. 1, 2, x, 10 124. 8, 0, x, 5 In Exercises 125 and 126, find y such that the distance between the given points is 17. Explain your results. 125. 0, 0, 8, y 126. 8, 4, 7, y In Exercises 127 and 128, consider an equation of the form x x a b, where a and b are constants. 3 mile 4 8x x 8 miles Not drawn to scale a Write the total cost C of running power lines in terms of x see figure . b Find the total cost when x 3. c Find the length x when C 1,098,662.40. d Use a graphing utility to graph the equation from part a . e Use your graph from part d to find the value of x that minimizes the cost. 127. Find a and b when the solution of the equation is x 9. There are many correct answers. 128. WRITING Write a short paragraph listing the steps required to solve this equation involving absolute values and explain why it is important to check your solutions. In Exercises 129 and 130, consider an equation of the form x x a b, where a and b are constants. 129. Find a and b when the solution of the equation is x 20. There are many correct answers. 130. WRITING Write a short paragraph listing the steps required to solve this equation involving radicals and explain why it is important to check your solutions. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 140 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.7 LINEAR INEQUALITIES IN ONE VARIABLE What you should learn Represent solutions of linear inequalities in one variable. Use properties of inequalities to create equivalent inequalities. Solve linear inequalities in one variable. Solve inequalities involving absolute values. Use inequalities to model and solve real life problems. Why you should learn it Inequalities can be used to model and solve real life problems. For instance, in Exercise 121 on page 148, you will use a linear inequality to analyze the average salary for elementary school teachers. Introduction Simple inequalities were discussed in Section P.1. There, you used the inequality symbols , , , and to compare two numbers and to denote subsets of real numbers. For instance, the simple inequality x 3 denotes all real numbers x that are greater than or equal to 3. Now, you will expand your work with inequalities to include more involved statements such as 5x 7 3x 9 and 3 6x 1 3. As with an equation, you solve an inequality in the variable x by finding all values of x for which the inequality is true. Such values are solutions and are said to satisfy the inequality. The set of all real numbers that are solutions of an inequality is the solution set of the inequality. For instance, the solution set of Jose Luis Pelaez, Inc.Corbis x1 4 is all real numbers that are less than 3. The set of all points on the real number line that represents the solution set is the graph of the inequality. Graphs of many types of inequalities consist of intervals on the real number line. See Section P.1 to review the nine basic types of intervals on the real number line. Note that each type of interval can be classified as bounded or unbounded. Example 1 Intervals and Inequalities Write an inequality to represent each interval, and state whether the interval is bounded or unbounded. a. 3, 5 b. 3, c. 0, 2 d. , Solution a. 3, 5 corresponds to 3 x 5. b. 3, corresponds to 3 x. c. 0, 2 corresponds to 0 x 2. d. , corresponds to x . Bounded Unbounded Bounded Unbounded Now try Exercise 9. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.7 Linear Inequalities in One Variable 141 Properties of Inequalities The procedures for solving linear inequalities in one variable are much like those for solving linear equations. To isolate the variable, you can make use of the Properties of Inequalities. These properties are similar to the properties of equality, but there are two important exceptions. When each side of an inequality is multiplied or divided by a negative number, the direction of the inequality symbol must be reversed. Here is an example. 2 5 Original inequality 32 35 Multiply each side by 3 and reverse inequality. 6 15 Simplify. Notice that if the inequality was not reversed, you would obtain the false statement 6 15. Two inequalities that have the same solution set are equivalent. For instance, the inequalities x2 5 and x 3 are equivalent. To obtain the second inequality from the first, you can subtract 2 from each side of the inequality. The following list describes the operations that can be used to create equivalent inequalities. Properties of Inequalities Let a, b, c, and d be real numbers. 1. Transitive Property a b and b c a c 2. Addition of Inequalities ac bd a b and c d 3. Addition of a Constant a b ac bc 4. Multiplication by a Constant For c 0, a b ac bc For c 0, a b ac bc Reverse the inequality. Each of the properties above is true if the symbol is replaced by and the symbol is replaced by . For instance, another form of the multiplication property would be as follows. For c 0, a b ac bc For c 0, a b ac bc www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 142 Chapter 1 Equations, Inequalities, and Mathematical Modeling Solving a Linear Inequality in One Variable The simplest type of inequality is a linear inequality in one variable. For instance, 2x 3 4 is a linear inequality in x. In the following examples, pay special attention to the steps in which the inequality symbol is reversed. Remember that when you multiply or divide by a negative number, you must reverse the inequality symbol. Example 2 Solving a Linear Inequality Solve 5x 7 3x 9. Solution Checking the solution set of an inequality is not as simple as checking the solutions of an equation. You can, however, get an indication of the validity of a solution set by substituting a few convenient values of x. For instance, in Example 2, try substituting x 5 and x 10 into the original inequality. 5x 7 3x 9 Write original inequality. 2x 7 9 Subtract 3x from each side. 2x 16 Add 7 to each side. x 8 Divide each side by 2. The solution set is all real numbers that are greater than 8, which is denoted by 8, . The graph of this solution set is shown in Figure 1.24. Note that a parenthesis at 8 on the real number line indicates that 8 is not part of the solution set. x 6 7 8 9 10 Solution interval: 8, FIGURE 1.24 Now try Exercise 35. Example 3 Solving a Linear Inequality Solve 1 32 x x 4. Graphical Solution Algebraic Solution 3 3x 1 x4 2 Write original inequality. 2 3x 2x 8 Multiply each side by 2. 2 5x 8 Subtract 2x from each side. 5x 10 x 2 Use a graphing utility to graph y1 1 2 x and y2 x 4 in the same viewing window. In Figure 1.26, you can see that the graphs appear to intersect at the point 2, 2. Use the intersect feature of the graphing utility to confirm this. The graph of y1 lies above the graph of y2 to the left of their point of intersection, which implies that y1 y2 for all x 2. Subtract 2 from each side. Divide each side by 5 and reverse the inequality. The solution set is all real numbers that are less than or equal to 2, which is denoted by , 2. The graph of this solution set is shown in Figure 1.25. Note that a bracket at 2 on the real number line indicates that 2 is part of the solution set. 2 5 7 y1 = 1 32 x x 0 1 2 Solution interval: , 2 FIGURE 1.25 3 y2 = x 4 6 4 FIGURE 1.26 Now try Exercise 37. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.7 Linear Inequalities in One Variable 143 Sometimes it is possible to write two inequalities as a double inequality. For instance, you can write the two inequalities 4 5x 2 and 5x 2 7 more simply as 4 5x 2 7. Double inequality This form allows you to solve the two inequalities together, as demonstrated in Example 4. Example 4 Solving a Double Inequality To solve a double inequality, you can isolate x as the middle term. 3 6x 1 3 Original inequality 3 1 6x 1 1 3 1 Add 1 to each part. 2 6x 4 Simplify. 2 6x 4 6 6 6 Divide each part by 6. 1 2 x 3 3 Simplify. The solution set is all real numbers that are greater than or equal to 13 and less than 2 1 2 3 , which is denoted by 3 , 3 . The graph of this solution set is shown in Figure 1.27. 13 2 3 x 1 0 1 Solution interval: 13, 23 FIGURE 1.27 Now try Exercise 47. The double inequality in Example 4 could have been solved in two parts, as follows. 3 6x 1 and 6x 1 3 2 6x 6x 4 1 x 3 x 2 3 The solution set consists of all real numbers that satisfy both inequalities. In other words, the solution set is the set of all values of x for which 2 1 x . 3 3 When combining two inequalities to form a double inequality, be sure that the inequalities satisfy the Transitive Property. For instance, it is incorrect to combine the inequalities 3 x and x 1 as 3 x 1. This inequality is wrong because 3 is not less than 1. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 144 Chapter 1 Equations, Inequalities, and Mathematical Modeling Inequalities Involving Absolute Values Solving an Absolute Value Inequality T E C H N O LO G Y Let x be a variable or an algebraic expression and let a be a real number such that a 0. A graphing utility can be used to identify the solution set of the graph of an inequality. For instance, to find the solution set of x 5 2 see Example 5a , rewrite the inequality as x 5 2 0, enter 1. The solutions of x a are all values of x that lie between a and a. if and only if x a if and only if a x a. Double inequality 2. The solutions of x a are all values of x that are less than a or greater than a. Y1 abs X 5 2 and press the graph key. The graph should look like the one shown below. x a x a or x a. Compound inequality These rules are also valid if is replaced by and is replaced by . Example 5 Solving an Absolute Value Inequality 6 Solve each inequality. 1 10 a. x 5 2 b. x 3 7 Solution 4 a. x 5 2 Write original inequality. 2 x 5 2 Notice that the graph is below the x axis on the interval 3, 7. Write equivalent inequalities. 2 5 x 5 5 2 5 Add 5 to each part. 3 x 7 Simplify. The solution set is all real numbers that are greater than 3 and less than 7, which is denoted by 3, 7. The graph of this solution set is shown in Figure 1.28. b. x 3 7 Write original inequality. x 3 7 x3 7 or x 3 3 7 3 x 33 73 x 10 Note that the graph of the inequality x 5 2 can be described as all real numbers within two units of 5, as shown in Figure 1.28. Write equivalent inequalities. Subtract 3 from each side. x 4 Simplify. The solution set is all real numbers that are less than or equal to 10 or greater than or equal to 4. The interval notation for this solution set is , 10 4, . The symbol is called a union symbol and is used to denote the combining of two sets. The graph of this solution set is shown in Figure 1.29. 2 units 2 units 7 units 7 units x x 2 3 4 5 6 7 8 x 5 2: Solutions lie inside 3, 7. FIGURE 1.28 12 10 8 6 4 2 0 2 4 6 x 3 7: Solutions lie outside 10, 4. FIGURE 1.29 Now try Exercise 61. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.7 145 Linear Inequalities in One Variable Applications A problem solving plan can be used to model and solve real life problems that involve inequalities, as illustrated in Example 6. Example 6 Comparative Shopping You are choosing between two different cell phone plans. Plan A costs 49.99 per month for 500 minutes plus 0.40 for each additional minute. Plan B costs 45.99 per month for 500 minutes plus 0.45 for each additional minute. How many additional minutes must you use in one month for plan B to cost more than plan A Solution Verbal Model: Monthly cost for plan B Monthly cost for plan A Minutes used over 500 in one month m Monthly cost for plan A 0.40m 49.99 Monthly cost for plan B 0.45m 45.99 Labels: minutes dollars dollars Inequality: 0.45m 45.99 0.40m 49.99 0.05m 4 m 80 minutes Plan B costs more if you use more than 80 additional minutes in one month. Now try Exercise 111. Example 7 Accuracy of a Measurement You go to a candy store to buy chocolates that cost 9.89 per pound. The scale that is used in the store has a state seal of approval that indicates the scale is accurate to 1 within half an ounce or 32 of a pound . According to the scale, your purchase weighs one half pound and costs 4.95. How much might you have been undercharged or overcharged as a result of inaccuracy in the scale Solution Let x represent the true weight of the candy. Because the scale is accurate 1 to within half an ounce or 32 of a pound , the difference between the exact weight x and the scale weight 12 is less than or equal to 321 of a pound. That is, x 12 You can solve this inequality as follows. 1 1 32 x2 15 32 x 1 32 . 1 32 17 32 0.46875 x 0.53125 In other words, your one half pound of candy could have weighed as little as 0.46875 pound which would have cost 4.64 or as much as 0.53125 pound which would have cost 5.25 . So, you could have been overcharged by as much as 0.31 or undercharged by as much as 0.30. Now try Exercise 125. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 146 Chapter 1 1.7 Equations, Inequalities, and Mathematical Modeling EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The set of all real numbers that are solutions of an inequality is the ________ ________ of the inequality. 2. The set of all points on the real number line that represents the solution set of an inequality is the ________ of the inequality. 3. To solve a linear inequality in one variable, you can use the properties of inequalities, which are identical to those used to solve equations, with the exception of multiplying or dividing each side by a ________ number. 4. Two inequalities that have the same solution set are ________. 5. It is sometimes possible to write two inequalities as one inequality, called a ________ inequality. 6. The symbol is called a ________ symbol and is used to denote the combining of two sets. SKILLS AND APPLICATIONS In Exercises 714, a write an inequality that represents the interval and b state whether the interval is bounded or unbounded. 7. 9. 11. 13. 0, 9 1, 5 11, , 2 7, 4 2, 10 5, , 7 8. 10. 12. 14. 24. 2x 1 3 a x 4 3 2 1 0 1 2 3 4 5 b 2 3 4 5 2 1 0 1 2 3 4 5 1 2 3 4 28. 2x 3 15 x 2 1 0 1 2 3 4 5 6 f x 5 4 3 2 1 0 1 2 3 4 5 5 6 g x 3 2 1 0 1 2 3 4 h x 4 15. 17. 19. 20. 21. 22. x 5 e 3 5 x 3 3 x 4 x 3 x 4 5 1 x 2 1 x 52 6 7 16. x 5 18. 0 x 26. 5 2x 1 1 27. x 10 3 x 0 x2 2 4 6 d 1 25. 0 x 6 c 3 Inequality 23. 5x 12 0 In Exercises 1522, match the inequality with its graph. The graphs are labeled a h . 5 In Exercises 2328, determine whether each value of x is a solution of the inequality. 8 9 2 a c a c a c a c a c a c Values b x 3 x3 3 x 52 d x 2 1 x0 b x 4 3 x 4 d x 2 x4 b x 10 7 x0 d x 2 1 5 x 2 b x 2 x 43 d x 0 x 13 b x 1 x 14 d x 9 x 6 b x 0 x 12 d x 7 In Exercises 2956, solve the inequality and sketch the solution on the real number line. Some inequalities have no solutions. 10x 40 6x 15 x 7 12 3x 1 2 x 6x 4 2 8x 4x 1 2x 3 3 27 x x 2 9x 1 3416x 2 29. 31. 33. 35. 37. 39. 41. 43. 4x 12 2x 3 x5 7 2x 7 3 4x 2x 1 1 5x 4 2x 33 x 3 4x 6 x 7 1 5 2 8x 1 3x 2 45. 47. 48. 49. 50. 3.6x 11 3.4 46. 15.6 1.3x 5.2 1 2x 3 9 8 3x 5 13 8 1 3x 2 13 0 2 3x 1 20 www.elsolucionario.net 30. 32. 34. 36. 38. 40. 42. 44. http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.7 2x 3 4 3 3 1 53. x 1 4 4 51. 4 55. 3.2 0.4x 1 4.4 52. 0 x3 5 2 x 1 3 1.5x 6 56. 4.5 10.5 2 54. 1 2 In Exercises 5772, solve the inequality and sketch the solution on the real number line. Some inequalities have no solution. 57. x 5 x 59. 1 2 61. x 5 1 63. x 20 6 65. 3 4x 9 x3 67. 4 2 69. 9 2x 2 1 71. 2x 10 9 58. x 8 x 60. 3 5 6x 12 5 2x 1 4x 3 8 x x 8 14 2x 7 13 97. x 3 2 1 0 1 2 3 2 3 x 1 0 1 x 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 100. x 7 3x 1 5 20 6x 1 3x 1 x 7 12x 9 13 2 x 1 3 Inequalities y 1 b y b y y 5 0 y 3 b y 1 y 3 b y y 2 b y 2 99. GRAPHICAL ANALYSIS In Exercises 83 88, use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. Equation y 2x 3 y 23x 1 y 12x 2 y 3x 8 In Exercises 97104, use absolute value notation to define the interval or pair of intervals on the real number line. 3 62. x 7 5 64. x 8 0 66. 1 2x 5 2x 68. 1 1 3 70. x 14 3 17 72. 34 5x 9 74. 76. 78. 80. 82. 95. THINK ABOUT IT The graph of x 5 3 can be described as all real numbers within three units of 5. Give a similar description of x 10 8. 96. THINK ABOUT IT The graph of x 2 5 can be described as all real numbers more than five units from 2. Give a similar description of x 8 4. 98. GRAPHICAL ANALYSIS In Exercises 73 82, use a graphing utility to graph the inequality and identify the solution set. 73. 75. 77. 79. 81. 147 Linear Inequalities in One Variable 101. 102. 103. 104. 6 5 4 3 2 1 All real numbers within 10 units of 12 All real numbers at least five units from 8 All real numbers more than four units from 3 All real numbers no more than seven units from 6 In Exercises 105108, use inequality notation to describe the subset of real numbers. 105. A company expects its earnings per share E for the next quarter to be no less than 4.10 and no more than 4.25. 106. The estimated daily oil production p at a refinery is greater than 2 million barrels but less than 2.4 million barrels. 107. According to a survey, the percent p of U.S. citizens that now conduct most of their banking transactions online is no more than 45. 108. The net income I of a company is expected to be no less than 239 million. In Exercises 8994, find the interval s on the real number line for which the radicand is nonnegative. PHYSIOLOGY In Exercises 109 and 110, use the following information. The maximum heart rate of a person in normal health is related to the persons age by the equation r 220 A, where r is the maximum heart rate in beats per minute and A is the persons age in years. Some physiologists recommend that during physical activity a sedentary person should strive to increase his or her heart rate to at least 50 of the maximum heart rate, and a highly fit person should strive to increase his or her heart rate to at most 85 of the maximum heart rate. Source: American Heart Association 89. x 5 91. x 3 4 7 2x 93. 109. Express as an interval the range of the target heart rate for a 20 year old. 110. Express as an interval the range of the target heart rate for a 40 year old. 83. 84. 85. 86. 87. y x 3 88. y 1 2x 1 a a a a a a y 4 0 0 0 0 4 b y 1 90. x 10 92. 3 x 4 6x 15 94. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 148 Chapter 1 Equations, Inequalities, and Mathematical Modeling 111. JOB OFFERS You are considering two job offers. The first job pays 13.50 per hour. The second job pays 9.00 per hour plus 0.75 per unit produced per hour. Write an inequality yielding the number of units x that must be produced per hour to make the second job pay the greater hourly wage. Solve the inequality. 112. JOB OFFERS You are considering two job offers. The first job pays 3000 per month. The second job pays 1000 per month plus a commission of 4 of your gross sales. Write an inequality yielding the gross sales x per month for which the second job will pay the greater monthly wage. Solve the inequality. 113. INVESTMENT In order for an investment of 1000 to grow to more than 1062.50 in 2 years, what must the annual interest rate be A P1 rt 114. INVESTMENT In order for an investment of 750 to grow to more than 825 in 2 years, what must the annual interest rate be A P1 rt 115. COST, REVENUE, AND PROFIT The revenue from selling x units of a product is R 115.95x. The cost of producing x units is C 95x 750. To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit 116. COST, REVENUE, AND PROFIT The revenue from selling x units of a product is R 24.55x. The cost of producing x units is C 15.4x 150,000. To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit 117. DAILY SALES A doughnut shop sells a dozen doughnuts for 4.50. Beyond the fixed costs rent, utilities, and insurance of 220 per day, it costs 2.75 for enough materials flour, sugar, and so on and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies from 60 to 270. Between what levels in dozens do the daily sales vary 118. WEIGHT LOSS PROGRAM A person enrolls in a diet and exercise program that guarantees a loss of at 1 least 12 pounds per week. The persons weight at the beginning of the program is 164 pounds. Find the maximum number of weeks before the person attains a goal weight of 128 pounds. 119. DATA ANALYSIS: IQ SCORES AND GPA The admissions office of a college wants to determine whether there is a relationship between IQ scores x and grade point averages y after the first year of school. An equation that models the data the admissions office obtained is y 0.067x 5.638. a Use a graphing utility to graph the model. b Use the graph to estimate the values of x that predict a grade point average of at least 3.0. 120. DATA ANALYSIS: WEIGHTLIFTING You want to determine whether there is a relationship between an athletes weight x in pounds and the athletes maximum bench press weight y in pounds . The table shows a sample of data from 12 athletes. Athletes weight, x Bench press weight, y 165 184 150 210 196 240 202 170 185 190 230 160 170 185 200 255 205 295 190 175 195 185 250 155 a Use a graphing utility to plot the data. b A model for the data is y 1.3x 36. Use a graphing utility to graph the model in the same viewing window used in part a . c Use the graph to estimate the values of x that predict a maximum bench press weight of at least 200 pounds. d Verify your estimate from part c algebraically. e Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athletes weight is not a particularly good indicator of the athletes maximum bench press weight, list other factors that might influence an individuals maximum bench press weight. 121. TEACHERS SALARIES The average salaries S in thousands of dollars for elementary school teachers in the United States from 1990 through 2005 are approximated by the model S 1.09t 30.9, 0 t 15 where t represents the year, with t 0 corresponding to 1990. Source: National Education Association a According to this model, when was the average salary at least 32,500, but not more than 42,000 b According to this model, when will the average salary exceed 54,000 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.7 E 1.52t 68.0, 0 t 16 123. 124. 125. 126. 127. where t represents the year, with t 0 corresponding to 1990. Source: U.S. Department of Agriculture a According to this model, when was the annual egg production 70 billion, but no more than 80 billion b According to this model, when will the annual egg production exceed 100 billion GEOMETRY The side of a square is measured as 1 10.4 inches with a possible error of 16 inch. Using these measurements, determine the interval containing the possible areas of the square. GEOMETRY The side of a square is measured as 24.2 centimeters with a possible error of 0.25 centimeter. Using these measurements, determine the interval containing the possible areas of the square. ACCURACY OF MEASUREMENT You stop at a self service gas station to buy 15 gallons of 87 octane gasoline at 2.09 a gallon. The gas pump is accurate to 1 within 10 of a gallon. How much might you be undercharged or overcharged ACCURACY OF MEASUREMENT You buy six T bone steaks that cost 14.99 per pound. The weight that is listed on the package is 5.72 pounds. The scale 1 that weighed the package is accurate to within 2 ounce. How much might you be undercharged or overcharged TIME STUDY A time study was conducted to determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two thirds of the workers in the study satisfied the inequality t 15.6 1 1.9 where t is time in minutes. Determine the interval on the real number line in which these times lie. 128. HEIGHT The heights h of two thirds of the members of a population satisfy the inequality 130. MUSIC Michael Kasha of Florida State University used physics and mathematics to design a new classical guitar. The model he used for the frequency of the vibrations on a circular plate was v 2.6td 2E, where v is the frequency in vibrations per second , t is the plate thickness in millimeters , d is the diameter of the plate, E is the elasticity of the plate material, and is the density of the plate material. For fixed values of d, E, and , the graph of the equation is a line see figure . Frequency vibrations per second 122. EGG PRODUCTION The numbers of eggs E in billions produced in the United States from 1990 through 2006 can be modeled by v 700 600 500 400 300 200 100 t 1 2 4 a Estimate the frequency when the plate thickness is 2 millimeters. b Estimate the plate thickness when the frequency is 600 vibrations per second. c Approximate the interval for the plate thickness when the frequency is between 200 and 400 vibrations per second. d Approximate the interval for the frequency when the plate thickness is less than 3 millimeters. EXPLORATION TRUE OR FALSE In Exercises 131 and 132, determine whether the statement is true or false. Justify your answer. 131. If a, b, and c are real numbers, and a b, then ac bc. 132. If 10 x 8, then 10 x and x 8. 133. Identify the graph of the inequality x a 2. a b x a2 a 2 x a2 a+2 x 2a where h is measured in inches. Determine the interval on the real number line in which these heights lie. 129. METEOROLOGY An electronic device is to be operated in an environment with relative humidity h in the interval defined by h 50 30. What are the minimum and maximum relative humidities for the operation of this device 3 Plate thickness in millimeters c h 68.5 1 2.7 149 Linear Inequalities in One Variable 2+a a a+2 d x 2a 2 2+a 134. Find sets of values of a, b, and c such that 0 x 10 is a solution of the inequality ax b c. 135. Give an example of an inequality with an unbounded solution set. 136. CAPSTONE Describe any differences between properties of equalities and properties of inequalities. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 150 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.8 OTHER TYPES OF INEQUALITIES What you should learn Solve polynomial inequalities. Solve rational inequalities. Use inequalities to model and solve real life problems. Why you should learn it Inequalities can be used to model and solve real life problems. For instance, in Exercise 77 on page 158, a polynomial inequality is used to model school enrollment in the United States. Polynomial Inequalities To solve a polynomial inequality such as x 2 2x 3 0, you can use the fact that a polynomial can change signs only at its zeros the x values that make the polynomial equal to zero . Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real number line into intervals in which the polynomial has no sign changes. These zeros are the key numbers of the inequality, and the resulting intervals are the test intervals for the inequality. For instance, the polynomial above factors as x 2 2x 3 x 1x 3 and has two zeros, x 1 and x 3. These zeros divide the real number line into three test intervals: , 1, 1, 3, and 3, . See Figure 1.30. So, to solve the inequality 2x 3 0, you need only test one value from each of these test intervals to determine whether the value satisfies the original inequality. If so, you can conclude that the interval is a solution of the inequality. Spencer Grant PhotoEdit x2 Zero x = 1 Test Interval , 1 Zero x=3 Test Interval 1, 3 Test Interval 3, x 4 FIGURE 3 2 1 0 1 2 3 4 5 1.30 Three test intervals for x2 2x 3 You can use the same basic approach to determine the test intervals for any polynomial. Finding Test Intervals for a Polynomial To determine the intervals on which the values of a polynomial are entirely negative or entirely positive, use the following steps. 1. Find all real zeros of the polynomial, and arrange the zeros in increasing order from smallest to largest . These zeros are the key numbers of the polynomial. 2. Use the key numbers of the polynomial to determine its test intervals. 3. Choose one representative x value in each test interval and evaluate the polynomial at that value. If the value of the polynomial is negative, the polynomial will have negative values for every x value in the interval. If the value of the polynomial is positive, the polynomial will have positive values for every x value in the interval. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 Example 1 You can review the techniques for factoring polynomials in Section P.4. 151 Other Types of Inequalities Solving a Polynomial Inequality Solve x 2 x 6 0. Solution By factoring the polynomial as x 2 x 6 x 2x 3 you can see that the key numbers are x 2 and x 3. So, the polynomials test intervals are , 2, 2, 3, and 3, . Test intervals In each test interval, choose a representative x value and evaluate the polynomial. Test Interval x Value Polynomial Value Conclusion , 2 x 3 3 3 6 6 Positive 2, 3 x0 0 0 6 6 Negative 3, x4 4 4 6 6 Positive 2 2 2 From this you can conclude that the inequality is satisfied for all x values in 2, 3. This implies that the solution of the inequality x 2 x 6 0 is the interval 2, 3, as shown in Figure 1.31. Note that the original inequality contains a less than symbol. This means that the solution set does not contain the endpoints of the test interval 2, 3. Choose x = 3. x + 2 x 3 0 Choose x = 4. x + 2 x 3 0 x 6 5 4 3 2 1 0 1 2 3 4 5 6 7 Choose x = 0. x + 2 x 3 0 FIGURE 1.31 Now try Exercise 21. As with linear inequalities, you can check the reasonableness of a solution by substituting x values into the original inequality. For instance, to check the solution found in Example 1, try substituting several x values from the interval 2, 3 into the inequality y 2 1 x 4 3 1 1 2 4 5 2 3 6 7 FIGURE 1.32 y = x2 x 6 x 2 x 6 0. Regardless of which x values you choose, the inequality should be satisfied. You can also use a graph to check the result of Example 1. Sketch the graph of y x 2 x 6, as shown in Figure 1.32. Notice that the graph is below the x axis on the interval 2, 3. In Example 1, the polynomial inequality was given in general form with the polynomial on one side and zero on the other . Whenever this is not the case, you should begin the solution process by writing the inequality in general form. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 152 Chapter 1 Equations, Inequalities, and Mathematical Modeling Example 2 Solving a Polynomial Inequality Solve 2x 3 3x 2 32x 48. Solution 2x 3 3x 2 32x 48 0 Write in general form. x 4x 42x 3 0 Factor. The key numbers are x 4, x , 4, 4, , 4, and 4, . 3 2 3 2, and x 4, and the test intervals are 3 2, Test Interval x Value Polynomial Value Conclusion , 4 x 5 25 35 325 48 Negative 4, 32, 4 x0 20 30 320 48 Positive x2 22 32 322 48 Negative 4, x5 25 35 325 48 Positive 3 2 3 2 3 2 3 2 3 2 From this you can conclude that the inequality is satisfied on the open intervals 4, 32 and 4, . So, the solution set is 4, 32 4, , as shown in Figure 1.33. Choose x = 0. x 4 x + 4 2x 3 0 Choose x = 5. x 4 x + 4 2x 3 0 x 7 5 6 4 3 2 1 0 Choose x = 5. x 4 x + 4 2x 3 0 FIGURE 1 2 3 4 5 6 Choose x = 2. x 4 x + 4 2x 3 0 1.33 Now try Exercise 27. Example 3 Solving a Polynomial Inequality Solve 4x2 5x 6. Algebraic Solution 4x2 Graphical Solution 5x 6 0 x 24x 3 0 Key Numbers: x 34, Test Intervals: , Test: 34 First write the polynomial inequality 4x2 5x 6 as 4x2 5x 6 0. Then use a graphing utility to graph y 4x2 5x 6. In Figure 1.34, you can see that the graph is above the x axis when x is less than 34 or when x is greater than 2. So, you can graphically approximate the solution set to be , 34 2, . Write in general form. Factor. x2 , 34, 2, 2, 6 Is x 24x 3 0 After testing these intervals, you can see that the polynomial 4x2 5x 6 is positive on the open intervals , 34 and 2, . So, the solution set of the inequality is , 34 2, . 2 34 , 0 2, 0 3 y = 4x 2 5x 6 10 FIGURE 1.34 Now try Exercise 23. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 Other Types of Inequalities 153 You may find it easier to determine the sign of a polynomial from its factored form. For instance, in Example 3, if the test value x 1 is substituted into the factored form x 24x 3 you can see that the sign pattern of the factors is which yields a negative result. Try using the factored forms of the polynomials to determine the signs of the polynomials in the test intervals of the other examples in this section. When solving a polynomial inequality, be sure you have accounted for the particular type of inequality symbol given in the inequality. For instance, in Example 3, note that the original inequality contained a greater than symbol and the solution consisted of two open intervals. If the original inequality had been 4x 2 5x 6 the solution would have consisted of the intervals , 34 and 2, . Each of the polynomial inequalities in Examples 1, 2, and 3 has a solution set that consists of a single interval or the union of two intervals. When solving the exercises for this section, watch for unusual solution sets, as illustrated in Example 4. Example 4 Unusual Solution Sets a. The solution set of the following inequality consists of the entire set of real numbers, , . In other words, the value of the quadratic x 2 2x 4 is positive for every real value of x. x 2 2x 4 0 b. The solution set of the following inequality consists of the single real number 1, because the quadratic x 2 2x 1 has only one key number, x 1, and it is the only value that satisfies the inequality. x 2 2x 1 0 c. The solution set of the following inequality is empty. In other words, the quadratic x2 3x 5 is not less than zero for any value of x. x 2 3x 5 0 d. The solution set of the following inequality consists of all real numbers except x 2. In interval notation, this solution set can be written as , 2 2, . x 2 4x 4 0 Now try Exercise 29. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 154 Chapter 1 Equations, Inequalities, and Mathematical Modeling Rational Inequalities The concepts of key numbers and test intervals can be extended to rational inequalities. To do this, use the fact that the value of a rational expression can change sign only at its zeros the x values for which its numerator is zero and its undefined values the x values for which its denominator is zero . These two types of numbers make up the key numbers of a rational inequality. When solving a rational inequality, begin by writing the inequality in general form with the rational expression on the left and zero on the right. Example 5 In Example 5, if you write 3 as 3 1 , you should be able to see that the LCD least common denominator is x 51 x 5. So, you can rewrite the general form as Solve 2x 7 3. x5 Solution 2x 7 3 x5 2x 7 3x 5 0, x5 x5 which simplifies as shown. Solving a Rational Inequality Write original inequality. 2x 7 3 0 x5 Write in general form. 2x 7 3x 15 0 x5 Find the LCD and subtract fractions. x 8 0 x5 Simplify. Key Numbers: x 5, x 8 Zeros and undefined values of rational expression Test Intervals: , 5, 5, 8, 8, Test: Is x 8 0 x5 After testing these intervals, as shown in Figure 1.35, you can see that the inequality is x 8 satisfied on the open intervals , 5 and 8, . Moreover, because 0 x5 when x 8, you can conclude that the solution set consists of all real numbers in the intervals , 5 8, . Be sure to use a closed interval to indicate that x can equal 8. Choose x = 6. x + 8 0 x5 x 4 5 6 Choose x = 4. x + 8 0 x5 FIGURE 7 8 9 Choose x = 9. x + 8 0 x5 1.35 Now try Exercise 45. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 Other Types of Inequalities 155 Applications One common application of inequalities comes from business and involves profit, revenue, and cost. The formula that relates these three quantities is Profit Revenue Cost P R C. Example 6 The marketing department of a calculator manufacturer has determined that the demand for a new model of calculator is Calculators Revenue in millions of dollars R p 100 0.00001x, 0 x 10,000,000 250 Demand equation where p is the price per calculator in dollars and x represents the number of calculators sold. If this model is accurate, no one would be willing to pay 100 for the calculator. At the other extreme, the company couldnt sell more than 10 million calculators. The revenue for selling x calculators is 200 150 100 R xp x 100 0.00001x 50 x 0 2 6 4 8 Revenue equation as shown in Figure 1.36. The total cost of producing x calculators is 10 per calculator plus a development cost of 2,500,000. So, the total cost is C 10x 2,500,000. 10 Number of units sold in millions FIGURE Increasing the Profit for a Product Cost equation What price should the company charge per calculator to obtain a profit of at least 190,000,000 1.36 Solution Verbal Model: Profit Revenue Cost Equation: P R C P 100x 0.00001x 2 10x 2,500,000 P 0.00001x 2 90x 2,500,000 Calculators Profit in millions of dollars P To answer the question, solve the inequality P 190,000,000 200 0.00001x 2 90x 2,500,000 190,000,000. 150 100 When you write the inequality in general form, find the key numbers and the test intervals, and then test a value in each test interval, you can find the solution to be 50 x 0 50 as shown in Figure 1.37. Substituting the x values in the original price equation shows that prices of 100 0 2 4 6 8 Number of units sold in millions FIGURE 1.37 3,500,000 x 5,500,000 10 45.00 p 65.00 will yield a profit of at least 190,000,000. Now try Exercise 75. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 156 Chapter 1 Equations, Inequalities, and Mathematical Modeling Another common application of inequalities is finding the domain of an expression that involves a square root, as shown in Example 7. Example 7 Finding the Domain of an Expression Find the domain of 64 4x 2. Algebraic Solution Graphical Solution Remember that the domain of an expression is the set of all x values for which the expression is defined. Because 64 4x 2 is defined has real values only if 64 4x 2 is nonnegative, the domain is given by 64 4x 2 0. Begin by sketching the graph of the equation y 64 4x2, as shown in Figure 1.38. From the graph, you can determine that the x values extend from 4 to 4 including 4 and 4 . So, the domain of the expression 64 4x2 is the interval 4, 4. 64 4x 2 0 Write in general form. 16 x 0 Divide each side by 4. 2 4 x4 x 0 y Write in factored form. 10 So, the inequality has two key numbers: x 4 and x 4. You can use these two numbers to test the inequality, as follows. 6 Key numbers: x 4, x 4 4 Test intervals: , 4, 4, 4, 4, 2 For what values of x is 64 Test: y = 64 4x 2 4x2 0 A test shows that the inequality is satisfied in the closed interval 4, 4. So, the domain of the expression 64 4x 2 is the interval 4, 4. x 6 4 FIGURE 2 2 4 6 2 1.38 Now try Exercise 59. Complex Number 4 FIGURE 1.39 Nonnegative Radicand Complex Number 4 To analyze a test interval, choose a representative x value in the interval and evaluate the expression at that value. For instance, in Example 7, if you substitute any number from the interval 4, 4 into the expression 64 4x2, you will obtain a nonnegative number under the radical symbol that simplifies to a real number. If you substitute any number from the intervals , 4 and 4, , you will obtain a complex number. It might be helpful to draw a visual representation of the intervals, as shown in Figure 1.39. CLASSROOM DISCUSSION Profit Analysis Consider the relationship PRC described on page 155. Write a paragraph discussing why it might be beneficial to solve P 0 if you owned a business. Use the situation described in Example 6 to illustrate your reasoning. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 1.8 EXERCISES 157 Other Types of Inequalities See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. Between two consecutive zeros, a polynomial must be entirely ________ or entirely ________. 2. To solve a polynomial inequality, find the ________ numbers of the polynomial, and use these numbers to create ________ ________ for the inequality. 3. The key numbers of a rational expression are its ________ and its ________ ________. 4. The formula that relates cost, revenue, and profit is ________. SKILLS AND APPLICATIONS In Exercises 58, determine whether each value of x is a solution of the inequality. Inequality 2 5. x 3 0 6. x 2 x 12 0 7. 8. x2 3 x4 3x2 1 x2 4 a c a c Values x3 b x 32 d x5 b x 4 d x0 x 5 x0 x 3 a x 5 c x 92 b x 4 d x 92 a x 2 c x 0 b x 1 d x 3 In Exercises 912, find the key numbers of the expression. 9. 3x 2 x 2 1 1 11. x5 10. 9x3 25x 2 x 2 12. x2 x1 In Exercises 3136, solve the inequality and write the solution set in interval notation. 31. 4x 3 6x 2 0 33. x3 4x 0 35. x 12x 23 0 GRAPHICAL ANALYSIS In Exercises 3740, use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. 37. 38. 39. 40. 13. 15. 17. 19. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. x2 9 14. 16. 18. 20. 43. x 2 16 x 22 25 x 32 1 2 x 4x 4 9 x 2 6x 9 16 x2 x 6 x 2 2x 3 2 2x 3 0 x x 2 2x 8 3x2 11x 20 2x 2 6x 15 0 x2 3x 18 0 x 3 2x 2 4x 8 0 x 3 3x 2 x 3 2x 3 13x 2 8x 46 6 4x 2 4x 1 0 x2 3x 8 0 y y y y Equation x 2 2x 3 12x 2 2x 1 18x 3 12x x 3 x 2 16x 16 a a a a y y y y Inequalities b y 0 b y 0 b y 0 b y 0 3 7 6 36 In Exercises 4154, solve the inequality and graph the solution on the real number line. 41. In Exercises 1330, solve the inequality and graph the solution on the real number line. 32. 4x 3 12x 2 0 34. 2x 3 x 4 0 36. x 4x 3 0 45. 47. 49. 51. 52. 53. 54. 4x 1 0 x 3x 5 0 x5 x6 2 0 x1 2 1 x5 x3 9 1 x3 4x 3 x2 2x 0 x2 9 x2 x 6 0 x 3 2x 1 x1 x1 3x x 3 x1 x4 www.elsolucionario.net 42. 44. 46. 48. 50. x2 1 0 x 5 7x 4 1 2x x 12 3 0 x2 3 5 x6 x2 1 1 x x3 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 158 Chapter 1 Equations, Inequalities, and Mathematical Modeling GRAPHICAL ANALYSIS In Exercises 5558, use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. 55. 56. 57. 58. Equation 3x y x2 2x 2 y x1 2x 2 y 2 x 4 5x y 2 x 4 Inequalities a y 0 b y 6 a y 0 b y 8 a y 1 b y 2 a y 1 b y 0 In Exercises 5964, find the domain of x in the expression. Use a graphing utility to verify your result. 59. 4 x 2 61. x 2 9x 20 63. x 2 x 2x 35 60. x 2 4 62. 81 4x 2 x 64. x2 9 74. GEOMETRY A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie 75. COST, REVENUE, AND PROFIT The revenue and cost equations for a product are R x75 0.0005x and C 30x 250,000, where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to obtain a profit of at least 750,000 What is the price per unit 76. COST, REVENUE, AND PROFIT The revenue and cost equations for a product are R x50 0.0002x and C 12x 150,000 where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to obtain a profit of at least 1,650,000 What is the price per unit 77. SCHOOL ENROLLMENT The numbers N in millions of students enrolled in schools in the United States from 1995 through 2006 are shown in the table. Source: U.S. Census Bureau In Exercises 6570, solve the inequality. Round your answers to two decimal places. 0.4x 2 5.26 10.2 1.3x 2 3.78 2.12 0.5x 2 12.5x 1.6 0 1.2x 2 4.8x 3.1 5.3 1 2 69. 70. 3.4 5.8 2.3x 5.2 3.1x 3.7 65. 66. 67. 68. HEIGHT OF A PROJECTILE In Exercises 71 and 72, use the position equation s 16t2 v0t s0, where s represents the height of an object in feet , v0 represents the initial velocity of the object in feet per second , s0 represents the initial height of the object in feet , and t represents the time in seconds . 71. A projectile is fired straight upward from ground level s0 0 with an initial velocity of 160 feet per second. a At what instant will it be back at ground level b When will the height exceed 384 feet 72. A projectile is fired straight upward from ground level s0 0 with an initial velocity of 128 feet per second. a At what instant will it be back at ground level b When will the height be less than 128 feet 73. GEOMETRY A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie Year Number, N 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 69.8 70.3 72.0 72.1 72.4 72.2 73.1 74.0 74.9 75.5 75.8 75.2 a Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t 5 corresponding to 1995. b Use the regression feature of a graphing utility to find a quartic model for the data. c Graph the model and the scatter plot in the same viewing window. How well does the model fit the data d According to the model, during what range of years will the number of students enrolled in schools exceed 74 million e Is the model valid for long term predictions of student enrollment in schools Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 78. SAFE LOAD The maximum safe load uniformly distributed over a one foot section of a two inch wide wooden beam is approximated by the model Load 168.5d 2 472.1, where d is the depth of the beam. a Evaluate the model for d 4, d 6, d 8, d 10, and d 12. Use the results to create a bar graph. b Determine the minimum depth of the beam that will safely support a load of 2000 pounds. 79. RESISTORS When two resistors of resistances R1 and R2 are connected in parallel see figure , the total resistance R satisfies the equation 1 1 1 . R R1 R2 Find R1 for a parallel circuit in which R2 2 ohms and R must be at least 1 ohm. + _ E R1 R2 80. TEACHERS SALARIES The mean salaries S in thousands of dollars of classroom teachers in the United States from 2000 through 2007 are shown in the table. Year Salary, S 2000 2001 2002 2003 2004 2005 2006 2007 42.2 43.7 43.8 45.0 45.6 45.9 48.2 49.3 159 Other Types of Inequalities c According to the model, in what year will the salary for classroom teachers exceed 60,000 d Is the model valid for long term predictions of classroom teacher salaries Explain. EXPLORATION TRUE OR FALSE In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. 81. The zeros of the polynomial x 3 2x 2 11x 12 0 divide the real number line into four test intervals. 3 82. The solution set of the inequality 2x 2 3x 6 0 is the entire set of real numbers. In Exercises 8386, a find the interval s for b such that the equation has at least one real solution and b write a conjecture about the interval s based on the values of the coefficients. 83. x 2 bx 4 0 85. 3x 2 bx 10 0 84. x 2 bx 4 0 86. 2x 2 bx 5 0 87. GRAPHICAL ANALYSIS You can use a graphing utility to verify the results in Example 4. For instance, the graph of y x 2 2x 4 is shown below. Notice that the y values are greater than 0 for all values of x, as stated in Example 4 a . Use the graphing utility to graph y x 2 2x 1, y x 2 3x 5, and y x 2 4x 4. Explain how you can use the graphs to verify the results of parts b , c , and d of Example 4. 10 9 9 2 88. CAPSTONE Consider the polynomial A model that approximates these data is given by x ax b 42.6 1.95t S 1 0.06t and the real number line shown below. x where t represents the year, with t 0 corresponding to 2000. Source: Educational Research Service, Arlington, VA a Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. b How well does the model fit the data Explain. a b a Identify the points on the line at which the polynomial is zero. b In each of the three subintervals of the line, write the sign of each factor and the sign of the product. c At what x values does the polynomial change signs www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 160 Chapter 1 Equations, Inequalities, and Mathematical Modeling Section 1.4 Section 1.3 Section 1.2 Section 1.1 1 CHAPTER SUMMARY What Did You Learn ExplanationExamples Review Exercises Sketch graphs of equations p. 76 , and find x and y intercepts of graphs of equations p. 79 . To graph an equation, make a table of values, plot the points, and connect the points with a smooth curve or line. The points at which a graph intersects or touches the x or y axis are called intercepts. 1 4 Use symmetry to sketch graphs of equations p. 80 . Graphs can have symmetry with respect to one of the coordinate axes or with respect to the origin. You can test for symmetry algebraically and graphically. 512 Find equations of and sketch graphs of circles p. 82 . The point x, y lies on the circle of radius r and center h, k if and only if x h2 y k2 r2. 1318 Use graphs of equations in solving real life problems p. 83 . The graph of an equation can be used to estimate the recommended weight for a man. See Example 9. 19, 20 Identify different types of equations p. 87 . Identity: true for every real number in the domain Conditional equation: true for just some or even none of the real numbers in the domain 2124 Solve linear equations in one variable p. 87 , and solve equations that lead to linear equations p. 90 . Linear equation in one variable: An equation that can be written in the standard form ax b 0, where a and b are real numbers with a 0. 2530 Find x and y intercepts algebraically p. 91 . To find x intercepts, set y equal to zero and solve for x. To find y intercepts, set x equal to zero and solve for y. 3136 Use linear equations to model and solve real life problems p. 91 . A linear equation can be used to model the number of female participants in athletic programs. See Example 5. 37, 38 Use a verbal model in a problemsolving plan p. 96 . Verbal Description 39, 40 Use mathematical models to solve real life problems p. 97 . Mathematical models can be used to find the percent of a raise, and a buildings height. See Examples 2 and 6. 41, 42 Solve mixture problems p. 100 . Mixture problems include simple interest problems and inventory problems. See Examples 7 and 8. 43, 44 Use common formulas to solve real life problems p. 101 . A literal equation contains more than one variable. A formula is an example of a literal equation. See Example 9. 45, 46 Solve quadratic equations by factoring p. 107 . The method of factoring is based on the Zero Factor Property, which states if ab 0, then a 0 or b 0. 47, 48 Solve quadratic equations by extracting square roots p. 108 . The equation u2 d, where d 0, has exactly two solutions: u d and u d. 4952 Solve quadratic equations by completing the square p. 109 and using the Quadratic Formula p. 111 . To complete the square for x2 bx, add b22. 5356 To solve an equation involving fractional expressions, find the LCD of all terms and multiply every term by the LCD. Verbal Model Quadratic Formula: x b Algebraic Equation b2 www.elsolucionario.net 4ac 2a http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 1.8 Section 1.7 Section 1.6 Section 1.5 Section 1.4 Chapter Summary 161 What Did You Learn ExplanationExamples Use quadratic equations to model and solve real life problems p. 113 . A quadratic equation can be used to model the number of Internet users in the United States from 2000 through 2008. See Example 9. 57, 58 Use the imaginary unit i to write complex numbers p. 122 , and add, subtract, and multiply complex numbers p. 123 . If a and b are real numbers, a bi is a complex number. Sum: a bi c di a c b di Difference: a bi c di a c b di The Distributive Property can be used to multiply. 5966 Use complex conjugates to write the quotient of two complex numbers in standard form p. 125 . To write a bic di in standard form, multiply the numerator and denominator by the complex conjugate of the denominator, c di. 6770 Find complex solutions of quadratic equations p. 126 . If a is a positive number, the principal square root of the negative number a is defined as a ai. 7174 Solve polynomial equations of degree three or greater p. 129 . Factoring is the most common method used to solve polynomial equations of degree three or greater. 7578 Solve equations involving radicals p. 131 . Solving equations involving radicals usually involves squaring or cubing each side of the equation. 7982 Solve equations involving fractions or absolute values p. 132 . To solve an equation involving fractions, multiply each side of the equation by the LCD of all terms in the equation. To solve an equation involving an absolute value, remember that the expression inside the absolute value signs can be positive or negative. 83 88 Use polynomial equations and equations involving radicals to model and solve real life problems p. 134 . Polynomial equations can be used to find the number of ski club members going on a ski trip, and the annual interest rate for an investment. See Examples 8 and 9. 89, 90 Represent solutions of linear inequalities in one variable p. 140 . Bounded 1, 2 1 x 2 4, 5 4 x 5 9194 Use properties of inequalities to create equivalent inequalities p. 141 and solve linear inequalities in one variable p. 142 . Solving linear inequalities is similar to solving linear equations. Use the Properties of Inequalities to isolate the variable. Just remember to reverse the inequality symbol when you multiply or divide by a negative number. 9598 Solve inequalities involving absolute values p. 144 . Let x be a variable or an algebraic expression and let a be a real number such that a 0. 1. Solutions of x a: All values of x that lie between a and a; x a if and only if a x a. 2. Solutions of x a: All values of x that are less than a or greater than a; x a if and only if x a or x a. 99, 100 Use inequalities to model and solve real life problems p. 145 . An inequality can be used to determine the accuracy of a measurement. See Example 7. 101, 102 Solve polynomial p. 150 and rational inequalities p. 154 . Use the concepts of key numbers and test intervals to solve both polynomial and rational inequalities. 103108 Use inequalities to model and solve real life problems p. 155 . A common application of inequalities involves profit P, revenue R, and cost C. See Example 6. 109, 110 www.elsolucionario.net Review Exercises Unbounded 3, 3 x , x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 162 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1 REVIEW EXERCISES 1.1 In Exercises 1 and 2, complete a table of values. Use the resulting solution points to sketch the graph of the equation. 1. y 4x 1 See www.CalcChat.com for worked out solutions to odd numbered exercises. 20. PHYSICS The force F in pounds required to stretch a spring x inches from its natural length see figure is 5 F x, 0 x 20. 4 2. y x2 2x In Exercises 3 and 4, graphically estimate the x and y intercepts of the graph. 4. y x 1 3 3. y x 32 4 y y 2 Natural length 6 4 2 6 4 2 2 4 6 8 F x 4 x x in. 2 4 6 4 6 4 a Use the model to complete the table. x In Exercises 512, use the algebraic tests to check for symmetry with respect to both axes and the origin. Then sketch the graph of the equation. 5. 7. 9. 11. y 4x 1 y 7 x2 y x3 3 y x 2 6. 8. 10. 12. y 5x 6 y x2 2 y 6 x 3 y x 9 4 8 12 16 20 Force, F b Sketch a graph of the model. c Use the graph to estimate the force necessary to stretch the spring 10 inches. 1.2 In Exercises 2124, determine whether the equation is an identity or a conditional equation. In Exercises 1316, find the center and radius of the circle and sketch its graph. 13. x 2 y 2 9 15. x 22 y 2 16 0 14. x 2 y 2 4 16. x 2 y 82 81 17. Find the standard form of the equation of the circle for which the endpoints of a diameter are 0, 0 and 4, 6. 18. Find the standard form of the equation of the circle for which the endpoints of a diameter are 2, 3 and 4, 10. 19. REVENUE The revenue R in billions of dollars for Target for the years 1998 through 2007 can be approximated by the model R 0.123t 2 0.43t 20.0, 8 t 17 where t represents the year, with t 8 corresponding to 1998. Source: Target Corp. a Sketch a graph of the model. b Use the graph to estimate the year in which the revenue was 50 billion dollars. 21. 22. 23. 24. 6 x 22 2 4x x 2 3x 2 2x 2x 3 x 3 x7 x 3 xx 2 x 7x 1 4 3x 2 4x 8 10x 2 3x 2 6 In Exercises 2530, solve the equation if possible and check your solution. 8x 5 3x 20 25. 26. 27. 28. 7x 3 3x 17 2x 5 7 3x 2 29. x x 3 1 5 3 3x 3 51 x 1 30. 4x 3 x x2 6 4 In Exercises 3136, find the x and y intercepts of the graph of the equation algebraically. 31. y 3x 1 33. y 2x 4 1 2 35. y 2 x 3 www.elsolucionario.net 32. y 5x 6 34. y 47x 1 3 1 36. y 4 x 4 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Review Exercises 37. GEOMETRY The surface area S of the cylinder shown in the figure is approximated by the equation S 23.1432 23.143h. The surface area is 244.92 square inches. Find the height h of the cylinder. 3 in. h 38. TEMPERATURE The Fahrenheit and Celsius temperature scales are related by the equation 5 160 C F . 9 9 Find the Fahrenheit temperature that corresponds to 100 Celsius. 1.3 39. PROFIT In October, a greeting card companys total profit was 12 more than it was in September. The total profit for the two months was 689,000. Write a verbal model, assign labels, and write an algebraic equation to find the profit for each month. 40. DISCOUNT The price of a digital camera has been discounted 85. The sale price is 340. Write a verbal model, assign labels, and write an algebraic equation to find the percent discount. 41. BUSINESS VENTURE You are planning to start a small business that will require an investment of 90,000. You have found some people who are willing to share equally in the venture. If you can find three more people, each persons share will decrease by 2500. How many people have you found so far 42. AVERAGE SPEED You commute 56 miles one way to work. The trip to work takes 10 minutes longer than the trip home. Your average speed on the trip home is 8 miles per hour faster. What is your average speed on the trip home 43. MIXTURE PROBLEM A car radiator contains 10 liters of a 30 antifreeze solution. How many liters will have to be replaced with pure antifreeze if the resulting solution is to be 50 antifreeze 1 1 44. INVESTMENT You invested 6000 at 42 and 52 simple interest. During the first year, the two accounts earned 305. How much did you invest in each fund 1 Note: The 52 account is more risky. 49. 51. 53. 55. 6 3x 2 x 132 25 x 2 12x 25 2x2 5x 27 0 50. 52. 54. 56. 16x 2 25 x 52 30 9x2 12x 14 20 3x 3x2 0 57. SIMPLY SUPPORTED BEAM A simply supported 20 foot beam supports a uniformly distributed load of 1000 pounds per foot. The bending moment M in footpounds x feet from one end of the beam is given by M 500x20 x. a Where is the bending moment zero b Use a graphing utility to graph the equation. c Use the graph to determine the point on the beam where the bending moment is the greatest. 58. SPORTS You throw a softball straight up into the air at a velocity of 30 feet per second. You release the softball at a height of 5.8 feet and catch it when it falls back to a height of 6.2 feet. a Use the position equation to write a mathematical model for the height of the softball. b What is the height of the softball after 1 second c How many seconds is the softball in the air 1.5 In Exercises 5962, write the complex number in standard form. 59. 4 9 61. i 2 3i 60. 3 16 62. 5i i 2 In Exercises 6366, perform the operation and write the result in standard form. 63. 7 5i 4 2i 2 2 2 2 64. i i 2 2 2 2 65. 6i5 2i 66. 1 6i5 2i In Exercises 67 and 68, write the quotient in standard form. 67. 6 5i i 68. 3 2i 5i In Exercises 69 and 70, perform the operation and write the result in standard form. 4 2 2 3i 1 i 1 5 2 i 1 4i In Exercises 45 and 46, solve for the indicated variable. 69. 45. Volume of a Cone 46. Kinetic Energy 1 1 Solve for h : V 3 r 2 h Solve for m: E 2 mv 2 In Exercises 7174, find all solutions of the equation. 1.4 In Exercises 4756, use any method to solve the quadratic equation. 71. 3x 2 1 0 73. x 2 2x 10 0 47. 15 x 2x 2 0 163 70. 72. 2 8x2 0 74. 6x 2 3x 27 0 48. 2x 2 x 28 0 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 164 Chapter 1 Equations, Inequalities, and Mathematical Modeling 1.6 In Exercises 75 88, find all solutions of the equation. Check your solutions in the original equation. 75. 77. 78. 79. 80. 81. 5x 4 12x 3 0 76. 4x 3 6x 2 0 x 4 5x 2 6 0 9x 4 27x 3 4x 2 12x 0 2x 3 x 2 2 5x x 1 6 x 123 25 0 82. x 234 27 83. 5 3 1 x x2 85. x 5 10 87. x 2 3 2x 84. 6 8 3 x x5 86. 2x 3 7 88. x 2 6 x 89. DEMAND The demand equation for a hair dryer is p 42 0.001x 2, where x is the number of units demanded per day and p is the price per unit. Find the demand if the price is set at 29.95. 90. DATA ANALYSIS: NEWSPAPERS The total numbers N of daily evening newspapers in the United States from 1970 through 2005 can be approximated by the model N 1465 4.2t 32, 0 t 35, where t represents the year, with t 0 corresponding to 1970. The actual numbers of newspapers for selected years are shown in the table. Source: Editor Publisher Co. Year Newspapers, N 1970 1975 1980 1985 1990 1995 2000 2005 1429 1436 1388 1220 1084 891 727 645 1.7 In Exercises 9194, write an inequality that represents the interval and state whether the interval is bounded or unbounded. 92. 4, 94. 2, 2 95. 96. 97. 99. 3x 2 7 2x 5 2x 7 4 5x 3 98. 123 x 132 3x 45 2x 128 x 100. x 32 32 x 3 4 101. GEOMETRY The side of a square is measured as 19.3 centimeters with a possible error of 0.5 centimeter. Using these measurements, determine the interval containing the area of the square. 102. COST, REVENUE, AND PROFIT The revenue for selling x units of a product is R 125.33x. The cost of producing x units is C 92x 1200. To obtain a profit, the revenue must be greater than the cost. Determine the smallest value of x for which this product returns a profit. 1.8 In Exercises 103108, solve the inequality. 103. x 2 6x 27 0 105. 6x 2 5x 4 107. 2 3 x1 x1 104. x 2 2x 3 106. 2x 2 x 15 x5 108. 0 3x 109. INVESTMENT P dollars invested at interest rate r compounded annually increases to an amount A P1 r2 in 2 years. An investment of 5000 is to increase to an amount greater than 5500 in 2 years. The interest rate must be greater than what percent 110. POPULATION OF A SPECIES A biologist introduces 200 ladybugs into a crop field. The population P of the ladybugs is approximated by the model P 10001 3t5 t, where t is the time in days. Find the time required for the population to increase to at least 2000 ladybugs. EXPLORATION a Use a graphing utility to plot the data and graph the model in the same viewing window. How well does the model fit the data b Use the graph in part a to estimate the year in which there were 800 daily evening newspapers. c Use the model to verify algebraically the estimate from part b . 91. 7, 2 93. , 10 In Exercises 95100, solve the inequality. TRUE OR FALSE In Exercises 111 and 112, determine whether the statement is true or false. Justify your answer. 111. 182 182 112. The equation 325x 2 717x 398 0 has no solution. 113. WRITING Explain why it is essential to check your solutions to radical, absolute value, and rational equations. 114. ERROR ANALYSIS What is wrong with the following solution 11x 4 26 11x 4 26 11x 22 x 2 www.elsolucionario.net or 11x 4 26 11x 22 x 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Test 1 CHAPTER TEST 165 See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 16, check for symmetry with respect to both axes and the origin. Then sketch the graph of the equation. Identify any x and y intercepts. 1. y 4 34x 4. y x x 3 2. y 4 34x 5. y 5 x 3. y 4 x 22 6. x 32 y 2 9 In Exercises 712, solve the equation if possible . 7. 23x 1 14x 10 x2 4 9. 40 x2 x2 11. 2x 2x 1 1 8. x 4x 2 7 10. x 4 x 2 6 0 12. 3x 1 7 In Exercises 1316, solve the inequality. Sketch the solution set on the real number line. 13. 3 2x 4 14 15. 2x 2 5x 12 2 5 x x6 16. 3x 5 10 14. 17. Perform each operation and write the result in standard form. a 10i 3 25 b 1 5i1 5i 5 . 2i 19. The sales y in billions of dollars for Dell, Inc. from 1999 through 2008 can be approximated by the model 18. Write the quotient in standard form: y 4.41t 14.6, 9 t 18 where t represents the year, with t 9 corresponding to 1999. Source: Dell, Inc. a Sketch a graph of the model. b Assuming that the pattern continues, use the graph in part a to estimate the sales in 2013. b a 40 FIGURE FOR 22 c Use the model to verify algebraically the estimate from part b . 20. A basketball has a volume of about 455.9 cubic inches. Find the radius of the basketball accurate to three decimal places . 21. On the first part of a 350 kilometer trip, a salesperson travels 2 hours and 15 minutes at an average speed of 100 kilometers per hour. The salesperson needs to arrive at the destination in another hour and 20 minutes. Find the average speed required for the remainder of the trip. 22. The area of the ellipse in the figure at the left is A ab. If a and b satisfy the constraint a b 100, find a and b such that the area of the ellipse equals the area of the circle. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS Conditional Statements Many theorems are written in the if then form if p, then q, which is denoted by pq Conditional statement where p is the hypothesis and q is the conclusion. Here are some other ways to express the conditional statement p q. p implies q. p, only if q. p is sufficient for q. Conditional statements can be either true or false. The conditional statement p q is false only when p is true and q is false. To show that a conditional statement is true, you must prove that the conclusion follows for all cases that fulfill the hypothesis. To show that a conditional statement is false, you need to describe only a single counterexample that shows that the statement is not always true. For instance, x 4 is a counterexample that shows that the following statement is false. If x2 16, then x 4. The hypothesis x2 16 is true because 42 16. However, the conclusion x 4 is false. This implies that the given conditional statement is false. For the conditional statement p q, there are three important associated conditional statements. 1. The converse of p q: q p 2. The inverse of p q: p q 3. The contrapositive of p q: q p The symbol means the negation of a statement. For instance, the negation of The engine is running is The engine is not running. Example Writing the Converse, Inverse, and Contrapositive Write the converse, inverse, and contrapositive of the conditional statement If I get a B on my test, then I will pass the course. Solution a. Converse: If I pass the course, then I got a B on my test. b. Inverse: If I do not get a B on my test, then I will not pass the course. c. Contrapositive: If I do not pass the course, then I did not get a B on my test. In the example above, notice that neither the converse nor the inverse is logically equivalent to the original conditional statement. On the other hand, the contrapositive is logically equivalent to the original conditional statement. 166 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROBLEM SOLVING This collection of thought provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Let x represent the time in seconds and let y represent the distance in feet between you and a tree. Sketch a possible graph that shows how x and y are related if you are walking toward the tree. 2. a Find the following sums 12345 12345678 123456 7 8 9 10 b Use the following formula for the sum of the first n natural numbers to verify your answers to part a . 1 1 2 3 . . . n nn 1 2 c Use the formula in part b to find n if the sum of the first n natural numbers is 210. 3. The area of an ellipse is given by A ab see figure . For a certain ellipse, it is required that a b 20. a Show that A a20 a. b Complete the table. 7 a A two story library is designed. Buildings this tall are often required to withstand wind pressure of 20 pounds per square foot. Under this requirement, how fast can the wind be blowing before it produces excessive stress on the building b To be safe, the library is designed so that it can withstand wind pressure of 40 pounds per square foot. Does this mean that the library can survive wind blowing at twice the speed you found in part a Justify your answer. c Use the pressure formula to explain why even a relatively small increase in the wind speed could have potentially serious effects on a building. 5. For a bathtub with a rectangular base, Toricellis Law implies that the height h of water in the tub t seconds after it begins draining is given by a 4 P 0.00256s2. h h0 b a 4. A building code requires that a building be able to withstand a certain amount of wind pressure. The pressure P in pounds per square foot from wind blowing at s miles per hour is given by 10 13 16 A c Find two values of a such that A 300. d Use a graphing utility to graph the area equation. e Find the a intercepts of the graph of the area equation. What do these values represent f What is the maximum area What values of a and b yield the maximum area 2d 2 3 t lw 2 where l and w are the tubs length and width, d is the diameter of the drain, and h0 is the waters initial height. All measurements are in inches. You completely fill a tub with water. The tub is 60 inches long by 30 inches wide by 25 inches high and has a drain with a two inch diameter. a Find the time it takes for the tub to go from being full to half full. b Find the time it takes for the tub to go from being half full to empty. c Based on your results in parts a and b , what general statement can you make about the speed at which the water drains 6. a Consider the sum of squares x2 9. If the sum can be factored, then there are integers m and n such that x 2 9 x mx n. Write two equations relating the sum and the product of m and n to the coefficients in x 2 9. b Show that there are no integers m and n that satisfy both equations you wrote in part a . What can you conclude 167 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 7. A Pythagorean Triple is a group of three integers, such as 3, 4, and 5, that could be the lengths of the sides of a right triangle. a Find two other Pythagorean Triples. b Notice that 3 4 5 60. Is the product of the three numbers in each Pythagorean Triple evenly divisible by 3 by 4 by 5 c Write a conjecture involving Pythagorean Triples and divisibility by 60. 8. Determine the solutions x1 and x2 of each quadratic equation. Use the values of x1 and x2 to fill in the boxes. Equation x1, x2 x1 x2 x1 x2 a x2 x 6 0 b 2x2 5x 3 0 c 4x2 9 0 d x2 10x 34 0 9. Consider a general quadratic equation 13. A fractal is a geometric figure that consists of a pattern that is repeated infinitely on a smaller and smaller scale. The most famous fractal is called the Mandelbrot Set, named after the Polish born mathematician Benoit Mandelbrot. To draw the Mandelbrot Set, consider the following sequence of numbers. c, c2 c, c2 c2 c, c2 c2 c2 c, . . . The behavior of this sequence depends on the value of the complex number c. If the sequence is bounded the absolute value of each number in the sequence, a bi a2 b2, is less than some fixed number N , the complex number c is in the Mandelbrot Set, and if the sequence is unbounded the absolute value of the terms of the sequence become infinitely large , the complex number c is not in the Mandelbrot Set. Determine whether the complex number c is in the Mandelbrot Set. a c i b c 1 i c c 2 The figure below shows a black and yellow photo of the Mandelbrot Set. ax2 bx c 0 i x 5 53i 2 ii x 5 53i 2 3 b The principal cube root of 27, 27, is 3. Evaluate the expression x 3 for each value of x. i x 3 33i 2 ii x 3 33i 2 c Use the results of parts a and b to list possible cube roots of i 1, ii 8, and iii 64. Verify your results algebraically. 11. The multiplicative inverse of z is a complex number z m such that z z m 1. Find the multiplicative inverse of each complex number. a z 1 i b z 3 i c z 2 8i 12. Prove that the product of a complex number a bi and its complex conjugate is a real number. American Mathematical Society whose solutions are x1 and x2. Use the results of Exercise 8 to determine a relationship among the coefficients a, b, and c and the sum x1 x2 and the product x1 x2 of the solutions. 3 10. a The principal cube root of 125, 125, is 5. 3 Evaluate the expression x for each value of x. 14. Use the equation 4x 2x k to find three different values of k such that the equation has two solutions, one solution, and no solution. Describe the process you used to find the values. 15. Use the graph of y x 4 x 3 6x2 4x 8 to solve the inequality x 4 x 3 6x2 4x 8 0. 16. When you buy a 16 ounce bag of chips, you expect to get precisely 16 ounces. The actual weight w in ounces of a 16 ounce bag of chips is given by 1 w 16 2. You buy four 16 ounce bags. What is the greatest amount you can expect to get What is the smallest amount Explain. 168 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Functions and Their Graphs 2.1 Linear Equations in Two Variables 2.2 Functions 2.3 Analyzing Graphs of Functions 2.4 A Library of Parent Functions 2.5 Transformations of Functions 2.6 Combinations of Functions: Composite Functions 2.7 Inverse Functions 2 In Mathematics Functions show how one variable is related to another variable. Functions are used to estimate values, stimulate processes, and discover relationships. You can model the enrollment rate of children in preschool and estimate the year in which the rate will reach a certain number. This estimate can be used to plan for future needs, such as adding teachers and buying books. See Exercise 113, page 210. Jose Luis PelaezGetty Images In Real Life IN CAREERS There are many careers that use functions. Several are listed below. Roofing Contractor Exercise 131, page 182 Sociologist Exercise 80, page 228 Financial Analyst Exercise 95, page 197 Biologist Exercise 73, page 237 169 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 170 Chapter 2 Functions and Their Graphs 2.1 LINEAR EQUATIONS IN TWO VARIABLES What you should learn Use slope to graph linear equations in two variables. Find the slope of a line given two points on the line. Write linear equations in two variables. Use slope to identify parallel and perpendicular lines. Use slope and linear equations in two variables to model and solve real life problems. Why you should learn it Linear equations in two variables can be used to model and solve real life problems. For instance, in Exercise 129 on page 182, you will use a linear equation to model student enrollment at the Pennsylvania State University. Using Slope The simplest mathematical model for relating two variables is the linear equation in two variables y mx b. The equation is called linear because its graph is a line. In mathematics, the term line means straight line. By letting x 0, you obtain y m0 b Substitute 0 for x. b. So, the line crosses the y axis at y b, as shown in Figure 2.1. In other words, the y intercept is 0, b. The steepness or slope of the line is m. y mx b Slope y Intercept The slope of a nonvertical line is the number of units the line rises or falls vertically for each unit of horizontal change from left to right, as shown in Figure 2.1 and Figure 2.2. y y y intercept 1 unit y = mx + b m units, m0 0, b m units, m0 0, b y intercept 1 unit y = mx + b Courtesy of Pennsylvania State University x Positive slope, line rises. FIGURE 2.1 x Negative slope, line falls. 2.2 FIGURE A linear equation that is written in the form y mx b is said to be written in slope intercept form. The Slope Intercept Form of the Equation of a Line The graph of the equation y mx b is a line whose slope is m and whose y intercept is 0, b. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 y Once you have determined the slope and the y intercept of a line, it is a relatively simple matter to sketch its graph. In the next example, note that none of the lines is vertical. A vertical line has an equation of the form 3, 5 5 171 Linear Equations in Two Variables 4 x a. x=3 Vertical line The equation of a vertical line cannot be written in the form y mx b because the slope of a vertical line is undefined, as indicated in Figure 2.3. 3 2 3, 1 1 Example 1 Graphing a Linear Equation x 1 FIGURE 2 4 5 Sketch the graph of each linear equation. 2.3 Slope is undefined. a. y 2x 1 b. y 2 c. x y 2 Solution a. Because b 1, the y intercept is 0, 1. Moreover, because the slope is m 2, the line rises two units for each unit the line moves to the right, as shown in Figure 2.4. b. By writing this equation in the form y 0x 2, you can see that the y intercept is 0, 2 and the slope is zero. A zero slope implies that the line is horizontalthat is, it doesnt rise or fall, as shown in Figure 2.5. c. By writing this equation in slope intercept form xy2 Write original equation. y x 2 Subtract x from each side. y 1x 2 Write in slope intercept form. you can see that the y intercept is 0, 2. Moreover, because the slope is m 1, the line falls one unit for each unit the line moves to the right, as shown in Figure 2.6. y y 5 y 5 5 y = 2x + 1 4 4 4 3 y=2 3 3 m=2 2 0, 2 2 m=0 0, 2 x 1 m = 1 1 1 0, 1 y = x + 2 2 3 4 5 When m is positive, the line rises. FIGURE 2.4 x x 1 2 3 4 5 When m is 0, the line is horizontal. FIGURE 2.5 1 2 3 4 5 When m is negative, the line falls. FIGURE 2.6 Now try Exercise 17. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 172 Chapter 2 Functions and Their Graphs Finding the Slope of a Line y y2 y1 Given an equation of a line, you can find its slope by writing the equation in slopeintercept form. If you are not given an equation, you can still find the slope of a line. For instance, suppose you want to find the slope of the line passing through the points x1, y1 and x2, y2 , as shown in Figure 2.7. As you move from left to right along this line, a change of y2 y1 units in the vertical direction corresponds to a change of x2 x1 units in the horizontal direction. x 2, y 2 y2 y1 x 1, y 1 x 2 x1 x1 FIGURE 2.7 x2 y2 y1 the change in y rise x and x2 x1 the change in x run The ratio of y2 y1 to x2 x1 represents the slope of the line that passes through the points x1, y1 and x2, y2 . Slope change in y change in x rise run y2 y1 x2 x1 The Slope of a Line Passing Through Two Points The slope m of the nonvertical line through x1, y1 and x2, y2 is m y2 y1 x2 x1 where x1 x2. When this formula is used for slope, the order of subtraction is important. Given two points on a line, you are free to label either one of them as x1, y1 and the other as x2, y2 . However, once you have done this, you must form the numerator and denominator using the same order of subtraction. m y2 y1 x2 x1 Correct m y1 y2 x1 x2 Correct m y2 y1 x1 x2 Incorrect For instance, the slope of the line passing through the points 3, 4 and 5, 7 can be calculated as m 74 3 53 2 or, reversing the subtraction order in both the numerator and denominator, as m 4 7 3 3 . 3 5 2 2 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 Example 2 Linear Equations in Two Variables 173 Finding the Slope of a Line Through Two Points Find the slope of the line passing through each pair of points. a. 2, 0 and 3, 1 b. 1, 2 and 2, 2 c. 0, 4 and 1, 1 d. 3, 4 and 3, 1 Solution a. Letting x1, y1 2, 0 and x2, y2 3, 1, you obtain a slope of To find the slopes in Example 2, you must be able to evaluate rational expressions. You can review the techniques for evaluating rational expressions in Section P.5. m y2 y1 10 1 . x2 x1 3 2 5 See Figure 2.8. b. The slope of the line passing through 1, 2 and 2, 2 is m 22 0 0. 2 1 3 See Figure 2.9. c. The slope of the line passing through 0, 4 and 1, 1 is m 1 4 5 5. 10 1 See Figure 2.10. d. The slope of the line passing through 3, 4 and 3, 1 is m 1 4 3 . 33 0 See Figure 2.11. Because division by 0 is undefined, the slope is undefined and the line is vertical. y y 4 In Figures 2.8 to 2.11, note the relationships between slope and the orientation of the line. a. Positive slope: line rises from left to right b. Zero slope: line is horizontal c. Negative slope: line falls from left to right d. Undefined slope: line is vertical 4 3 m= 2 3, 1 2, 0 2 1 FIGURE 1, 2 1 x 1 1 2 3 2.8 2 1 0, 4 1 2 3 3, 4 4 3 m = 5 2 2 Slope is undefined. 3, 1 1 1 x 2 1, 1 1 FIGURE x 1 y 3 1 2, 2 1 2.9 FIGURE y 4 m=0 3 1 5 3 4 2.10 1 x 1 1 FIGURE 2 4 2.11 Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 174 Chapter 2 Functions and Their Graphs Writing Linear Equations in Two Variables If x1, y1 is a point on a line of slope m and x, y is any other point on the line, then y y1 m. x x1 This equation, involving the variables x and y, can be rewritten in the form y y1 mx x1 which is the point slope form of the equation of a line. Point Slope Form of the Equation of a Line The equation of the line with slope m passing through the point x1, y1 is y y1 mx x1. The point slope form is most useful for finding the equation of a line. You should remember this form. Example 3 y y = 3x 5 Find the slope intercept form of the equation of the line that has a slope of 3 and passes through the point 1, 2. 1 2 x 1 1 3 1 2 3 3 4 Solution Use the point slope form with m 3 and x1, y1 1, 2. y y1 mx x1 1 1, 2 4 5 FIGURE Using the Point Slope Form 2.12 y 2 3x 1 y 2 3x 3 Point slope form Substitute for m, x1, and y1. Simplify. y 3x 5 Write in slope intercept form. The slope intercept form of the equation of the line is y 3x 5. The graph of this line is shown in Figure 2.12. Now try Exercise 51. The point slope form can be used to find an equation of the line passing through two points x1, y1 and x2, y2 . To do this, first find the slope of the line When you find an equation of the line that passes through two given points, you only need to substitute the coordinates of one of the points in the point slope form. It does not matter which point you choose because both points will yield the same result. m y2 y1 x2 x1 , x1 x2 and then use the point slope form to obtain the equation y y1 y2 y1 x2 x1 x x1. Two point form This is sometimes called the two point form of the equation of a line. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 Linear Equations in Two Variables 175 Parallel and Perpendicular Lines Slope can be used to decide whether two nonvertical lines in a plane are parallel, perpendicular, or neither. Parallel and Perpendicular Lines 1. Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is, m1 m2. 2. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 1m2. Example 4 y 2x 3y = 5 3 2 Finding Parallel and Perpendicular Lines Find the slope intercept forms of the equations of the lines that pass through the point 2, 1 and are a parallel to and b perpendicular to the line 2x 3y 5. y = 23 x + 2 Solution 1 By writing the equation of the given line in slope intercept form x 1 4 5 1 2, 1 FIGURE y = 23 x 7 3 2.13 2x 3y 5 Write original equation. 3y 2x 5 y 2 3x Subtract 2x from each side. 5 3 Write in slope intercept form. you can see that it has a slope of m 2 3, as shown in Figure 2.13. a. Any line parallel to the given line must also have a slope of 23. So, the line through 2, 1 that is parallel to the given line has the following equation. y 1 23x 2 3 y 1 2x 2 T E C H N O LO G Y On a graphing utility, lines will not appear to have the correct slope unless you use a viewing window that has a square setting. For instance, try graphing the lines in Example 4 using the standard setting 10 x 10 and 10 y 10. Then reset the viewing window with the square setting 9 x 9 and 6 y 6. On which setting do the lines y 23 x 53 and y 32 x 2 appear to be perpendicular 3y 3 2x 4 y 2 3x 7 3 Write in point slope form. Multiply each side by 3. Distributive Property Write in slope intercept form. 3 3 b. Any line perpendicular to the given line must have a slope of 2 because 2 2 is the negative reciprocal of 3 . So, the line through 2, 1 that is perpendicular to the given line has the following equation. y 1 2x 2 3 2 y 1 3x 2 2y 2 3x 6 y 32x 2 Write in point slope form. Multiply each side by 2. Distributive Property Write in slope intercept form. Now try Exercise 87. Notice in Example 4 how the slope intercept form is used to obtain information about the graph of a line, whereas the point slope form is used to write the equation of a line. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 176 Chapter 2 Functions and Their Graphs Applications In real life problems, the slope of a line can be interpreted as either a ratio or a rate. If the x axis and y axis have the same unit of measure, then the slope has no units and is a ratio. If the x axis and y axis have different units of measure, then the slope is a rate or rate of change. Example 5 Using Slope as a Ratio 1 The maximum recommended slope of a wheelchair ramp is 12 . A business is installing a wheelchair ramp that rises 22 inches over a horizontal length of 24 feet. Is the ramp steeper than recommended Source: Americans with Disabilities Act Handbook Solution The horizontal length of the ramp is 24 feet or 1224 288 inches, as shown in Figure 2.14. So, the slope of the ramp is Slope 22 in. vertical change 0.076. horizontal change 288 in. 1 Because 12 0.083, the slope of the ramp is not steeper than recommended. y 22 in. x 24 ft FIGURE 2.14 Now try Exercise 115. Example 6 A kitchen appliance manufacturing company determines that the total cost in dollars of producing x units of a blender is Manufacturing Cost in dollars C 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 C 25x 3500. C = 25x + 3500 Cost equation Describe the practical significance of the y intercept and slope of this line. Marginal cost: m = 25 Solution Fixed cost: 3500 x 50 100 Number of units FIGURE Using Slope as a Rate of Change 2.15 Production cost 150 The y intercept 0, 3500 tells you that the cost of producing zero units is 3500. This is the fixed cost of productionit includes costs that must be paid regardless of the number of units produced. The slope of m 25 tells you that the cost of producing each unit is 25, as shown in Figure 2.15. Economists call the cost per unit the marginal cost. If the production increases by one unit, then the margin, or extra amount of cost, is 25. So, the cost increases at a rate of 25 per unit. Now try Exercise 119. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 Linear Equations in Two Variables 177 Most business expenses can be deducted in the same year they occur. One exception is the cost of property that has a useful life of more than 1 year. Such costs must be depreciated decreased in value over the useful life of the property. If the same amount is depreciated each year, the procedure is called linear or straight line depreciation. The book value is the difference between the original value and the total amount of depreciation accumulated to date. Example 7 Straight Line Depreciation A college purchased exercise equipment worth 12,000 for the new campus fitness center. The equipment has a useful life of 8 years. The salvage value at the end of 8 years is 2000. Write a linear equation that describes the book value of the equipment each year. Solution Let V represent the value of the equipment at the end of year t. You can represent the initial value of the equipment by the data point 0, 12,000 and the salvage value of the equipment by the data point 8, 2000. The slope of the line is m 2000 12,000 1250 80 which represents the annual depreciation in dollars per year. Using the point slope form, you can write the equation of the line as follows. V 12,000 1250t 0 Write in point slope form. V 1250t 12,000 Write in slope intercept form. The table shows the book value at the end of each year, and the graph of the equation is shown in Figure 2.16. Useful Life of Equipment Year, t Value, V 0 12,000 1 10,750 2 9500 3 8250 4 7000 5 5750 6 4500 7 3250 8 2000 V Value in dollars 12,000 0, 12,000 V = 1250t +12,000 10,000 8,000 6,000 4,000 2,000 8, 2000 t 2 4 6 8 10 Number of years FIGURE 2.16 Straight line depreciation Now try Exercise 121. In many real life applications, the two data points that determine the line are often given in a disguised form. Note how the data points are described in Example 7. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 178 Chapter 2 Functions and Their Graphs Example 8 Predicting Sales The sales for Best Buy were approximately 35.9 billion in 2006 and 40.0 billion in 2007. Using only this information, write a linear equation that gives the sales in billions of dollars in terms of the year. Then predict the sales for 2010. Source: Best Buy Company, Inc. Solution Let t 6 represent 2006. Then the two given values are represented by the data points 6, 35.9 and 7, 40.0. The slope of the line through these points is Sales in billions of dollars y = 4.1t + 11.3 60 50 40 30 m Best Buy y 4.1. 10, 52.3 Using the point slope form, you can find the equation that relates the sales y and the year t to be 7, 40.0 6, 35.9 y 35.9 4.1t 6 20 Write in point slope form. y 4.1t 11.3. 10 t 6 7 8 9 10 11 12 Year 6 2006 FIGURE 40.0 35.9 76 Write in slope intercept form. According to this equation, the sales for 2010 will be y 4.110 11.3 41 11.3 52.3 billion. See Figure 2.17. Now try Exercise 129. 2.17 The prediction method illustrated in Example 8 is called linear extrapolation. Note in Figure 2.18 that an extrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in Figure 2.19, the procedure is called linear interpolation. Because the slope of a vertical line is not defined, its equation cannot be written in slope intercept form. However, every line has an equation that can be written in the general form y Given points Estimated point Ax By C 0 x Linear extrapolation FIGURE 2.18 where A and B are not both zero. For instance, the vertical line given by x a can be represented by the general form x a 0. Summary of Equations of Lines y Given points 1. General form: Ax By C 0 2. Vertical line: xa 3. Horizontal line: yb 4. Slope intercept form: y mx b Estimated point 5. Point slope form: y y1 mx x1 6. Two point form: y y1 x Linear interpolation FIGURE 2.19 General form y2 y1 x x1 x2 x1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 2.1 EXERCISES 179 Linear Equations in Two Variables See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 17, fill in the blanks. The simplest mathematical model for relating two variables is the ________ equation in two variables y mx b. For a line, the ratio of the change in y to the change in x is called the ________ of the line. Two lines are ________ if and only if their slopes are equal. Two lines are ________ if and only if their slopes are negative reciprocals of each other. When the x axis and y axis have different units of measure, the slope can be interpreted as a ________. The prediction method ________ ________ is the method used to estimate a point on a line when the point does not lie between the given points. 7. Every line has an equation that can be written in ________ form. 8. Match each equation of a line with its form. a Ax By C 0 i Vertical line b x a ii Slope intercept form c y b iii General form d y mx b iv Point slope form e y y1 mx x1 v Horizontal line 1. 2. 3. 4. 5. 6. SKILLS AND APPLICATIONS In Exercises 9 and 10, identify the line that has each slope. 2 9. a m 3 b m is undefined. c m 2 6 6 4 4 2 2 x y 4 L1 L3 L1 L3 L2 x x L2 In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point 11. 2, 3 12. 4, 1 Slopes a 0 b 1 c 2 d 3 1 a 3 b 3 c 2 d Undefined In Exercises 1316, estimate the slope of the line. y 13. y 14. 8 8 6 6 4 4 2 2 x 2 4 6 8 x 2 4 y 16. 8 10. a m 0 3 b m 4 c m 1 y y 15. 6 8 6 x 8 2 4 6 In Exercises 1728, find the slope and y intercept if possible of the equation of the line. Sketch the line. 17. y 5x 3 19. y 12x 4 21. 5x 2 0 23. 7x 6y 30 25. y 3 0 27. x 5 0 18. y x 10 20. y 32x 6 22. 3y 5 0 24. 2x 3y 9 26. y 4 0 28. x 2 0 In Exercises 2940, plot the points and find the slope of the line passing through the pair of points. 29. 31. 33. 35. 37. 39. 40. 0, 9, 6, 0 30. 32. 3, 2, 1, 6 34. 5, 7, 8, 7 36. 6, 1, 6, 4 11 4 3 1 38. 2 , 3 , 2, 3 4.8, 3.1, 5.2, 1.6 1.75, 8.3, 2.25, 2.6 www.elsolucionario.net 12, 0, 0, 8 2, 4, 4, 4 2, 1, 4, 5 0, 10, 4, 0 78, 34 , 54, 14 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 180 Chapter 2 Functions and Their Graphs In Exercises 4150, use the point on the line and the slope m of the line to find three additional points through which the line passes. There are many correct answers. 41. 43. 45. 46. 47. 49. 2, 1, m 0 42. 5, 6, m 1 44. 8, 1, m is undefined. 1, 5, m is undefined. 5, 4, m 2 48. 1 7, 2, m 2 50. 3, 2, m 0 10, 6, m 1 0, 9, m 2 1, 6, m 12 In Exercises 51 64, find the slope intercept form of the equation of the line that passes through the given point and has the indicated slope m. Sketch the line. 51. 0, 2, m 3 53. 3, 6, m 2 55. 4, 0, m 13 57. 59. 60. 61. 63. 52. 0, 10, m 1 54. 0, 0, m 4 56. 8, 2, m 14 2, 3, m 12 58. 2, 5, m 34 6, 1, m is undefined. 10, 4, m is undefined. 1 3 62. 2, 2 , m 0 4, 52 , m 0 5.1, 1.8, m 5 64. 2.3, 8.5, m 2.5 In Exercises 6578, find the slope intercept form of the equation of the line passing through the points. Sketch the line. 65. 67. 69. 71. 73. 75. 77. 5, 1, 5, 5 8, 1, 8, 7 2, 12 , 12, 54 101 , 35 , 109 , 95 1, 0.6, 2, 0.6 2, 1, 13, 1 73, 8, 73, 1 66. 68. 70. 72. 74. 76. 78. 4, 3, 4, 4 1, 4, 6, 4 1, 1, 6, 23 34, 32 , 43, 74 8, 0.6, 2, 2.4 15, 2, 6, 2 1.5, 2, 1.5, 0.2 In Exercises 79 82, determine whether the lines are parallel, perpendicular, or neither. 1 79. L1: y 3 x 2 L2: y 1 3x 3 81. L1: y 12 x 3 L2: y 12 x 1 80. L1: y 4x 1 L2: y 4x 7 82. L1: y 45 x 5 L2: y 54 x 1 In Exercises 83 86, determine whether the lines L1 and L2 passing through the pairs of points are parallel, perpendicular, or neither. 83. L1: 0, 1, 5, 9 L2: 0, 3, 4, 1 84. L1: 2, 1, 1, 5 L2: 1, 3, 5, 5 85. L1: 3, 6, 6, 0 L2: 0, 1, 5, 73 86. L1: 4, 8, 4, 2 L2: 3, 5, 1, 13 In Exercises 8796, write the slope intercept forms of the equations of the lines through the given point a parallel to the given line and b perpendicular to the given line. 87. 89. 91. 93. 95. 96. 88. 4x 2y 3, 2, 1 2 7 90. 3x 4y 7, 3, 8 92. y 3 0, 1, 0 94. x 4 0, 3, 2 x y 4, 2.5, 6.8 6x 2y 9, 3.9, 1.4 x y 7, 3, 2 5x 3y 0, 78, 34 y 2 0, 4, 1 x 2 0, 5, 1 In Exercises 97102, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts a, 0 and 0, b is x y 1 1, a a b 0, b 0. 97. x intercept: 2, 0 y intercept: 0, 3 99. x intercept: 16, 0 y intercept: 0, 23 101. Point on line: 1, 2 x intercept: c, 0 y intercept: 0, c, c 102. Point on line: 3, 4 x intercept: d, 0 y intercept: 0, d, d 98. x intercept: 3, 0 y intercept: 0, 4 100. x intercept: 23, 0 y intercept: 0, 2 0 0 GRAPHICAL ANALYSIS In Exercises 103106, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correctthat is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. 103. 104. 105. 106. a a a a y 2x y 23x y 12x yx8 b b b b c y 2x c y 32x 1 y 2x 3 c c yx1 y 12x y 23x 2 y 2x 4 y x 3 In Exercises 107110, find a relationship between x and y such that x, y is equidistant the same distance from the two points. 107. 4, 1, 2, 3 109. 3, 52 , 7, 1 www.elsolucionario.net 108. 6, 5, 1, 8 110. 12, 4, 72, 54 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 111. SALES The following are the slopes of lines representing annual sales y in terms of time x in years. Use the slopes to interpret any change in annual sales for a one year increase in time. a The line has a slope of m 135. b The line has a slope of m 0. c The line has a slope of m 40. 112. REVENUE The following are the slopes of lines representing daily revenues y in terms of time x in days. Use the slopes to interpret any change in daily revenues for a one day increase in time. a The line has a slope of m 400. b The line has a slope of m 100. c The line has a slope of m 0. 113. AVERAGE SALARY The graph shows the average salaries for senior high school principals from 1996 through 2008. Source: Educational Research Service Salary in dollars 100,000 18, 97,486 95,000 16, 90,260 90,000 12, 83,944 85,000 80,000 14, 86,160 10, 79,839 8, 74,380 6, 69,277 75,000 70,000 65,000 6 8 10 12 14 16 18 Year 6 1996 a Use the slopes of the line segments to determine the time periods in which the average salary increased the greatest and the least. Sales in billions of dollars b Find the slope of the line segment connecting the points for the years 1996 and 2008. c Interpret the meaning of the slope in part b in the context of the problem. 114. SALES The graph shows the sales in billions of dollars for Apple Inc. for the years 2001 through 2007. Source: Apple Inc. 28 7, 24.01 24 6, 19.32 20 16 5, 13.93 12 2, 5.74 8 4 3, 6.21 1, 5.36 1 2 3 4 5 6 181 a Use the slopes of the line segments to determine the years in which the sales showed the greatest increase and the least increase. b Find the slope of the line segment connecting the points for the years 2001 and 2007. c Interpret the meaning of the slope in part b in the context of the problem. 115. ROAD GRADE You are driving on a road that has a 6 uphill grade see figure . This means that the slope 6 of the road is 100 . Approximate the amount of vertical change in your position if you drive 200 feet. 116. ROAD GRADE From the top of a mountain road, a surveyor takes several horizontal measurements x and several vertical measurements y, as shown in the table x and y are measured in feet . x 300 600 900 1200 1500 1800 2100 y 25 50 75 100 125 150 175 a Sketch a scatter plot of the data. b Use a straightedge to sketch the line that you think best fits the data. c Find an equation for the line you sketched in part b . d Interpret the meaning of the slope of the line in part c in the context of the problem. e The surveyor needs to put up a road sign that indicates the steepness of the road. For instance, a surveyor would put up a sign that states 8 grade on a road with a downhill grade that has a 8 slope of 100 . What should the sign state for the road in this problem RATE OF CHANGE In Exercises 117 and 118, you are given the dollar value of a product in 2010 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. Let t 10 represent 2010. 2010 Value 117. 2540 118. 156 4, 8.28 Linear Equations in Two Variables Rate 125 decrease per year 4.50 increase per year 7 Year 1 2001 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 182 Chapter 2 Functions and Their Graphs 119. DEPRECIATION The value V of a molding machine t years after it is purchased is V 4000t 58,500, 0 t 5. Explain what the V intercept and the slope measure. 120. COST The cost C of producing n computer laptop bags is given by C 1.25n 15,750, 121. 122. 123. 124. 125. 126. 127. 128. 0 n. Explain what the C intercept and the slope measure. DEPRECIATION A sub shop purchases a used pizza oven for 875. After 5 years, the oven will have to be replaced. Write a linear equation giving the value V of the equipment during the 5 years it will be in use. DEPRECIATION A school district purchases a high volume printer, copier, and scanner for 25,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be 2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use. SALES A discount outlet is offering a 20 discount on all items. Write a linear equation giving the sale price S for an item with a list price L. HOURLY WAGE A microchip manufacturer pays its assembly line workers 12.25 per hour. In addition, workers receive a piecework rate of 0.75 per unit produced. Write a linear equation for the hourly wage W in terms of the number of units x produced per hour. MONTHLY SALARY A pharmaceutical salesperson receives a monthly salary of 2500 plus a commission of 7 of sales. Write a linear equation for the salespersons monthly wage W in terms of monthly sales S. BUSINESS COSTS A sales representative of a company using a personal car receives 120 per day for lodging and meals plus 0.55 per mile driven. Write a linear equation giving the daily cost C to the company in terms of x, the number of miles driven. CASH FLOW PER SHARE The cash flow per share for the Timberland Co. was 1.21 in 1999 and 1.46 in 2007. Write a linear equation that gives the cash flow per share in terms of the year. Let t 9 represent 1999. Then predict the cash flows for the years 2012 and 2014. Source: The Timberland Co. NUMBER OF STORES In 2003 there were 1078 J.C. Penney stores and in 2007 there were 1067 stores. Write a linear equation that gives the number of stores in terms of the year. Let t 3 represent 2003. Then predict the numbers of stores for the years 2012 and 2014. Are your answers reasonable Explain. Source: J.C. Penney Co. 129. COLLEGE ENROLLMENT The Pennsylvania State University had enrollments of 40,571 students in 2000 and 44,112 students in 2008 at its main campus in University Park, Pennsylvania. Source: Penn State Fact Book a Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the year t, where t 0 corresponds to 2000. b Use your model from part a to predict the enrollments in 2010 and 2015. c What is the slope of your model Explain its meaning in the context of the situation. 130. COLLEGE ENROLLMENT The University of Florida had enrollments of 46,107 students in 2000 and 51,413 students in 2008. Source: University of Florida a What was the average annual change in enrollment from 2000 to 2008 b Use the average annual change in enrollment to estimate the enrollments in 2002, 2004, and 2006. c Write the equation of a line that represents the given data in terms of the year t, where t 0 corresponds to 2000. What is its slope Interpret the slope in the context of the problem. d Using the results of parts a c , write a short paragraph discussing the concepts of slope and average rate of change. 131. COST, REVENUE, AND PROFIT A roofing contractor purchases a shingle delivery truck with a shingle elevator for 42,000. The vehicle requires an average expenditure of 6.50 per hour for fuel and maintenance, and the operator is paid 11.50 per hour. a Write a linear equation giving the total cost C of operating this equipment for t hours. Include the purchase cost of the equipment. b Assuming that customers are charged 30 per hour of machine use, write an equation for the revenue R derived from t hours of use. c Use the formula for profit PRC to write an equation for the profit derived from t hours of use. d Use the result of part c to find the break even pointthat is, the number of hours this equipment must be used to yield a profit of 0 dollars. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.1 132. RENTAL DEMAND A real estate office handles an apartment complex with 50 units. When the rent per unit is 580 per month, all 50 units are occupied. However, when the rent is 625 per month, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear. a Write the equation of the line giving the demand x in terms of the rent p. b Use this equation to predict the number of units occupied when the rent is 655. c Predict the number of units occupied when the rent is 595. 133. GEOMETRY The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width x surrounds the garden. a Draw a diagram that gives a visual representation of the problem. b Write the equation for the perimeter y of the walkway in terms of x. c Use a graphing utility to graph the equation for the perimeter. d Determine the slope of the graph in part c . For each additional one meter increase in the width of the walkway, determine the increase in its perimeter. 134. AVERAGE ANNUAL SALARY The average salaries in millions of dollars of Major League Baseball players from 2000 through 2007 are shown in the scatter plot. Find the equation of the line that you think best fits these data. Let y represent the average salary and let t represent the year, with t 0 corresponding to 2000. Source: Major League Baseball Players Association Average salary in millions of dollars y Linear Equations in Two Variables 183 135. DATA ANALYSIS: NUMBER OF DOCTORS The numbers of doctors of osteopathic medicine y in thousands in the United States from 2000 through 2008, where x is the year, are shown as data points x, y. Source: American Osteopathic Association 2000, 44.9, 2001, 47.0, 2002, 49.2, 2003, 51.7, 2004, 54.1, 2005, 56.5, 2006, 58.9, 2007, 61.4, 2008, 64.0 a Sketch a scatter plot of the data. Let x 0 correspond to 2000. b Use a straightedge to sketch the line that you think best fits the data. c Find the equation of the line from part b . Explain the procedure you used. d Write a short paragraph explaining the meanings of the slope and y intercept of the line in terms of the data. e Compare the values obtained using your model with the actual values. f Use your model to estimate the number of doctors of osteopathic medicine in 2012. 136. DATA ANALYSIS: AVERAGE SCORES An instructor gives regular 20 point quizzes and 100 point exams in an algebra course. Average scores for six students, given as data points x, y, where x is the average quiz score and y is the average test score, are 18, 87, 10, 55, 19, 96, 16, 79, 13, 76, and 15, 82. Note: There are many correct answers for parts b d . a Sketch a scatter plot of the data. b Use a straightedge to sketch the line that you think best fits the data. c Find an equation for the line you sketched in part b . d Use the equation in part c to estimate the average test score for a person with an average quiz score of 17. 3.0 2.8 2.6 e The instructor adds 4 points to the average test score of each student in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. 2.4 2.2 2.0 1.8 t 1 2 3 4 5 6 7 Year 0 2000 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 184 Chapter 2 Functions and Their Graphs EXPLORATION TRUE OR FALSE In Exercises 137 and 138, determine whether the statement is true or false. Justify your answer. 137. A line with a slope of 57 is steeper than a line with a slope of 67. 138. The line through 8, 2 and 1, 4 and the line through 0, 4 and 7, 7 are parallel. 139. Explain how you could show that the points A2, 3, B2, 9, and C4, 3 are the vertices of a right triangle. 140. Explain why the slope of a vertical line is said to be undefined. 141. With the information shown in the graphs, is it possible to determine the slope of each line Is it possible that the lines could have the same slope Explain. a b y 146. CAPSTONE Match the description of the situation with its graph. Also determine the slope and y intercept of each graph and interpret the slope and y intercept in the context of the situation. The graphs are labeled i , ii , iii , and iv . y y i ii 40 200 30 150 20 100 10 50 x 2 4 6 y iii 24 800 18 600 12 400 200 y x x 2 x 2 2 4 6 8 10 y iv 6 x 2 8 x 4 2 4 142. The slopes of two lines are 4 and 52. Which is steeper Explain. 143. Use a graphing utility to compare the slopes of the lines y mx, where m 0.5, 1, 2, and 4. Which line rises most quickly Now, let m 0.5, 1, 2, and 4. Which line falls most quickly Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the rate at which the line rises or falls 144. Find d1 and d2 in terms of m1 and m2, respectively see figure . Then use the Pythagorean Theorem to find a relationship between m1 and m2. 4 6 8 2 4 6 8 a A person is paying 20 per week to a friend to repay a 200 loan. b An employee is paid 8.50 per hour plus 2 for each unit produced per hour. c A sales representative receives 30 per day for food plus 0.32 for each mile traveled. d A computer that was purchased for 750 depreciates 100 per year. PROJECT: BACHELORS DEGREES To work an extended application analyzing the numbers of bachelors degrees earned by women in the United States from 1996 through 2007, visit this texts website at academic.cengage.com. Data Source: U.S. National Center for Education Statistics y d1 0, 0 1, m1 x d2 1, m 2 145. THINK ABOUT IT Is it possible for two lines with positive slopes to be perpendicular Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 Functions 185 2.2 FUNCTIONS What you should learn Determine whether relations between two variables are functions. Use function notation and evaluate functions. Find the domains of functions. Use functions to model and solve real life problems. Evaluate difference quotients. Introduction to Functions Many everyday phenomena involve two quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. In mathematics, relations are often represented by mathematical equations and formulas. For instance, the simple interest I earned on 1000 for 1 year is related to the annual interest rate r by the formula I 1000r. The formula I 1000r represents a special kind of relation that matches each item from one set with exactly one item from a different set. Such a relation is called a function. Why you should learn it Functions can be used to model and solve real life problems. For instance, in Exercise 100 on page 198, you will use a function to model the force of water against the face of a dam. Definition of Function A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain or set of inputs of the function f, and the set B contains the range or set of outputs . To help understand this definition, look at the function that relates the time of day to the temperature in Figure 2.20. Time of day P.M. 1 Temperature in degrees C 1 9 Lester LefkowitzCorbis 15 5 7 6 14 12 10 6 Set A is the domain. Inputs: 1, 2, 3, 4, 5, 6 3 4 4 3 FIGURE 2 13 2 16 5 8 11 Set B contains the range. Outputs: 9, 10, 12, 13, 15 2.20 This function can be represented by the following ordered pairs, in which the first coordinate x value is the input and the second coordinate y value is the output. 1, 9, 2, 13, 3, 15, 4, 15, 5, 12, 6, 10 Characteristics of a Function from Set A to Set B 1. Each element in A must be matched with an element in B. 2. Some elements in B may not be matched with any element in A. 3. Two or more elements in A may be matched with the same element in B. 4. An element in A the domain cannot be matched with two different elements in B. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 186 Chapter 2 Functions and Their Graphs Functions are commonly represented in four ways. Four Ways to Represent a Function 1. Verbally by a sentence that describes how the input variable is related to the output variable 2. Numerically by a table or a list of ordered pairs that matches input values with output values 3. Graphically by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis 4. Algebraically by an equation in two variables To determine whether or not a relation is a function, you must decide whether each input value is matched with exactly one output value. If any input value is matched with two or more output values, the relation is not a function. Example 1 Testing for Functions Determine whether the relation represents y as a function of x. a. The input value x is the number of representatives from a state, and the output value y is the number of senators. y b. c. Input, x Output, y 2 11 2 10 3 8 4 5 5 1 3 2 1 3 2 1 x 1 2 3 2 3 FIGURE 2.21 Solution a. This verbal description does describe y as a function of x. Regardless of the value of x, the value of y is always 2. Such functions are called constant functions. b. This table does not describe y as a function of x. The input value 2 is matched with two different y values. c. The graph in Figure 2.21 does describe y as a function of x. Each input value is matched with exactly one output value. Now try Exercise 11. Representing functions by sets of ordered pairs is common in discrete mathematics. In algebra, however, it is more common to represent functions by equations or formulas involving two variables. For instance, the equation y x2 y is a function of x. represents the variable y as a function of the variable x. In this equation, x is www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 HISTORICAL NOTE BettmannCorbis 187 the independent variable and y is the dependent variable. The domain of the function is the set of all values taken on by the independent variable x, and the range of the function is the set of all values taken on by the dependent variable y. Example 2 Leonhard Euler 17071783 , a Swiss mathematician, is considered to have been the most prolific and productive mathematician in history. One of his greatest influences on mathematics was his use of symbols, or notation. The function notation y f x was introduced by Euler. Functions Testing for Functions Represented Algebraically Which of the equations represent s y as a function of x a. x 2 y 1 b. x y 2 1 Solution To determine whether y is a function of x, try to solve for y in terms of x. a. Solving for y yields x2 y 1 Write original equation. y1 x 2. Solve for y. To each value of x there corresponds exactly one value of y. So, y is a function of x. b. Solving for y yields x y 2 1 Write original equation. y2 1 x Add x to each side. y 1 x. Solve for y. The indicates that to a given value of x there correspond two values of y. So, y is not a function of x. Now try Exercise 21. Function Notation When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easily. For example, you know that the equation y 1 x 2 describes y as a function of x. Suppose you give this function the name f. Then you can use the following function notation. Input Output Equation x f x f x 1 x 2 The symbol f x is read as the value of f at x or simply f of x. The symbol f x corresponds to the y value for a given x. So, you can write y f x. Keep in mind that f is the name of the function, whereas f x is the value of the function at x. For instance, the function given by f x 3 2x has function values denoted by f 1, f 0, f 2, and so on. To find these values, substitute the specified input values into the given equation. For x 1, f 1 3 21 3 2 5. For x 0, f 0 3 20 3 0 3. For x 2, f 2 3 22 3 4 1. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 188 Chapter 2 Functions and Their Graphs Although f is often used as a convenient function name and x is often used as the independent variable, you can use other letters. For instance, f x x 2 4x 7, f t t 2 4t 7, and gs s 2 4s 7 all define the same function. In fact, the role of the independent variable is that of a placeholder. Consequently, the function could be described by f 4 7. 2 WARNING CAUTION In Example 3, note that gx 2 is not equal to gx g2. In general, gu v gu gv. Example 3 Evaluating a Function Let gx x 2 4x 1. Find each function value. a. g2 b. gt c. gx 2 Solution a. Replacing x with 2 in gx x2 4x 1 yields the following. g2 22 42 1 4 8 1 5 b. Replacing x with t yields the following. gt t2 4t 1 t 2 4t 1 c. Replacing x with x 2 yields the following. gx 2 x 22 4x 2 1 x 2 4x 4 4x 8 1 x 2 4x 4 4x 8 1 x 2 5 Now try Exercise 41. A function defined by two or more equations over a specified domain is called a piecewise defined function. Example 4 A Piecewise Defined Function Evaluate the function when x 1, 0, and 1. f x xx 1,1, 2 x 0 x 0 Solution Because x 1 is less than 0, use f x x 2 1 to obtain f 1 12 1 2. For x 0, use f x x 1 to obtain f 0 0 1 1. For x 1, use f x x 1 to obtain f 1 1 1 0. Now try Exercise 49. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 Example 5 Functions 189 Finding Values for Which f x 0 Find all real values of x such that f x 0. a. f x 2x 10 b. f x x2 5x 6 Solution For each function, set f x 0 and solve for x. a. 2x 10 0 2x 10 x5 Set f x equal to 0. Subtract 10 from each side. Divide each side by 2. So, f x 0 when x 5. b. x2 5x 6 0 x 2x 3 0 x20 x2 Set 1st factor equal to 0. x30 x3 Set 2nd factor equal to 0. Set f x equal to 0. Factor. So, f x 0 when x 2 or x 3. Now try Exercise 59. Example 6 Finding Values for Which f x g x Find the values of x for which f x gx. a. f x x2 1 and gx 3x x2 b. f x x2 1 and gx x2 x 2 Solution x2 1 3x x2 a. Set f x equal to gx. 3x 1 0 2x 1x 1 0 2x 1 0 x 12 Set 1st factor equal to 0. x10 x1 Set 2nd factor equal to 0. 2x2 So, f x gx when x b. Write in general form. Factor. 1 or x 1. 2 x2 1 x2 x 2 2x2 x 3 0 2x 3x 1 0 2x 3 0 Write in general form. Factor. x10 So, f x gx when x Set f x equal to gx. x 32 Set 1st factor equal to 0. x 1 Set 2nd factor equal to 0. 3 or x 1. 2 Now try Exercise 67. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 190 Chapter 2 Functions and Their Graphs The Domain of a Function T E C H N O LO G Y Use a graphing utility to graph the functions given by y 4 x 2 and y x 2 4. What is the domain of each function Do the domains of these two functions overlap If so, for what values do the domains overlap The domain of a function can be described explicitly or it can be implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined. For instance, the function given by f x 1 x2 4 Domain excludes x values that result in division by zero. has an implied domain that consists of all real x other than x 2. These two values are excluded from the domain because division by zero is undefined. Another common type of implied domain is that used to avoid even roots of negative numbers. For example, the function given by Domain excludes x values that result in even roots of negative numbers. f x x is defined only for x 0. So, its implied domain is the interval 0, . In general, the domain of a function excludes values that would cause division by zero or that would result in the even root of a negative number. Example 7 Finding the Domain of a Function Find the domain of each function. 1 x5 a. f : 3, 0, 1, 4, 0, 2, 2, 2, 4, 1 b. gx 4 c. Volume of a sphere: V 3 r 3 d. hx 4 3x Solution a. The domain of f consists of all first coordinates in the set of ordered pairs. Domain 3, 1, 0, 2, 4 b. Excluding x values that yield zero in the denominator, the domain of g is the set of all real numbers x except x 5. c. Because this function represents the volume of a sphere, the values of the radius r must be positive. So, the domain is the set of all real numbers r such that r 0. d. This function is defined only for x values for which 4 3x 0. Using the methods described in Section 1.8, you can conclude that x 43. So, the domain is the interval , 43. Now try Exercise 73. In Example 7 c , note that the domain of a function may be implied by the physical context. For instance, from the equation 4 V 3 r 3 you would have no reason to restrict r to positive values, but the physical context implies that a sphere cannot have a negative or zero radius. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 Functions 191 Applications h r =4 r Example 8 The Dimensions of a Container You work in the marketing department of a soft drink company and are experimenting with a new can for iced tea that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4, as shown in Figure 2.22. h a. Write the volume of the can as a function of the radius r. b. Write the volume of the can as a function of the height h. Solution a. Vr r 2h r 24r 4 r 3 b. Vh FIGURE 4 h h 2 h3 16 Write V as a function of r. Write V as a function of h. Now try Exercise 87. 2.22 Example 9 The Path of a Baseball A baseball is hit at a point 3 feet above ground at a velocity of 100 feet per second and an angle of 45o. The path of the baseball is given by the function f x 0.0032x 2 x 3 where x and f x are measured in feet. Will the baseball clear a 10 foot fence located 300 feet from home plate Algebraic Solution Graphical Solution When x 300, you can find the height of the baseball as follows. Use a graphing utility to graph the function y 0.0032x2 x 3. Use the value feature or the zoom and trace features of the graphing utility to estimate that y 15 when x 300, as shown in Figure 2.23. So, the ball will clear a 10 foot fence. f x 0.0032x2 x 3 Write original function. f 300 0.00323002 300 3 15 Substitute 300 for x. Simplify. When x 300, the height of the baseball is 15 feet, so the baseball will clear a 10 foot fence. 100 0 400 0 FIGURE 2.23 Now try Exercise 93. In the equation in Example 9, the height of the baseball is a function of the distance from home plate. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 192 Chapter 2 Functions and Their Graphs Example 10 The number V in thousands of alternative fueled vehicles in the United States increased in a linear pattern from 1995 to 1999, as shown in Figure 2.24. Then, in 2000, the number of vehicles took a jump and, until 2006, increased in a different linear pattern. These two patterns can be approximated by the function Number of Alternative Fueled Vehicles in the U.S. Number of vehicles in thousands V 650 600 550 500 450 400 350 300 250 200 Vt t 7 9 11 13 15 Year 5 1995 2.24 155.3, 18.08t 34.75t 74.9, 5 t 9 10 t 16 where t represents the year, with t 5 corresponding to 1995. Use this function to approximate the number of alternative fueled vehicles for each year from 1995 to 2006. Source: Science Applications International Corporation; Energy Information Administration 5 FIGURE Alternative Fueled Vehicles Solution From 1995 to 1999, use Vt 18.08t 155.3. 245.7 263.8 281.9 299.9 318.0 1995 1996 1997 1998 1999 From 2000 to 2006, use Vt 34.75t 74.9. 422.4 457.2 491.9 526.7 561.4 596.2 630.9 2000 2001 2002 2003 2004 2005 2006 Now try Exercise 95. Difference Quotients One of the basic definitions in calculus employs the ratio f x h f x , h h 0. This ratio is called a difference quotient, as illustrated in Example 11. Example 11 Evaluating a Difference Quotient For f x x 2 4x 7, find Solution f x h f x h f x h f x . h x h2 4x h 7 x 2 4x 7 h x 2 2xh h2 4x 4h 7 x 2 4x 7 h 2xh h2 4h h2x h 4 2x h 4, h h h 0 Now try Exercise 103. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 193 Functions You may find it easier to calculate the difference quotient in Example 11 by first finding f x h, and then substituting the resulting expression into the difference quotient, as follows. f x h x h2 4x h 7 x2 2xh h2 4x 4h 7 f x h f x x2 2xh h2 4x 4h 7 x2 4x 7 h h 2xh h2 4h h2x h 4 2x h 4, h h h 0 Summary of Function Terminology Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable. Function Notation: y f x f is the name of the function. y is the dependent variable. x is the independent variable. f x is the value of the function at x. Domain: The domain of a function is the set of all values inputs of the independent variable for which the function is defined. If x is in the domain of f, f is said to be defined at x. If x is not in the domain of f, f is said to be undefined at x. Range: The range of a function is the set of all values outputs assumed by the dependent variable that is, the set of all function values . Implied Domain: If f is defined by an algebraic expression and the domain is not specified, the implied domain consists of all real numbers for which the expression is defined. CLASSROOM DISCUSSION Everyday Functions In groups of two or three, identify common real life functions. Consider everyday activities, events, and expenses, such as long distance telephone calls and car insurance. Here are two examples. a. The statement, Your happiness is a function of the grade you receive in this course is not a correct mathematical use of the word function. The word happiness is ambiguous. b. The statement, Your federal income tax is a function of your adjusted gross income is a correct mathematical use of the word function. Once you have determined your adjusted gross income, your income tax can be determined. Describe your functions in words. Avoid using ambiguous words. Can you find an example of a piecewise defined function www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 194 Chapter 2 2.2 Functions and Their Graphs EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. A relation that assigns to each element x from a set of inputs, or ________, exactly one element y in a set of outputs, or ________, is called a ________. 2. Functions are commonly represented in four different ways, ________, ________, ________, and ________. 3. For an equation that represents y as a function of x, the set of all values taken on by the ________ variable x is the domain, and the set of all values taken on by the ________ variable y is the range. 4. The function given by f x 2xx 4,1, 2 x 0 x 0 is an example of a ________ function. 5. If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the ________ ________. 6. In calculus, one of the basic definitions is that of a ________ ________, given by f x h f x , h h 0. SKILLS AND APPLICATIONS 12. In Exercises 710, is the relationship a function 7. Domain 2 1 0 1 2 9. Domain National League American League Range Range 8. Domain 2 1 0 1 2 5 6 7 8 Range 3 4 5 10. Domain Cubs Pirates Dodgers Range Number of North Atlantic tropical storms and hurricanes Year 10 12 15 16 21 27 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Orioles Yankees Twins 13. Input, x 2 1 0 1 2 Output, y 8 1 0 1 8 0 1 2 1 0 Output, y 4 2 0 2 4 Input, x 10 7 4 7 10 Output, y 3 6 9 12 15 Input, x 0 3 9 12 15 Output, y 3 3 3 3 3 In Exercises 15 and 16, which sets of ordered pairs represent functions from A to B Explain. 15. A 0, 1, 2, 3 and B 2, 1, 0, 1, 2 a 0, 1, 1, 2, 2, 0, 3, 2 b 0, 1, 2, 2, 1, 2, 3, 0, 1, 1 c 0, 0, 1, 0, 2, 0, 3, 0 d 0, 2, 3, 0, 1, 1 In Exercises 1114, determine whether the relation represents y as a function of x. 11. 14. Input, x 16. A a, b, c and B 0, 1, 2, 3 a a, 1, c, 2, c, 3, b, 3 b a, 1, b, 2, c, 3 c 1, a, 0, a, 2, c, 3, b d c, 0, b, 0, a, 3 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 Circulation in millions CIRCULATION OF NEWSPAPERS In Exercises 17 and 18, use the graph, which shows the circulation in millions of daily newspapers in the United States. Source: Editor Publisher Company 50 40 Morning Evening 30 20 10 1997 1999 2001 2003 2005 2007 Year 17. Is the circulation of morning newspapers a function of the year Is the circulation of evening newspapers a function of the year Explain. 18. Let f x represent the circulation of evening newspapers in year x. Find f 2002. In Exercises 1936, determine whether the equation represents y as a function of x. x2 y 2 4 20. x2 y 4 22. 2x 3y 4 24. x 22 y 12 25 x 22 y2 4 y2 x2 1 28. 2 y 16 x 30. y 4 x 32. x 14 34. 35. y 5 0 36. 19. 21. 23. 25. 26. 27. 29. 31. 33. x2 y2 16 y 4x2 36 2x 5y 10 x y2 4 y x 5 y 4 x y 75 x10 In Exercises 3752, evaluate the function at each specified value of the independent variable and simplify. 37. f x 2x 3 a f 1 b f 3 38. g y 7 3y 7 a g0 b g 3 4 39. Vr 3 r 3 3 a V3 b V 2 40. Sr 4r2 1 a S2 b S2 2 41. gt 4t 3t 5 a g2 b gt 2 c f x 1 c gs 2 c V 2r c S3r c gt g2 42. ht t 2 2t a h2 b 43. f y 3 y a f 4 b 44. f x x 8 2 a f 8 b 2 45. qx 1x 9 a q0 b 46. qt 2t2 3t2 a q2 b 47. f x xx a f 2 b 48. f x x 4 a f 2 b 49. f x 2x2x 1,2, Functions h1.5 c hx 2 f 0.25 c f 4x 2 f 1 c f x 8 q3 c q y 3 q0 c qx f 2 c f x 1 f 2 c f x2 x 0 x 0 b f 0 a f 1 x 2 2, x 1 50. f x 2x 2 2, x 1 a f 2 b f 1 3x 1, x 1 51. f x 4, 1 x 1 x2, x 1 a f 2 b f 12 4 5x, x 2 52. f x 0, 2 x 2 x2 1, x 2 a f 3 b f 4 195 c f 2 c f 2 c f 3 c f 1 In Exercises 5358, complete the table. 53. f x x 2 3 x 2 1 0 1 6 7 2 f x 54. gx x 3 x 3 4 5 gx 55. ht 12t 3 t 5 4 3 2 1 ht www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 196 Chapter 2 56. f s s 2 s Functions and Their Graphs In Exercises 83 86, assume that the domain of f is the set A 2, 1, 0, 1, 2 . Determine the set of ordered pairs that represents the function f. s2 0 3 2 1 5 2 83. f x x 2 85. f x x 2 4 f s 12x 4, 57. f x x 22, x 2 87. GEOMETRY Write the area A of a square as a function of its perimeter P. 88. GEOMETRY Write the area A of a circle as a function of its circumference C. 89. MAXIMUM VOLUME An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides see figure . x 0 x 0 1 0 1 2 f x 58. f x x x 3, 9 x 2, 1 2 x 3 x 3 3 4 84. f x x 32 86. f x x 1 5 x f x 24 2x In Exercises 59 66, find all real values of x such that f x 0. 59. f x 15 3x 60. f x 5x 1 3x 4 12 x2 61. f x 62. f x 5 5 63. f x x 2 9 64. f x x 2 8x 15 65. f x x 3 x 66. f x x3 x 2 4x 4 In Exercises 6770, find the value s of x for which f x gx. 67. f x x2, gx x 2 68. f x x 2 2x 1, gx 7x 5 69. f x x 4 2x 2, gx 2x 2 70. f x x 4, gx 2 x In Exercises 7182, find the domain of the function. 71. f x 5x 2 2x 1 4 73. ht t 75. g y y 10 1 3 77. gx x x2 s 1 79. f s s4 81. f x x4 x 72. gx 1 2x 2 3y 74. s y y5 3 t 4 76. f t 10 78. hx 2 x 2x 80. f x 82. f x 24 2x x x a The table shows the volumes V in cubic centimeters of the box for various heights x in centimeters . Use the table to estimate the maximum volume. Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1024 980 864 b Plot the points x, V from the table in part a . Does the relation defined by the ordered pairs represent V as a function of x c If V is a function of x, write the function and determine its domain. 90. MAXIMUM PROFIT The cost per unit in the production of an MP3 player is 60. The manufacturer charges 90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 per MP3 player for each unit ordered in excess of 100 for example, there would be a charge of 87 per MP3 player for an order size of 120 . a The table shows the profits P in dollars for various numbers of units ordered, x. Use the table to estimate the maximum profit. x 6 6x x2 x 10 Units, x 110 120 130 140 Profit, P 3135 3240 3315 3360 Units, x 150 160 170 Profit, P 3375 3360 3315 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 b Plot the points x, P from the table in part a . Does the relation defined by the ordered pairs represent P as a function of x c If P is a function of x, write the function and determine its domain. 91. GEOMETRY A right triangle is formed in the first quadrant by the x and y axes and a line through the point 2, 1 see figure . Write the area A of the triangle as a function of x, and determine the domain of the function. y 4 Number of prescriptions in millions d 750 740 730 720 710 700 690 t y 0, b 8 0 4 2, 1 a, 0 1 x 1 FIGURE FOR 2 3 x, y 2 4 91 x 6 4 2 FIGURE FOR 2 4 6 92 92. GEOMETRY A rectangle is bounded by the x axis and the semicircle y 36 x 2 see figure . Write the area A of the rectangle as a function of x, and graphically determine the domain of the function. 93. PATH OF A BALL The height y in feet of a baseball thrown by a child is y FIGURE FOR pt 3 5 4 6 7 94 1.011t2 12.38t 170.5, 8 t 13 6.950t2 222.55t 1557.6, 14 t 17 where t represents the year, with t 8 corresponding to 1998. Use this model to find the median sale price of an existing one family home in each year from 1998 through 2007. Source: National Association of Realtors 1 2 x 3x 6 10 p 250 Median sale price in thousands of dollars 94. PRESCRIPTION DRUGS The numbers d in millions of drug prescriptions filled by independent outlets in the United States from 2000 through 2007 see figure can be approximated by the model 699, 10.6t 15.5t 637, 2 95. MEDIAN SALES PRICE The median sale prices p in thousands of dollars of an existing one family home in the United States from 1998 through 2007 see figure can be approximated by the model where x is the horizontal distance in feet from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet. dt 1 Year 0 2000 36 x 2 y= 3 2 197 Functions 200 150 100 50 0 t 4 5 t 7 t 8 where t represents the year, with t 0 corresponding to 2000. Use this model to find the number of drug prescriptions filled by independent outlets in each year from 2000 through 2007. Source: National Association of Chain Drug Stores 9 10 11 12 13 14 15 16 17 Year 8 1998 96. POSTAL REGULATIONS A rectangular package to be sent by the U.S. Postal Service can have a maximum combined length and girth perimeter of a cross section of 108 inches see figure . x x y www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 198 Chapter 2 Functions and Their Graphs a Write the volume V of the package as a function of x. What is the domain of the function b Use a graphing utility to graph your function. Be sure to use an appropriate window setting. c What dimensions will maximize the volume of the package Explain your answer. 97. COST, REVENUE, AND PROFIT A company produces a product for which the variable cost is 12.30 per unit and the fixed costs are 98,000. The product sells for 17.98. Let x be the number of units produced and sold. a The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost C as a function of the number of units produced. b Write the revenue R as a function of the number of units sold. c Write the profit P as a function of the number of units sold. Note: P R C 98. AVERAGE COST The inventor of a new game believes that the variable cost for producing the game is 0.95 per unit and the fixed costs are 6000. The inventor sells each game for 1.69. Let x be the number of games sold. a The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost C as a function of the number of games sold. b Write the average cost per unit C Cx as a function of x. 99. TRANSPORTATION For groups of 80 or more people, a charter bus company determines the rate per person according to the formula n 90 100 110 120 130 140 150 Rn 100. PHYSICS The force F in tons of water against the face of a dam is estimated by the function F y 149.7610y 52, where y is the depth of the water in feet . a Complete the table. What can you conclude from the table 10 20 30 40 F y b Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. c Find the depth at which the force against the dam is 1,000,000 tons algebraically. 101. HEIGHT OF A BALLOON A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. a Draw a diagram that gives a visual representation of the problem. Let h represent the height of the balloon and let d represent the distance between the balloon and the receiving station. b Write the height of the balloon as a function of d. What is the domain of the function 102. E FILING The table shows the numbers of tax returns in millions made through e file from 2000 through 2007. Let f t represent the number of tax returns made through e file in the year t. Source: Internal Revenue Service Rate 8 0.05n 80, n 80 where the rate is given in dollars and n is the number of people. a Write the revenue R for the bus company as a function of n. b Use the function in part a to complete the table. What can you conclude 5 y Year Number of tax returns made through e file 2000 35.4 2001 40.2 2002 46.9 2003 52.9 2004 61.5 2005 68.5 2006 73.3 2007 80.0 f 2007 f 2000 and interpret the result in 2007 2000 the context of the problem. b Make a scatter plot of the data. c Find a linear model for the data algebraically. Let N represent the number of tax returns made through e file and let t 0 correspond to 2000. d Use the model found in part c to complete the table. a Find t 0 1 2 3 4 5 6 7 N www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.2 e Compare your results from part d with the actual data. f Use a graphing utility to find a linear model for the data. Let x 0 correspond to 2000. How does the model you found in part c compare with the model given by the graphing utility In Exercises 103110, find the difference quotient and simplify your answer. f 2 h f 2 , h 0 h f 5 h f 5 f x 5x x 2, , h 0 h f x h f x f x x 3 3x, , h 0 h f x h f x f x 4x2 2x, , h 0 h 1 gx g3 g x 2, , x 3 x x3 1 f t f 1 f t , , t 1 t2 t1 103. f x x 2 x 1, 104. 105. 106. 107. 108. 109. f x 5x, f x f 5 , x5 x f x f 8 , x8 110. f x x23 1, 5 x 112. 113. 4 1 0 1 4 y 6 3 0 3 6 EXPLORATION TRUE OR FALSE In Exercises 115118, determine whether the statement is true or false. Justify your answer. 115. Every relation is a function. 116. Every function is a relation. 117. The domain of the function given by f x x 4 1 is , , and the range of f x is 0, . 118. The set of ordered pairs 8, 2, 6, 0, 4, 0, 2, 2, 0, 4, 2, 2 represents a function. 119. THINK ABOUT IT f x x 1 and Consider gx 1 x 1 . Why are the domains of f and g different 120. THINK ABOUT IT Consider f x x 2 and 3 gx x 2. Why are the domains of f and g different 121. THINK ABOUT IT Given f x x2, is f the independent variable Why or why not 122. CAPSTONE a Describe any differences between a relation and a function. b In your own words, explain the meanings of domain and range. In Exercises 123 and 124, determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. x 4 1 0 1 4 y 32 2 0 2 32 x 4 1 0 1 4 y 1 4 1 0 1 4 1 x 4 1 0 1 4 y 8 32 Undefined 32 8 The symbol x 199 8 In Exercises 111114, match the data with one of the following functions c f x cx, g x cx 2, h x cx, and r x x and determine the value of the constant c that will make the function fit the data in the table. 111. 114. Functions 123. a The sales tax on a purchased item is a function of the selling price. b Your score on the next algebra exam is a function of the number of hours you study the night before the exam. 124. a The amount in your savings account is a function of your salary. b The speed at which a free falling baseball strikes the ground is a function of the height from which it was dropped. indicates an example or exercise that highlights algebraic techniques specifically used in calculus. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 200 Chapter 2 Functions and Their Graphs 2.3 ANALYZING GRAPHS OF FUNCTIONS What you should learn The Graph of a Function Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions. Determine the average rate of change of a function. Identify even and odd functions. In Section 2.2, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective. The graph of a function f is the collection of ordered pairs x, f x such that x is in the domain of f. As you study this section, remember that x the directed distance from the y axis y f x the directed distance from the x axis as shown in Figure 2.25. y Why you should learn it 2 Graphs of functions can help you visualize relationships between variables in real life. For instance, in Exercise 110 on page 210, you will use the graph of a function to represent visually the temperature of a city over a 24 hour period. 1 FIGURE Example 1 1 5 y = f x 0, 3 1 x 2 3 4 2, 3 5 FIGURE 2.26 x 2.25 Finding the Domain and Range of a Function Solution 5, 2 1, 1 3 2 2 Use the graph of the function f, shown in Figure 2.26, to find a the domain of f, b the function values f 1 and f 2, and c the range of f. y Range f x x 1 1 4 y = f x Domain 6 a. The closed dot at 1, 1 indicates that x 1 is in the domain of f, whereas the open dot at 5, 2 indicates that x 5 is not in the domain. So, the domain of f is all x in the interval 1, 5. b. Because 1, 1 is a point on the graph of f, it follows that f 1 1. Similarly, because 2, 3 is a point on the graph of f, it follows that f 2 3. c. Because the graph does not extend below f 2 3 or above f 0 3, the range of f is the interval 3, 3. Now try Exercise 9. The use of dots open or closed at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 201 Analyzing Graphs of Functions By the definition of a function, at most one y value corresponds to a given x value. This means that the graph of a function cannot have two or more different points with the same x coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions. Vertical Line Test for Functions A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point. Example 2 Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure 2.27 represent y as a function of x. y y y 4 4 4 3 3 3 2 2 1 1 1 x 1 1 1 4 5 x x 1 2 3 4 1 2 a FIGURE b 1 2 3 4 1 c 2.27 Solution a. This is not a graph of y as a function of x, because you can find a vertical line that intersects the graph twice. That is, for a particular input x, there is more than one output y. b. This is a graph of y as a function of x, because every vertical line intersects the graph at most once. That is, for a particular input x, there is at most one output y. c. This is a graph of y as a function of x. Note that if a vertical line does not intersect the graph, it simply means that the function is undefined for that particular value of x. That is, for a particular input x, there is at most one output y. Now try Exercise 17. T E C H N O LO G Y Most graphing utilities are designed to graph functions of x more easily than other types of equations. For instance, the graph shown in Figure 2.27 a represents the equation x y 12 0. To use a graphing utility to duplicate this graph, you must first solve the equation for y to obtain y 1 x, and then graph the two equations y1 1 1 x and y2 1 x in the same viewing window. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 202 Chapter 2 Functions and Their Graphs Zeros of a Function If the graph of a function of x has an x intercept at a, 0, then a is a zero of the function. Zeros of a Function The zeros of a function f of x are the x values for which f x 0. f x = 3x 2 + x 10 y x 3 1 1 2 2, 0 Finding the Zeros of a Function Find the zeros of each function. 53 , 0 4 Example 3 2 a. f x 3x 2 x 10 6 b. gx 10 x 2 c. ht 2t 3 t5 Solution 8 To find the zeros of a function, set the function equal to zero and solve for the independent variable. Zeros of f: x 2, x 53 FIGURE 2.28 a. 3x 2 x 10 0 3x 5x 2 0 y 2 6 4 2 2 b. 10 x 2 0 6 10 c. 32 , 0 2 2 2 h t = 4 8 3 2 x2 Set gx equal to 0. Square each side. Add x 2 to each side. Extract square roots. t 4 6 2t 3 t+5 2t 3 0 t5 Set ht equal to 0. 2t 3 0 Multiply each side by t 5. 2t 3 t 6 Set 2nd factor equal to 0. The zeros of g are x 10 and x 10. In Figure 2.29, note that the graph of g has 10, 0 and 10, 0 as its x intercepts. y Zero of h: t FIGURE 2.30 0 10 x Zeros of g: x 10 FIGURE 2.29 4 x2 10 4 2 x 2 Set 1st factor equal to 0. The zeros of f are x and x 2. In Figure 2.28, note that the graph of f 5 has 3, 0 and 2, 0 as its x intercepts. 10, 0 4 5 3 5 3 x 2 x x20 g x = 10 x 2 4 10, 0 Factor. 3x 5 0 8 6 Set f x equal to 0. Add 3 to each side. 3 2 Divide each side by 2. The zero of h is t 32. In Figure 2.30, note that the graph of h has its t intercept. 32, 0 as Now try Exercise 23. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 203 Analyzing Graphs of Functions Increasing and Decreasing Functions y The more you know about the graph of a function, the more you know about the function itself. Consider the graph shown in Figure 2.31. As you move from left to right, this graph falls from x 2 to x 0, is constant from x 0 to x 2, and rises from x 2 to x 4. cre 3 g Inc n asi rea De sin g 4 1 Constant Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 x2 implies f x1 f x 2 . x 2 FIGURE 1 1 2 3 4 1 A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 x2 implies f x1 f x 2 . 2.31 A function f is constant on an interval if, for any x1 and x2 in the interval, f x1 f x 2 . Example 4 Increasing and Decreasing Functions Use the graphs in Figure 2.32 to describe the increasing or decreasing behavior of each function. Solution a. This function is increasing over the entire real line. b. This function is increasing on the interval , 1, decreasing on the interval 1, 1, and increasing on the interval 1, . c. This function is increasing on the interval , 0, constant on the interval 0, 2, and decreasing on the interval 2, . y y f x = x 3 3x y 1, 2 f x = x 3 2 2 1 0, 1 2, 1 1 x 1 1 x 2 1 1 t 2 1 1 f t = 1 a FIGURE 1 2 2 1, 2 b 2 3 t + 1, t 0 1, 0 t 2 t + 3, t 2 c 2.32 Now try Exercise 41. To help you decide whether a function is increasing, decreasing, or constant on an interval, you can evaluate the function for several values of x. However, calculus is needed to determine, for certain, all intervals on which a function is increasing, decreasing, or constant. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 204 Chapter 2 Functions and Their Graphs The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative minimum or relative maximum values of the function. A relative minimum or relative maximum is also referred to as a local minimum or local maximum. Definitions of Relative Minimum and Relative Maximum A function value f a is called a relative minimum of f if there exists an interval x1, x2 that contains a such that x1 x x2 implies y A function value f a is called a relative maximum of f if there exists an interval x1, x2 that contains a such that Relative maxima x1 x x2 Relative minima x FIGURE f a f x. 2.33 implies f a f x. Figure 2.33 shows several different examples of relative minima and relative maxima. In Section 3.1, you will study a technique for finding the exact point at which a second degree polynomial function has a relative minimum or relative maximum. For the time being, however, you can use a graphing utility to find reasonable approximations of these points. Example 5 Approximating a Relative Minimum Use a graphing utility to approximate the relative minimum of the function given by f x 3x 2 4x 2. Solution f x = 3x 2 The graph of f is shown in Figure 2.34. By using the zoom and trace features or the minimum feature of a graphing utility, you can estimate that the function has a relative minimum at the point 4x 2 2 4 5 0.67, 3.33. Relative minimum Later, in Section 3.1, you will be able to determine that the exact point at which the relative minimum occurs is 23, 10 3 . 4 FIGURE 2.34 Now try Exercise 57. You can also use the table feature of a graphing utility to approximate numerically the relative minimum of the function in Example 5. Using a table that begins at 0.6 and increments the value of x by 0.01, you can approximate that the minimum of f x 3x 2 4x 2 occurs at the point 0.67, 3.33. T E C H N O LO G Y If you use a graphing utility to estimate the x and y values of a relative minimum or relative maximum, the zoom feature will often produce graphs that are nearly flat. To overcome this problem, you can manually change the vertical setting of the viewing window. The graph will stretch vertically if the values of Ymin and Ymax are closer together. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 Analyzing Graphs of Functions 205 Average Rate of Change y In Section 2.1, you learned that the slope of a line can be interpreted as a rate of change. For a nonlinear graph whose slope changes at each point, the average rate of change between any two points x1, f x1 and x2, f x2 is the slope of the line through the two points see Figure 2.35 . The line through the two points is called the secant line, and the slope of this line is denoted as msec. x2, f x2 x1, f x1 x2 x1 x1 FIGURE Secant line f Average rate of change of f from x1 to x2 f x2 f x 1 2.35 Example 6 y f x = x3 change in y change in x msec x x2 f x2 f x1 x2 x1 Average Rate of Change of a Function Find the average rates of change of f x x3 3x a from x1 2 to x2 0 and b from x1 0 to x2 1 see Figure 2.36 . 3x Solution 2 a. The average rate of change of f from x1 2 to x2 0 is 0, 0 3 2 1 x 1 2 1 2, 2 3 FIGURE 3 f x2 f x1 f 0 f 2 0 2 1. x2 x1 0 2 2 Secant line has positive slope. b. The average rate of change of f from x1 0 to x2 1 is 1, 2 f x2 f x1 f 1 f 0 2 0 2. x2 x1 10 1 Secant line has negative slope. Now try Exercise 75. 2.36 Example 7 Finding Average Speed The distance s in feet a moving car is from a stoplight is given by the function st 20t 32, where t is the time in seconds . Find the average speed of the car a from t1 0 to t2 4 seconds and b from t1 4 to t2 9 seconds. Solution a. The average speed of the car from t1 0 to t2 4 seconds is s t2 s t1 s 4 s 0 160 0 40 feet per second. t2 t1 4 0 4 b. The average speed of the car from t1 4 to t2 9 seconds is s t2 s t1 s 9 s 4 540 160 76 feet per second. t2 t1 94 5 Now try Exercise 113. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 206 Chapter 2 Functions and Their Graphs Even and Odd Functions In Section 1.1, you studied different types of symmetry of a graph. In the terminology of functions, a function is said to be even if its graph is symmetric with respect to the y axis and to be odd if its graph is symmetric with respect to the origin. The symmetry tests in Section 1.1 yield the following tests for even and odd functions. Tests for Even and Odd Functions A function y f x is even if, for each x in the domain of f, f x f x. A function y f x is odd if, for each x in the domain of f, f x f x. Example 8 Even and Odd Functions a. The function gx x 3 x is odd because gx gx, as follows. gx x 3 x Substitute x for x. x 3 x Simplify. x 3 x Distributive Property gx Test for odd function b. The function hx x2 1 is even because hx hx, as follows. hx x 1 2 Substitute x for x. x2 1 Simplify. hx Test for even function The graphs and symmetry of these two functions are shown in Figure 2.37. y y 6 3 g x = x 3 x 5 x, y 1 3 x 2 x, y 4 1 2 3 3 x, y 1 x, y 2 h x = x 2 + 1 2 3 a Symmetric to origin: Odd Function FIGURE 3 2 1 x 1 2 3 b Symmetric to y axis: Even Function 2.37 Now try Exercise 83. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 2.3 EXERCISES 207 Analyzing Graphs of Functions See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The graph of a function f is the collection of ________ ________ x, f x such that x is in the domain of f. 2. The ________ ________ ________ is used to determine whether the graph of an equation is a function of y in terms of x. 3. The ________ of a function f are the values of x for which f x 0. 4. A function f is ________ on an interval if, for any x1 and x2 in the interval, x1 x2 implies f x1 f x2 . 5. A function value f a is a relative ________ of f if there exists an interval x1, x2 containing a such that x1 x x2 implies f a f x. 6. The ________ ________ ________ ________ between any two points x1, f x1 and x2, f x2 is the slope of the line through the two points, and this line is called the ________ line. 7. A function f is ________ if, for each x in the domain of f, f x f x. 8. A function f is ________ if its graph is symmetric with respect to the y axis. SKILLS AND APPLICATIONS In Exercises 9 12, use the graph of the function to find the domain and range of f. y 9. 6 15. a f 2 c f 3 y 10. y b f 1 d f 1 y = f x 16. a f 2 c f 0 y = f x 2 y = f x 4 4 2 2 x 2 2 4 2 y 11. 6 4 y = f x x 2 4 6 4 4 y = f x 4 2 4 6 In Exercises 1722, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. y = f x x 2 4 17. y 12x 2 2 x 2 2 x 2 4 x 2 2 y 12. 2 2 2 2 2 4 y 6 2 4 b f 1 d f 2 4 2 18. y 14x 3 y y 4 4 6 2 In Exercises 1316, use the graph of the function to find the domain and range of f and the indicated function values. 13. a f 2 c f 12 b f 1 d f 1 y = f x y 14. a f 1 c f 0 4 x 2 2 x 2 4 4 19. x y 2 1 4 x 2 2 4 4 20. x 2 y 2 25 y 4 6 4 2 2 x 4 2 www.elsolucionario.net 4 2 y 3 4 4 4 2 2 x 3 y y = f x 4 3 2 b f 2 d f 1 4 6 2 x 2 4 6 4 6 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 208 Chapter 2 Functions and Their Graphs 22. x y 2 21. x 2 2xy 1 y 43. f x x 1 x 1 44. f x y x2 x 1 x1 y y 2 4 x 2 4 2 2 2 2 x 4 4 6 6 8 0, 1 4 4 23. f x 2x 2 7x 30 24. f x 3x 2 22x 16 x 9x 2 4 25. f x 27. 28. 29. 30. 31. 26. f x x 9x 14 4x 2 f x 12 x 3 x f x x 3 4x 2 9x 36 f x 4x 3 24x 2 x 6 f x 9x 4 25x 2 f x 2x 1 32. f x 3x 2 38. f x 4 2 46. f x 2xx 2,1, x 1 x 1 2 y 2 2x 2 9 3x y 4 2 x 2 4 2 4 x 2 41. f x x3 3x 2 2 6 2 2, 4 4 42. f x x 2 1 y y x 2 2 4 4 40. f x x 2 4x y In Exercises 4756, a use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and b make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part a . 47. f x 3 s2 49. gs 4 51. f t t 4 53. f x 1 x 55. f x x 32 48. gx x 50. hx x2 4 52. f x 3x 4 6x 2 54. f x xx 3 56. f x x23 6 4 2 x 2 36. f x 3x 14 8 39. f x 32 x 2 y 34. f x xx 7 3x 1 x6 2 x 3, x 0 45. f x 3, 0 x 2 2x 1, x 2 4 In Exercises 39 46, determine the intervals over which the function is increasing, decreasing, or constant. 4 4 4 5 x 35. f x 2x 11 37. f x 2 6 In Exercises 3338, a use a graphing utility to graph the function and find the zeros of the function and b verify your results from part a algebraically. 33. f x 3 x 2 x 2 In Exercises 2332, find the zeros of the function algebraically. 2 2, 3 2 1, 2 1, 2 6 4 4 0, 2 4 x 2 2, 2 4 2 1, 0 1, 0 4 2 2 x 4 2 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 In Exercises 5766, use a graphing utility to graph the function and approximate to two decimal places any relative minimum or relative maximum values. 57. 59. 61. 62. 63. 64. 65. 66. f x x 4x 2 f x x2 3x 2 f x xx 2x 3 f x x3 3x 2 x 1 gx 2x3 3x2 12x hx x3 6x2 15 hx x 1x gx x4 x 58. f x 3x 2 2x 5 60. f x 2x2 9x f x 4 x f x 9 x2 f x x 1 f x 1 x f x x6 2x 2 3 gx x 3 5x hx xx 5 f s 4s32 x 2 + 4x 1 4 1, 2 1, 3 3 h 1 y 102. y= 2 h 2 3, 2 y = 4x x 2 1 x x x 3 1 68. 70. 72. 74. x1 x1 x1 x1 x1 x1 x1 x1 x Values 0, x2 0, x2 1, x2 1, x2 1, x2 1, x2 3, x2 3, x2 84. 86. 88. 90. 103. 3 3 5 5 3 6 11 8 hx x 3 5 f t t 2 2t 3 f x x1 x 2 gs 4s 23 92. f x 9 94. f x 5 3x 96. f x x2 8 y x1 4 2 3 4 8, 2 h 3 4 y 104. y = 4x x 2 2, 4 4 f x 4x 2 f x x 2 4x f x x 2 f x 122 x In Exercises 91100, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answers algebraically. 91. f x 5 93. f x 3x 2 95. hx x2 4 y 3 In Exercises 8390, determine whether the function is even, odd, or neither. Then describe the symmetry. 83. 85. 87. 89. 3 t1 98. gt 100. f x x 5 In Exercises 101104, write the height h of the rectangle as a function of x. 4 In Exercises 75 82, find the average rate of change of the function from x1 to x2. Function 75. f x 2x 15 76. f x 3x 8 77. f x x2 12x 4 78. f x x2 2x 8 79. f x x3 3x2 x 80. f x x3 6x2 x 81. f x x 2 5 82. f x x 1 3 97. f x 1 x 99. f x x 2 101. In Exercises 6774, graph the function and determine the interval s for which f x 0. 67. 69. 71. 73. 209 Analyzing Graphs of Functions h 2 x y = 2x 1 3 4 x 2 x 1x 2 2 6 8 y = 3x 4 In Exercises 105108, write the length L of the rectangle as a function of y. y 105. 6 106. L y x= 4 8, 4 4 x= 1 2 y 2 4 6 2 y x 2 L 8 2 1 y x= 2 y 1 L 1 2 3 4 x = 2y y 4, 2 2 3 12 , 4 4 y2 x 2 y 108. 4 3 2y 2, 4 3 y 107. 3 1, 2 L x x 4 1 2 3 4 109. ELECTRONICS The number of lumens time rate of flow of light L from a fluorescent lamp can be approximated by the model L 0.294x 2 97.744x 664.875, 20 x 90 where x is the wattage of the lamp. a Use a graphing utility to graph the function. b Use the graph from part a to estimate the wattage necessary to obtain 2000 lumens. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 210 Chapter 2 Functions and Their Graphs 110. DATA ANALYSIS: TEMPERATURE The table shows the temperatures y in degrees Fahrenheit in a certain city over a 24 hour period. Let x represent the time of day, where x 0 corresponds to 6 A.M. Time, x Temperature, y 0 2 4 6 8 10 12 14 16 18 20 22 24 34 50 60 64 63 59 53 46 40 36 34 37 45 112. GEOMETRY Corners of equal size are cut from a square with sides of length 8 meters see figure . x 8 x x x 8 x x x x a Write the area A of the resulting figure as a function of x. Determine the domain of the function. b Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. c Identify the figure that would result if x were chosen to be the maximum value in the domain of the function. What would be the length of each side of the figure 113. ENROLLMENT RATE The enrollment rates r of children in preschool in the United States from 1970 through 2005 can be approximated by the model A model that represents these data is given by y 0.026x3 1.03x2 10.2x 34, 0 x 24. a Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. b How well does the model fit the data c Use the graph to approximate the times when the temperature was increasing and decreasing. d Use the graph to approximate the maximum and minimum temperatures during this 24 hour period. e Could this model be used to predict the temperatures in the city during the next 24 hour period Why or why not 111. COORDINATE AXIS SCALE Each function described below models the specified data for the years 1998 through 2008, with t 8 corresponding to 1998. Estimate a reasonable scale for the vertical axis e.g., hundreds, thousands, millions, etc. of the graph and justify your answer. There are many correct answers. a f t represents the average salary of college professors. b f t represents the U.S. population. c f t represents the percent of the civilian work force that is unemployed. r 0.021t2 1.44t 39.3, 0 t 35 where t represents the year, with t 0 corresponding to 1970. Source: U.S. Census Bureau a Use a graphing utility to graph the model. b Find the average rate of change of the model from 1970 through 2005. Interpret your answer in the context of the problem. 114. VEHICLE TECHNOLOGY SALES The estimated revenues r in millions of dollars from sales of in vehicle technologies in the United States from 2003 through 2008 can be approximated by the model r 157.30t2 397.4t 6114, 3 t 8 where t represents the year, with t 3 corresponding to 2003. Source: Consumer Electronics Association a Use a graphing utility to graph the model. b Find the average rate of change of the model from 2003 through 2008. Interpret your answer in the context of the problem. PHYSICS In Exercises 115 120, a use the position equation s 16t2 1 v0t 1 s0 to write a function that represents the situation, b use a graphing utility to graph the function, c find the average rate of change of the function from t1 to t2, d describe the slope of the secant line through t1 and t2 , e find the equation of the secant line through t1 and t2, and f graph the secant line in the same viewing window as your position function. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.3 115. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. t1 0, t2 3 116. An object is thrown upward from a height of 6.5 feet at a velocity of 72 feet per second. t1 0, t2 4 117. An object is thrown upward from ground level at a velocity of 120 feet per second. t1 3, t2 5 132. CONJECTURE Use the results of Exercise 131 to make a conjecture about the graphs of y x 7 and y x 8. Use a graphing utility to graph the functions and compare the results with your conjecture. 133. Use the information in Example 7 to find the average speed of the car from t1 0 to t2 9 seconds. Explain why the result is less than the value obtained in part b of Example 7. 134. Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. gx 2x 3 1 hx x 5 2x3 x t1 2, t2 5 119. An object is dropped from a height of 120 feet. jx 2 x 6 x 8 kx x 5 2x 4 x 2 t1 0, t2 2 120. An object is dropped from a height of 80 feet. t1 1, t2 2 EXPLORATION TRUE OR FALSE In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. A function with a square root cannot have a domain that is the set of real numbers. 122. It is possible for an odd function to have the interval 0, as its domain. 123. If f is an even function, determine whether g is even, odd, or neither. Explain. a gx f x b gx f x c gx f x 2 d gx f x 2 124. THINK ABOUT IT Does the graph in Exercise 19 represent x as a function of y Explain. px x9 3x 5 x 3 x What do you notice about the equations of functions that are odd What do you notice about the equations of functions that are even Can you describe a way to identify a function as odd or even by inspecting the equation Can you describe a way to identify a function as neither odd nor even by inspecting the equation 135. WRITING Write a short paragraph describing three different functions that represent the behaviors of quantities between 1998 and 2009. Describe one quantity that decreased during this time, one that increased, and one that was constant. Present your results graphically. 136. CAPSTONE Use the graph of the function to answer a e . y y = f x 8 6 THINK ABOUT IT In Exercises 125130, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is a even and b odd. 125. 4 127. 4, 9 129. x, y 211 f x x 2 x 4 118. An object is thrown upward from ground level at a velocity of 96 feet per second. 32, Analyzing Graphs of Functions 126. 7 128. 5, 1 130. 2a, 2c 4 2 x 4 53, 131. WRITING Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. a y x b y x 2 c y x 3 4 5 d y x e y x f y x 6 2 2 4 6 a Find the domain and range of f. b Find the zero s of f. c Determine the intervals over which f is increasing, decreasing, or constant. d Approximate any relative minimum or relative maximum values of f. e Is f even, odd, or neither www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 212 Chapter 2 Functions and Their Graphs 2.4 A LIBRARY OF PARENT FUNCTIONS What you should learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal functions. Identify and graph step and other piecewise defined functions. Recognize graphs of parent functions. Why you should learn it Step functions can be used to model real life situations. For instance, in Exercise 69 on page 218, you will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this text is to enable you to recognize the basic shapes of the graphs of different types of functions. For instance, you know that the graph of the linear function f x ax b is a line with slope m a and y intercept at 0, b. The graph of the linear function has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an x intercept of bm, 0 and a y intercept of 0, b. The graph is increasing if m 0, decreasing if m 0, and constant if m 0. Example 1 Writing a Linear Function Write the linear function f for which f 1 3 and f 4 0. Solution To find the equation of the line that passes through x1, y1 1, 3 and x2, y2 4, 0, first find the slope of the line. m y2 y1 0 3 3 1 x2 x1 4 1 3 Next, use the point slope form of the equation of a line. Getty Images y y1 mx x1 Point slope form y 3 1x 1 Substitute for x1, y1, and m. y x 4 Simplify. f x x 4 Function notation The graph of this function is shown in Figure 2.38. y 5 4 f x = x + 4 3 2 1 1 x 1 1 FIGURE 2 3 4 5 2.38 Now try Exercise 11. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.4 213 A Library of Parent Functions There are two special types of linear functions, the constant function and the identity function. A constant function has the form f x c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line, as shown in Figure 2.39. The identity function has the form f x x. Its domain and range are the set of all real numbers. The identity function has a slope of m 1 and a y intercept at 0, 0. The graph of the identity function is a line for which each x coordinate equals the corresponding y coordinate. The graph is always increasing, as shown in Figure 2.40. y y 2 3 1 f x = c 2 2 1 x 1 1 2 1 x 1 FIGURE f x = x 2 2 3 2.39 FIGURE 2.40 The graph of the squaring function f x x2 is a U shaped curve with the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all nonnegative real numbers. The function is even. The graph has an intercept at 0, 0. The graph is decreasing on the interval , 0 and increasing on the interval 0, . The graph is symmetric with respect to the y axis. The graph has a relative minimum at 0, 0. The graph of the squaring function is shown in Figure 2.41. y f x = x 2 5 4 3 2 1 3 2 1 1 FIGURE x 1 2 3 0, 0 2.41 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 214 Chapter 2 Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below. 1. The graph of the cubic function f x x3 has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The function is odd. The graph has an intercept at 0, 0. The graph is increasing on the interval , . The graph is symmetric with respect to the origin. The graph of the cubic function is shown in Figure 2.42. 2. The graph of the square root function f x x has the following characteristics. The domain of the function is the set of all nonnegative real numbers. The range of the function is the set of all nonnegative real numbers. The graph has an intercept at 0, 0. The graph is increasing on the interval 0, . The graph of the square root function is shown in Figure 2.43. 1 has the following characteristics. x The domain of the function is , 0 0, . 3. The graph of the reciprocal function f x The range of the function is , 0 0, . The function is odd. The graph does not have any intercepts. The graph is decreasing on the intervals , 0 and 0, . The graph is symmetric with respect to the origin. The graph of the reciprocal function is shown in Figure 2.44. y y 3 f x = 0, 0 2 3 Cubic function FIGURE 2.42 f x = 3 1 1 3 4 2 3 2 y x 1 2 3 x 1 1 2 3 1 0, 0 1 1 x 2 2 x3 f x = x 1 2 3 4 1 x 1 5 2 Square root function FIGURE 2.43 www.elsolucionario.net Reciprocal function FIGURE 2.44 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.4 A Library of Parent Functions 215 Step and Piecewise Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted by x and defined as f x x the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. 1 greatest integer 1 1 y 12 greatest integer 12 1 101 greatest integer 101 0 3 2 1 x 4 3 2 1 1 2 3 4 The graph of the greatest integer function f x x f x = x 3 has the following characteristics, as shown in Figure 2.45. The domain of the function is the set of all real numbers. The range of the function is the set of all integers. The graph has a y intercept at 0, 0 and x intercepts in the interval 0, 1. The graph is constant between each pair of consecutive integers. The graph jumps vertically one unit at each integer value. 4 FIGURE 1.5 greatest integer 1.5 1 2.45 T E C H N O LO G Y Example 2 When graphing a step function, you should set your graphing utility to dot mode. Evaluating a Step Function Evaluate the function when x 1, 2, and 32. f x x 1 Solution For x 1, the greatest integer 1 is 1, so y f 1 1 1 1 1 0. 5 For x 2, the greatest integer 2 is 2, so 4 f 2 2 1 2 1 3. 3 2 f x = x + 1 1 3 2 1 2 FIGURE 2.46 x 1 2 3 4 5 3 For x 2, the greatest integer f 3 2 3 2 is 1, so 1 1 1 2. 3 2 You can verify your answers by examining the graph of f x x 1 shown in Figure 2.46. Now try Exercise 43. Recall from Section 2.2 that a piecewise defined function is defined by two or more equations over a specified domain. To graph a piecewise defined function, graph each equation separately over the specified domain, as shown in Example 3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 216 Chapter 2 Functions and Their Graphs Example 3 y 6 5 4 3 y = 2x + 3 Sketch the graph of y = x + 4 f x 1 5 4 3 x 1 1 2 3 4 6 x2x 3,4, x 1 . x 1 Solution This piecewise defined function is composed of two linear functions. At x 1 and to the left of x 1 the graph is the line y 2x 3, and to the right of x 1 the graph is the line y x 4, as shown in Figure 2.47. Notice that the point 1, 5 is a solid dot and the point 1, 3 is an open dot. This is because f 1 21 3 5. 2 3 4 5 6 FIGURE Graphing a Piecewise Defined Function Now try Exercise 57. 2.47 Parent Functions The eight graphs shown in Figure 2.48 represent the most commonly used functions in algebra. Familiarity with the basic characteristics of these simple graphs will help you analyze the shapes of more complicated graphsin particular, graphs obtained from these graphs by the rigid and nonrigid transformations studied in the next section. y y 3 f x = c 2 y f x = x 2 2 1 1 y f x = x 3 1 x 1 2 3 a Constant Function 1 2 2 1 1 1 1 2 2 b Identity Function 4 2 x 1 3 1 f x = 2 1 1 1 x 2 1 e Quadratic Function FIGURE 1 f x = x2 2 1 x 3 2 1 x 1 d Square Root Function 1 2 x 2 3 y 2 2 2 y 2 3 1 c Absolute Value Function y y x x x 2 1 f x = 2 1 2 3 3 2 1 f x = x 3 x 1 2 3 f x = x 3 f Cubic Function g Reciprocal Function h Greatest Integer Function 2.48 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.4 2.4 EXERCISES A Library of Parent Functions 217 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 19, match each function with its name. 1. f x x 2. f x x 3. f x 1x 4. f x 7. f x x a squaring function d linear function g greatest integer function 5. f x x 8. f x x3 b square root function e constant function h reciprocal function 6. f x c 9. f x ax b c cubic function f absolute value function i identity function x2 10. Fill in the blank: The constant function and the identity function are two special types of ________ functions. SKILLS AND APPLICATIONS In Exercises 1118, a write the linear function f such that it has the indicated function values and b sketch the graph of the function. 11. 13. 15. 16. 17. 18. f 1 4, f 0 6 12. f 3 8, f 1 2 14. f 3 9, f 1 11 f 5 4, f 2 17 f 5 1, f 5 1 f 10 12, f 16 1 f 12 6, f 4 3 f 23 15 2 , f 4 11 In Exercises 1942, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. 19. 21. 23. 25. 27. 29. 31. 33. f x 0.8 x f x 16 x 52 gx 2x2 f x 3x2 1.75 f x x3 1 f x x 13 2 f x 4x gx 2 x 4 20. 22. 24. 26. 28. 30. 32. 34. f x 2.5x 4.25 f x 56 23x hx 1.5 x2 f x 0.5x2 2 f x 8 x3 gx 2x 33 1 f x 4 2x hx x 2 3 35. f x 1x 36. f x 4 1x 37. hx 1x 2 38. kx 1x 3 39. gx x 5 41. f x x 4 40. hx 3 x 42. f x x 1 In Exercises 4350, evaluate the function for the indicated values. 43. f x x a f 2.1 b f 2.9 c f 3.1 d f 72 44. g x 2x a g 3 b g 0.25 c g 9.5 d g 11 3 45. h x x 3 a h 2 b h12 46. f x 4x 7 a f 0 b f 1.5 47. h x 3x 1 a h 2.5 b h 3.2 48. k x 12x 6 a k 5 b k 6.1 49. gx 3x 2 5 a g 2.7 b g 1 50. gx 7x 4 6 a g 18 b g9 c h 4.2 d h21.6 c f 6 d f 53 c h73 d h 21 3 c k 0.1 d k15 c g 0.8 d g14.5 c g4 d g 32 In Exercises 5156, sketch the graph of the function. 51. g x x 53. g x x 2 54. g x x 1 55. g x x 1 56. g x x 3 52. g x 4 x In Exercises 57 64, graph the function. 3 x, x 0 x 6, x 4 58. gx x 4, x 4 4 x, x 0 59. f x 4 x, x 0 1 x 1 , x 2 60. f x x 2, x 2 x 5, x 1 61. f x x 4x 3, x 1 57. f x www.elsolucionario.net 2x 3, x 0 1 2 2 2 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 218 Chapter 2 62. h x x3 x2,, x 0 x 0 x 2 2 x 0 x 0 2 2 4 x2, 63. hx 3 x, x2 1, Functions and Their Graphs 2x 1, 64. kx 2x2 1, 1 x2, 73. REVENUE The table shows the monthly revenue y in thousands of dollars of a landscaping business for each month of the year 2008, with x 1 representing January. x 1 1 x 1 x 1 Month, x Revenue, y 1 2 3 4 5 6 7 8 9 10 11 12 5.2 5.6 6.6 8.3 11.5 15.8 12.8 10.1 8.6 6.9 4.5 2.7 In Exercises 6568, a use a graphing utility to graph the function, b state the domain and range of the function, and c describe the pattern of the graph. 65. sx 214x 14x 67. hx 412x 12x 66. gx 214x 14x 2 68. kx 412x 12x 2 69. DELIVERY CHARGES The cost of sending an overnight package from Los Angeles to Miami is 23.40 for a package weighing up to but not including 1 pound and 3.75 for each additional pound or portion of a pound. A model for the total cost C in dollars of sending the package is C 23.40 3.75x, x 0, where x is the weight in pounds. a Sketch a graph of the model. b Determine the cost of sending a package that weighs 9.25 pounds. 70. DELIVERY CHARGES The cost of sending an overnight package from New York to Atlanta is 22.65 for a package weighing up to but not including 1 pound and 3.70 for each additional pound or portion of a pound. a Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing x pounds, x 0. b Sketch the graph of the function. 71. WAGES A mechanic is paid 14.00 per hour for regular time and time and a half for overtime. The weekly wage function is given by 14h, Wh 21h 40 560, 0 h 40 h 40 where h is the number of hours worked in a week. a Evaluate W30, W40, W45, and W50. b The company increased the regular work week to 45 hours. What is the new weekly wage function 72. SNOWSTORM During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm A mathematical model that represents these data is f x 26.3 . 1.97x 0.505x 1.47x 6.3 2 a Use a graphing utility to graph the model. What is the domain of each part of the piecewise defined function How can you tell Explain your reasoning. b Find f 5 and f 11, and interpret your results in the context of the problem. c How do the values obtained from the model in part a compare with the actual data values EXPLORATION TRUE OR FALSE In Exercises 74 and 75, determine whether the statement is true or false. Justify your answer. 74. A piecewise defined function will always have at least one x intercept or at least one y intercept. 75. A linear equation will always have an x intercept and a y intercept. 76. CAPSTONE For each graph of f shown in Figure 2.48, do the following. a Find the domain and range of f. b Find the x and y intercepts of the graph of f. c Determine the intervals over which f is increasing, decreasing, or constant. d Determine whether f is even, odd, or neither. Then describe the symmetry. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.5 219 Transformations of Functions 2.5 TRANSFORMATIONS OF FUNCTIONS What you should learn Use vertical and horizontal shifts to sketch graphs of functions. Use reflections to sketch graphs of functions. Use nonrigid transformations to sketch graphs of functions. Why you should learn it Transformations of functions can be used to model real life applications. For instance, Exercise 79 on page 227 shows how a transformation of a function can be used to model the total numbers of miles driven by vans, pickups, and sport utility vehicles in the United States. Shifting Graphs Many functions have graphs that are simple transformations of the parent graphs summarized in Section 2.4. For example, you can obtain the graph of hx x 2 2 by shifting the graph of f x x 2 upward two units, as shown in Figure 2.49. In function notation, h and f are related as follows. hx x 2 2 f x 2 Upward shift of two units Similarly, you can obtain the graph of gx x 22 by shifting the graph of f x x 2 to the right two units, as shown in Figure 2.50. In this case, the functions g and f have the following relationship. gx x 22 f x 2 Right shift of two units h x = x 2 + 2 y y 4 4 3 3 f x = x 2 g x = x 2 2 Transtock Inc.Alamy 2 1 2 FIGURE 1 1 f x = x2 x 1 2 2.49 x 1 FIGURE 1 2 3 2.50 The following list summarizes this discussion about horizontal and vertical shifts. Vertical and Horizontal Shifts Let c be a positive real number. Vertical and horizontal shifts in the graph of y f x are represented as follows. WARNING CAUTION In items 3 and 4, be sure you see that hx f x c corresponds to a right shift and hx f x c corresponds to a left shift for c 0. 1. Vertical shift c units upward: hx f x c 2. Vertical shift c units downward: hx f x c 3. Horizontal shift c units to the right: hx f x c 4. Horizontal shift c units to the left: www.elsolucionario.net hx f x c http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 220 Chapter 2 Functions and Their Graphs Some graphs can be obtained from combinations of vertical and horizontal shifts, as demonstrated in Example 1 b . Vertical and horizontal shifts generate a family of functions, each with the same shape but at different locations in the plane. Example 1 Shifts in the Graphs of a Function Use the graph of f x x3 to sketch the graph of each function. a. gx x 3 1 b. hx x 23 1 Solution a. Relative to the graph of f x x 3, the graph of gx x 3 1 is a downward shift of one unit, as shown in Figure 2.51. f x = x 3 y 2 1 2 In Example 1 a , note that gx f x 1 and that in Example 1 b , hx f x 2 1. x 1 1 g x = x 3 1 2 FIGURE 2 2.51 b. Relative to the graph of f x x3, the graph of hx x 23 1 involves a left shift of two units and an upward shift of one unit, as shown in Figure 2.52. 3 h x = x + 2 + 1 y f x = x 3 3 2 1 4 2 x 1 1 2 1 2 3 FIGURE 2.52 Now try Exercise 7. In Figure 2.52, notice that the same result is obtained if the vertical shift precedes the horizontal shift or if the horizontal shift precedes the vertical shift. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.5 221 Transformations of Functions Reflecting Graphs y The second common type of transformation is a reflection. For instance, if you consider the x axis to be a mirror, the graph of 2 hx x 2 is the mirror image or reflection of the graph of 1 f x = x 2 2 x 1 1 2 f x x 2, as shown in Figure 2.53. h x = x 2 1 Reflections in the Coordinate Axes 2 FIGURE Reflections in the coordinate axes of the graph of y f x are represented as follows. 2.53 1. Reflection in the x axis: hx f x 2. Reflection in the y axis: hx f x Example 2 Finding Equations from Graphs The graph of the function given by f x x 4 is shown in Figure 2.54. Each of the graphs in Figure 2.55 is a transformation of the graph of f. Find an equation for each of these functions. 3 3 f x = x 4 1 1 3 3 3 3 y = g x 1 1 a FIGURE 5 2.54 FIGURE 3 y = h x b 2.55 Solution a. The graph of g is a reflection in the x axis followed by an upward shift of two units of the graph of f x x 4. So, the equation for g is gx x 4 2. b. The graph of h is a horizontal shift of three units to the right followed by a reflection in the x axis of the graph of f x x 4. So, the equation for h is hx x 34. Now try Exercise 15. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 222 Chapter 2 Example 3 Functions and Their Graphs Reflections and Shifts Compare the graph of each function with the graph of f x x . a. gx x b. hx x c. kx x 2 Algebraic Solution Graphical Solution a. The graph of g is a reflection of the graph of f in the x axis because a. Graph f and g on the same set of coordinate axes. From the graph in Figure 2.56, you can see that the graph of g is a reflection of the graph of f in the x axis. b. Graph f and h on the same set of coordinate axes. From the graph in Figure 2.57, you can see that the graph of h is a reflection of the graph of f in the y axis. c. Graph f and k on the same set of coordinate axes. From the graph in Figure 2.58, you can see that the graph of k is a left shift of two units of the graph of f, followed by a reflection in the x axis. gx x f x. b. The graph of h is a reflection of the graph of f in the y axis because hx x f x. y y c. The graph of k is a left shift of two units followed by a reflection in the x axis because 2 f x = x 3 x h x = kx x 2 1 f x 2. x 1 1 2 FIGURE x 1 2 1 3 1 2 f x = x 2 1 g x = x 1 2.56 FIGURE 2.57 y 2 f x = x 1 x 1 1 2 k x = x + 2 2 FIGURE 2.58 Now try Exercise 25. When sketching the graphs of functions involving square roots, remember that the domain must be restricted to exclude negative numbers inside the radical. For instance, here are the domains of the functions in Example 3. Domain of gx x: Domain of hx x: x 0 x 0 Domain of kx x 2: x 2 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.5 y 3 2 f x = x 1 FIGURE 2.59 x 1 2 Horizontal shifts, vertical shifts, and reflections are rigid transformations because the basic shape of the graph is unchanged. These transformations change only the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortiona change in the shape of the original graph. For instance, a nonrigid transformation of the graph of y f x is represented by gx cf x, where the transformation is a vertical stretch if c 1 and a vertical shrink if 0 c 1. Another nonrigid transformation of the graph of y f x is represented by hx f cx, where the transformation is a horizontal shrink if c 1 and a horizontal stretch if 0 c 1. Example 4 y Nonrigid Transformations Compare the graph of each function with the graph of f x x. 4 g x = 13x 223 Nonrigid Transformations h x = 3x 4 2 Transformations of Functions a. hx 3x f x = x b. gx 13x Solution a. Relative to the graph of f x x, the graph of 2 hx 3x 3f x 1 x 2 1 FIGURE 2.60 1 2 is a vertical stretch each y value is multiplied by 3 of the graph of f. See Figure 2.59. b. Similarly, the graph of gx 13x 13 f x y is a vertical shrink each y value is multiplied by Figure 2.60. 6 Example 5 f x = 2 x 3 x 2 3 4 b. hx f 2 x 1 a. Relative to the graph of f x 2 x3, the graph of y gx f 2x 2 2x3 2 8x3 6 is a horizontal shrink c 1 of the graph of f. See Figure 2.61. 5 4 3 h x = 2 18 x 3 4 3 2 1 b. Similarly, the graph of hx f 12 x 2 12 x 2 18 x3 3 is a horizontal stretch 0 c 1 of the graph of f. See Figure 2.62. 1 FIGURE Compare the graph of each function with the graph of f x 2 x3. Solution 2.61 f x = 2 of the graph of f. See Nonrigid Transformations a. gx f 2x 2 FIGURE Now try Exercise 29. g x = 2 8x 3 4 3 2 1 1 1 3 x 1 2 3 4 Now try Exercise 35. x3 2.62 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 224 Chapter 2 2.5 Functions and Their Graphs EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 15, fill in the blanks. 1. Horizontal shifts, vertical shifts, and reflections are called ________ transformations. 2. A reflection in the x axis of y f x is represented by hx ________, while a reflection in the y axis of y f x is represented by hx ________. 3. Transformations that cause a distortion in the shape of the graph of y f x are called ________ transformations. 4. A nonrigid transformation of y f x represented by hx f cx is a ________ ________ if c 1 and a ________ ________ if 0 c 1. 5. A nonrigid transformation of y f x represented by gx cf x is a ________ ________ if c 1 and a ________ ________ if 0 c 1. 6. Match the rigid transformation of y f x with the correct representation of the graph of h, where c 0. a hx f x c i A horizontal shift of f, c units to the right b hx f x c ii A vertical shift of f, c units downward c hx f x c iii A horizontal shift of f, c units to the left d hx f x c iv A vertical shift of f, c units upward SKILLS AND APPLICATIONS 7. For each function, sketch on the same set of coordinate axes a graph of each function for c 1, 1, and 3. a f x x c b f x x c c f x x 4 c 8. For each function, sketch on the same set of coordinate axes a graph of each function for c 3, 1, 1, and 3. a f x x c b f x x c c f x x 3 c 9. For each function, sketch on the same set of coordinate axes a graph of each function for c 2, 0, and 2. a f x x c b f x x c c f x x 1 c 10. For each function, sketch on the same set of coordinate axes a graph of each function for c 3, 1, 1, and 3. xx c,c, xx 00 x c , x 0 b f x x c , x 0 a f x In Exercises 1114, use the graph of f to sketch each graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 11. a b c d e f g y f x 2 y f x 2 y 2 f x y f x y f x 3 y f x y f 12 x y 6 4 2 4 2 2 2 2 y 4 3, 1 1, 0 2 FIGURE FOR 8 4, 2 4, 2 f 6, 2 f x 2 4 0, 1 6 11 13. a y f x 1 b c d e f g y f x y f x 4 y 2 f x y f x 4 y f x 3 y f x 1 y f 2x 12. a b c d e f g y f x 1 y f x y f x 1 y f x 2 y 12 f x y f 2x www.elsolucionario.net 4 x 4 0, 2 2, 62 FIGURE FOR 8 12 14. a y f x 5 b c d e f g y f x 3 y 13 f x y f x 1 y f x y f x 10 y f 13 x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.5 y 2, 4 f 0, 5 3, 0 2 0, 3 2 10 6 1, 0 4 2 3, 0 x 6 2 f 6, 4 6 6, 4 x 4 2 2 6 3, 1 4 13 FIGURE FOR 17. Use the graph of f x x to write an equation for each function whose graph is shown. y y a b y 6 FIGURE FOR 225 Transformations of Functions x 6 10 4 14 2 4 14 15. Use the graph of f x x to write an equation for each function whose graph is shown. y y a b 4 2 y c 6 x 2 2 y d x 2 1 3 1 x 2 1 1 2 1 x 4 6 4 6 18. Use the graph of f x x to write an equation for each function whose graph is shown. y y a b y d 6 4 4 2 2 4 2 x 2 2 x 2 4 4 6 8 6 8 10 8 8 10 y c 2 2 x 2 x 1 2 2 x 4 x 4 6 4 x 2 4 6 8 10 8 10 In Exercises 1924, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. 4 2 4 x 4 2 4 3 y d 4 2 2 2 2 1 2 y 4 8 10 6 1 6 6 y d 8 4 c 4 4 6 3 1 2 4 3 1 x 2 x 2 6 16. Use the graph of f x x3 to write an equation for each function whose graph is shown. y y a b 2 12 3 y 2 8 4 2 x 1 2 2 c 4 2 4 8 y 19. y 20. 2 2 8 12 x 2 2 www.elsolucionario.net x 2 4 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 226 Chapter 2 Functions and Their Graphs y 21. y 22. 6 x 2 2 4 2 2 4 2 y 23. x 2 4 y 24. 59. The shape of f x x, but shifted 12 units upward and reflected in the x axis 60. The shape of f x x, but shifted four units to the left and eight units downward 61. The shape of f x x, but shifted six units to the left and reflected in both the x axis and the y axis 62. The shape of f x x, but shifted nine units downward and reflected in both the x axis and the y axis 63. Use the graph of f x x 2 to write an equation for each function whose graph is shown. y y a b 2 4 x 1 4 4 2 2 x 1, 7 x 3 2 1 1 2 3 1, 3 In Exercises 25 54, g is related to one of the parent functions described in Section 2.4. a Identify the parent function f. b Describe the sequence of transformations from f to g. c Sketch the graph of g. d Use function notation to write g in terms of f. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51. 53. g x 12 x 2 g x x 3 7 gx 23 x2 4 g x 2 x 52 gx 3 2x 4 2 gx 3x g x x 13 2 gx 3x 2 3 g x x 2 g x x 4 8 gx 2x 1 4 g x 3 x g x x 9 g x 7 x 2 g x 12 x 4 26. 28. 30. 32. 34. 36. 38. 40. 42. 44. 46. 48. 50. 52. 54. g x x 82 g x x 3 1 gx 2x 72 g x x 102 5 gx 14x 22 2 gx 14 x g x x 33 10 gx 12x 13 g x 6 x 5 g x x 3 9 gx 12x 2 3 g x 2x 5 g x x 4 8 g x 12x 3 1 g x 3x 1 In Exercises 5562, write an equation for the function that is described by the given characteristics. 55. The shape of f x x 2, but shifted three units to the right and seven units downward 56. The shape of f x x 2, but shifted two units to the left, nine units upward, and reflected in the x axis 57. The shape of f x x3, but shifted 13 units to the right 58. The shape of f x x3, but shifted six units to the left, six units downward, and reflected in the y axis 2 5 x 2 4 2 64. Use the graph of f x x 3 to write an equation for each function whose graph is shown. y y a b 6 3 2 4 2, 2 2 x 6 4 2 4 6 3 2 1 x 1 2 3 1, 2 2 3 4 6 65. Use the graph of f x x to write an equation for each function whose graph is shown. y y a b 8 4 6 2 x 4 6 4 6 4 2, 3 4, 2 4 2 8 x 2 4 6 4 66. Use the graph of f x x to write an equation for each function whose graph is shown. y a b y 20 16 12 8 4 1 4, 16 x 1 x 4 4 8 12 16 20 www.elsolucionario.net 1 4, 12 2 3 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.5 In Exercises 6772, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer. y 67. 1 4 3 2 4 3 2 1 2 3 x 2 1 1 2 2 x 3 2 1 y 69. 70. x 3 4 6 2 3 f 2 3 8 y 71. 4 2 4 2 1 6 4 2 x x 2 4 6 1 2 GRAPHICAL ANALYSIS In Exercises 73 76, use the viewing window shown to write a possible equation for the transformation of the parent function. 73. 74. 6 10 2 2 3 75. 76. 7 1 4 8 4 7 a gx f x 5 c gx f x e gx f 2x 1 1 b gx f x 2 d gx 4 f x 1 f gx f 4 x 2 79. MILES DRIVEN The total numbers of miles M in billions driven by vans, pickups, and SUVs sport utility vehicles in the United States from 1990 through 2006 can be approximated by the function M 527 128.0 t, 0 t 16 where t represents the year, with t 0 corresponding to 1990. Source: U.S. Federal Highway Administration 5 8 x 2 4 6 8 10 12 4 6 y 72. 2 4 3 2 1 1 b gx f x 1 d gx 2f x f gx f 12 x 6 4 x 1 x 1 2 3 4 5 y 78. 1 6 4 f a gx f x 2 c gx f x e gx f 4x 3 2 2 4 1 2 3 y 4 4 y 77. 5 4 2 227 GRAPHICAL REASONING In Exercises 77 and 78, use the graph of f to sketch the graph of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. y 68. Transformations of Functions 8 a Describe the transformation of the parent function f x x. Then use a graphing utility to graph the function over the specified domain. b Find the average rate of change of the function from 1990 to 2006. Interpret your answer in the context of the problem. c Rewrite the function so that t 0 represents 2000. Explain how you got your answer. d Use the model from part c to predict the number of miles driven by vans, pickups, and SUVs in 2012. Does your answer seem reasonable Explain. 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 228 Chapter 2 Functions and Their Graphs 80. MARRIED COUPLES The numbers N in thousands of married couples with stay at home mothers from 2000 through 2007 can be approximated by the function a The profits were only three fourths as large as expected. y 40,000 g 20,000 t N 24.70t 5.992 5617, 0 t 7 2 where t represents the year, with t 0 corresponding to 2000. Source: U.S. Census Bureau a Describe the transformation of the parent function f x x2. Then use a graphing utility to graph the function over the specified domain. b Find the average rate of the change of the function from 2000 to 2007. Interpret your answer in the context of the problem. c Use the model to predict the number of married couples with stay at home mothers in 2015. Does your answer seem reasonable Explain. EXPLORATION TRUE OR FALSE In Exercises 81 84, determine whether the statement is true or false. Justify your answer. 81. The graph of y f x is a reflection of the graph of y f x in the x axis. 82. The graph of y f x is a reflection of the graph of y f x in the y axis. 83. The graphs of f x x 6 and f x x 6 are identical. 84. If the graph of the parent function f x x 2 is shifted six units to the right, three units upward, and reflected in the x axis, then the point 2, 19 will lie on the graph of the transformation. 85. DESCRIBING PROFITS Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function f shown. The actual profits are shown by the function g along with a verbal description. Use the concepts of transformations of graphs to write g in terms of f. y f 40,000 20,000 t 2 4 b The profits were consistently 10,000 greater than predicted. 4 y 60,000 g 30,000 t 2 c There was a two year delay in the introduction of the product. After sales began, profits grew as expected. 4 y 40,000 g 20,000 t 2 4 6 86. THINK ABOUT IT You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function Explain. a f x 3x2 4x 1 b f x 2x 12 6 87. The graph of y f x passes through the points 0, 1, 1, 2, and 2, 3. Find the corresponding points on the graph of y f x 2 1. 88. Use a graphing utility to graph f, g, and h in the same viewing window. Before looking at the graphs, try to predict how the graphs of g and h relate to the graph of f. a f x x 2, gx x 42, hx x 42 3 b f x x 2, gx x 12, hx x 12 2 c f x x 2, gx x 42, hx x 42 2 89. Reverse the order of transformations in Example 2 a . Do you obtain the same graph Do the same for Example 2 b . Do you obtain the same graph Explain. 90. CAPSTONE Use the fact that the graph of y f x is increasing on the intervals , 1 and 2, and decreasing on the interval 1, 2 to find the intervals on which the graph is increasing and decreasing. If not possible, state the reason. a y f x b y f x c y 12 f x d y f x 1 e y f x 2 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.6 Combinations of Functions: Composite Functions 229 2.6 COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS What you should learn Arithmetic Combinations of Functions Add, subtract, multiply, and divide functions. Find the composition of one function with another function. Use combinations and compositions of functions to model and solve real life problems. Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. For example, the functions given by f x 2x 3 and gx x 2 1 can be combined to form the sum, difference, product, and quotient of f and g. f x gx 2x 3 x 2 1 Why you should learn it Compositions of functions can be used to model and solve real life problems. For instance, in Exercise 76 on page 237, compositions of functions are used to determine the price of a new hybrid car. x 2 2x 4 Sum f x gx 2x 3 x 1 2 x 2 2x 2 Difference f xgx 2x 3x 1 2 Jim WestThe Image Works 2x 3 3x 2 2x 3 2x 3 f x 2 , gx x 1 x Product 1 Quotient The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient f xgx, there is the further restriction that gx 0. Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows. 1. Sum: f gx f x gx 2. Difference: f gx f x gx 3. Product: fgx f x gx 4. Quotient: gx gx , Example 1 f f x gx 0 Finding the Sum of Two Functions Given f x 2x 1 and gx x 2 2x 1, find f gx. Then evaluate the sum when x 3. Solution f gx f x gx 2x 1 x 2 2x 1 x 2 4x When x 3, the value of this sum is f g3 32 43 21. Now try Exercise 9 a . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 230 Chapter 2 Functions and Their Graphs Example 2 Finding the Difference of Two Functions Given f x 2x 1 and gx x 2 2x 1, find f gx. Then evaluate the difference when x 2. Solution The difference of f and g is f gx f x gx 2x 1 x 2 2x 1 x 2 2. When x 2, the value of this difference is f g2 22 2 2. Now try Exercise 9 b . Example 3 Finding the Product of Two Functions Given f x x2 and gx x 3, find fgx. Then evaluate the product when x 4. Solution fg x f xgx x2x 3 x3 3x2 When x 4, the value of this product is fg4 43 342 16. Now try Exercise 9 c . In Examples 13, both f and g have domains that consist of all real numbers. So, the domains of f g, f g, and fg are also the set of all real numbers. Remember that any restrictions on the domains of f and g must be considered when forming the sum, difference, product, or quotient of f and g. Example 4 Finding the Quotients of Two Functions Find fgx and gf x for the functions given by f x x and gx 4 x 2 . Then find the domains of fg and gf. Solution The quotient of f and g is f x x gx gx 4 x f 2 and the quotient of g and f is Note that the domain of fg includes x 0, but not x 2, because x 2 yields a zero in the denominator, whereas the domain of gf includes x 2, but not x 0, because x 0 yields a zero in the denominator. gx f x f x g 4 x 2 x . The domain of f is 0, and the domain of g is 2, 2. The intersection of these domains is 0, 2. So, the domains of fg and gf are as follows. Domain of fg : 0, 2 Domain of gf : 0, 2 Now try Exercise 9 d . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.6 Combinations of Functions: Composite Functions 231 Composition of Functions Another way of combining two functions is to form the composition of one with the other. For instance, if f x x 2 and gx x 1, the composition of f with g is f gx f x 1 x 12. This composition is denoted as f g and reads as f composed with g. f g Definition of Composition of Two Functions g x x f g x f g Domain of g Domain of f FIGURE The composition of the function f with the function g is f gx f gx. The domain of f g is the set of all x in the domain of g such that gx is in the domain of f. See Figure 2.63. 2.63 Example 5 Composition of Functions Given f x x 2 and gx 4 x2, find the following. a. f gx b. g f x c. g f 2 Solution a. The composition of f with g is as follows. The following tables of values help illustrate the composition f gx given in Example 5. x gx 0 4 1 3 2 0 3 5 f gx f gx Definition of f g f 4 x 2 Definition of gx 4 Definition of f x 2 x2 x 6 2 Simplify. b. The composition of g with f is as follows. gx 4 3 0 5 f gx 6 5 2 3 x f gx 0 6 1 5 2 2 3 3 g f x g f x Definition of g f gx 2 Definition of f x 4 x 2 Definition of gx 4 Expand. 2 x2 4x 4 x 2 4x Note that the first two tables can be combined or composed to produce the values given in the third table. Note that, in this case, f gx Simplify. g f x. c. Using the result of part b , you can write the following. g f 2 22 42 Substitute. 4 8 Simplify. 4 Simplify. Now try Exercise 37. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 232 Chapter 2 Example 6 Functions and Their Graphs Finding the Domain of a Composite Function Find the domain of f gx for the functions given by f x x2 9 gx 9 x2. and Algebraic Solution Graphical Solution The composition of the functions is as follows. You can use a graphing utility to graph the composition of the functions 2 f gx as y 9 x2 9. Enter the functions as follows. f gx f gx y1 9 x2 f 9 x 2 y2 y12 9 Graph y2, as shown in Figure 2.64. Use the trace feature to determine that the x coordinates of points on the graph extend from 3 to 3. So, you can graphically estimate the domain of f g to be 3, 3. 9 x 2 9 2 9 x2 9 x 2 y= From this, it might appear that the domain of the composition is the set of all real numbers. This, however, is not true. Because the domain of f is the set of all real numbers and the domain of g is 3, 3, the domain of f g is 3, 3. 2 9 x2 9 0 4 4 12 FIGURE 2.64 Now try Exercise 41. In Examples 5 and 6, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For instance, the function h given by hx 3x 53 is the composition of f with g, where f x x3 and gx 3x 5. That is, hx 3x 53 gx3 f gx. Basically, to decompose a composite function, look for an inner function and an outer function. In the function h above, gx 3x 5 is the inner function and f x x3 is the outer function. Example 7 Decomposing a Composite Function Write the function given by hx 1 as a composition of two functions. x 22 Solution One way to write h as a composition of two functions is to take the inner function to be gx x 2 and the outer function to be f x 1 x2. x2 Then you can write hx 1 x 22 f x 2 f gx. x 22 Now try Exercise 53. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.6 Combinations of Functions: Composite Functions 233 Application Example 8 Bacteria Count The number N of bacteria in a refrigerated food is given by NT 20T 2 80T 500, 2 T 14 where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 4t 2, 0 t 3 where t is the time in hours. a Find the composition NTt and interpret its meaning in context. b Find the time when the bacteria count reaches 2000. Solution a. NTt 204t 22 804t 2 500 2016t 2 16t 4 320t 160 500 320t 2 320t 80 320t 160 500 320t 2 420 The composite function NTt represents the number of bacteria in the food as a function of the amount of time the food has been out of refrigeration. b. The bacteria count will reach 2000 when 320t 2 420 2000. Solve this equation to find that the count will reach 2000 when t 2.2 hours. When you solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. Now try Exercise 73. CLASSROOM DISCUSSION Analyzing Arithmetic Combinations of Functions a. Use the graphs of f and f 1 g in Figure 2.65 to make a table showing the values of gx when x 1, 2, 3, 4, 5, and 6. Explain your reasoning. b. Use the graphs of f and f h in Figure 2.65 to make a table showing the values of hx when x 1, 2, 3, 4, 5, and 6. Explain your reasoning. y y y 6 6 f 5 6 f+g 5 4 4 3 3 3 2 2 2 1 1 1 x 1 FIGURE 2 3 4 5 6 fh 5 4 x x 1 2 3 4 5 6 1 2 3 4 5 6 2.65 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 234 Chapter 2 2.6 Functions and Their Graphs EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. Two functions f and g can be combined by the arithmetic operations of ________, ________, ________, and _________ to create new functions. 2. The ________ of the function f with g is f gx f gx. 3. The domain of f g is all x in the domain of g such that _______ is in the domain of f. 4. To decompose a composite function, look for an ________ function and an ________ function. SKILLS AND APPLICATIONS In Exercises 5 8, use the graphs of f and g to graph hx f 1 gx. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. y 5. y 6. 2 f 2 g x 2 4 x 2 g 2 2 6 In Exercises 916, find a f 1 gx, b f gx, c fgx, and d fgx. What is the domain of fg x 2, gx x 2 2x 5, gx 2 x x 2, gx 4x 5 3x 1, gx 5x 4 x 2 6, gx 1 x x2 14. f x x2 4, gx 2 x 1 1 1 15. f x , gx 2 x x x , gx x 3 16. f x x1 9. 10. 11. 12. 13. In Exercises 2932, graph the functions f, g, and f 1 g on the same set of coordinate axes. 29. 30. 31. 32. f 20. 22. 24. 26. f x 12 x, f x 13 x, f x x 2, f x 4 gx x 1 gx x 4 gx 2x x 2, gx x 2 4 2 f y 8. 6 2 2 2 4 y 7. x 2 x g g 2 f 2 f g1 f gt 2 fg6 fg0 28. fg5 f 4 f g0 f g3t fg6 fg5 27. fg1 g3 19. 21. 23. 25. f x f x f x f x f x 18. f g1 33. f x 3x, gx x 34. f x , 2 x3 10 gx x 35. f x 3x 2, gx x 5 1 36. f x x2 2, gx 3x2 1 In Exercises 37 40, find a f g, b g f, and c g g. 37. f x x2, gx x 1 38. f x 3x 5, gx 5 x 3 x 1, gx x 3 1 39. f x 1 40. f x x 3, gx x In Exercises 1728, evaluate the indicated function for f x x 2 1 1 and gx x 4. 17. f g2 GRAPHICAL REASONING In Exercises 3336, use a graphing utility to graph f, g, and f 1 g in the same viewing window. Which function contributes most to the magnitude of the sum when 0 x 2 Which function contributes most to the magnitude of the sum when x 6 In Exercises 4148, find a f g and b g f. Find the domain of each function and each composite function. 41. f x x 4, gx x 2 3 x 5, 42. f x gx x 3 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.6 43. 44. 45. 46. R1 480 8t 0.8t 2, t 3, 4, 5, 6, 7, 8 where t 3 represents 2003. During the same six year period, the sales R 2 in thousands of dollars for the second restaurant can be modeled by 1 47. f x , gx x 3 x 3 , x2 1 gx x 1 R2 254 0.78t, t 3, 4, 5, 6, 7, 8. In Exercises 4952, use the graphs of f and g to evaluate the functions. y y = f x y 3 3 2 2 1 1 x x 1 49. 50. 51. 52. a a a a y = g x 4 4 2 3 f g3 f g1 f g2 f g1 1 4 b b b b 2 3 4 fg2 fg4 g f 2 g f 3 In Exercises 53 60, find two functions f and g such that f gx hx. There are many correct answers. 53. hx 2x 12 3 2 x 4 55. hx 1 57. hx x2 59. hx x 2 3 4 x2 235 62. SALES From 2003 through 2008, the sales R1 in thousands of dollars for one of two restaurants owned by the same parent company can be modeled by f x x 2 1, gx x f x x 23, gx x6 f x x, gx x 6 f x x 4, gx 3 x 48. f x Combinations of Functions: Composite Functions 54. hx 1 x3 56. hx 9 x 4 58. hx 5x 22 60. hx a Write a function R3 that represents the total sales of the two restaurants owned by the same parent company. b Use a graphing utility to graph R1, R2, and R3 in the same viewing window. 63. VITAL STATISTICS Let bt be the number of births in the United States in year t, and let dt represent the number of deaths in the United States in year t, where t 0 corresponds to 2000. a If pt is the population of the United States in year t, find the function ct that represents the percent change in the population of the United States. b Interpret the value of c5. 64. PETS Let dt be the number of dogs in the United States in year t, and let ct be the number of cats in the United States in year t, where t 0 corresponds to 2000. a Find the function pt that represents the total number of dogs and cats in the United States. b Interpret the value of p5. c Let nt represent the population of the United States in year t, where t 0 corresponds to 2000. Find and interpret 27x 3 6x 10 27x 3 ht pt . nt 61. STOPPING DISTANCE The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance in feet the car travels during the drivers reaction time is given by 3 Rx 4x, where x is the speed of the car in miles per hour. The distance in feet traveled while the driver is 1 braking is given by Bx 15 x 2. 65. MILITARY PERSONNEL The total numbers of Navy personnel N in thousands and Marines personnel M in thousands from 2000 through 2007 can be approximated by the models a Find the function that represents the total stopping distance T. b Graph the functions R, B, and T on the same set of coordinate axes for 0 x 60. where t represents the year, with t 0 corresponding to 2000. Source: Department of Defense a Find and interpret N Mt. Evaluate this function for t 0, 6, and 12. b Find and interpret N Mt Evaluate this function for t 0, 6, and 12. c Which function contributes most to the magnitude of the sum at higher speeds Explain. Nt 0.192t3 3.88t2 12.9t 372 and Mt 0.035t3 0.23t2 1.7t 172 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter 2 Functions and Their Graphs 66. SPORTS The numbers of people playing tennis T in millions in the United States from 2000 through 2007 can be approximated by the function Tt 0.0233t 4 0.3408t3 1.556t2 1.86t 22.8 and the U.S. population P in millions from 2000 through 2007 can be approximated by the function Pt 2.78t 282.5, where t represents the year, with t 0 corresponding to 2000. Source: Tennis Industry Association, U.S. Census Bureau a Find and interpret ht Tt . Pt b Evaluate the function in part a for t 0, 3, and 6. BIRTHS AND DEATHS In Exercises 67 and 68, use the table, which shows the total numbers of births B in thousands and deaths D in thousands in the United States from 1990 through 2006. Source: U.S. Census Bureau Year, t Births, B Deaths, D 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 4158 4111 4065 4000 3953 3900 3891 3881 3942 3959 4059 4026 4022 4090 4112 4138 4266 2148 2170 2176 2269 2279 2312 2315 2314 2337 2391 2403 2416 2443 2448 2398 2448 2426 69. GRAPHICAL REASONING An electronically controlled thermostat in a home is programmed to lower the temperature automatically during the night. The temperature in the house T in degrees Fahrenheit is given in terms of t, the time in hours on a 24 hour clock see figure . Temperature in F 236 T 80 70 60 50 t 3 6 9 12 15 18 21 24 Time in hours a Explain why T is a function of t. b Approximate T 4 and T 15. c The thermostat is reprogrammed to produce a temperature H for which Ht T t 1. How does this change the temperature d The thermostat is reprogrammed to produce a temperature H for which Ht T t 1. How does this change the temperature e Write a piecewise defined function that represents the graph. 70. GEOMETRY A square concrete foundation is prepared as a base for a cylindrical tank see figure . r x a Write the radius r of the tank as a function of the length x of the sides of the square. b Write the area A of the circular base of the tank as a function of the radius r. c Find and interpret A rx. The models for these data are Bt 0.197t3 1 8.96t2 90.0t 1 4180 and Dt 1.21t2 1 38.0t 1 2137 where t represents the year, with t 0 corresponding to 1990. 67. Find and interpret B Dt. 68. Evaluate Bt, Dt, and B Dt for the years 2010 and 2012. What does each function value represent 71. RIPPLES A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r in feet of the outer ripple is r t 0.6t, where t is the time in seconds after the pebble strikes the water. The area A of the circle is given by the function Ar r 2. Find and interpret A rt. 72. POLLUTION The spread of a contaminant is increasing in a circular pattern on the surface of a lake. The radius of the contaminant can be modeled by rt 5.25t, where r is the radius in meters and t is the time in hours since contamination. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.6 a Find a function that gives the area A of the circular leak in terms of the time t since the spread began. b Find the size of the contaminated area after 36 hours. c Find when the size of the contaminated area is 6250 square meters. 73. BACTERIA COUNT The number N of bacteria in a refrigerated food is given by NT 10T 2 20T 600, 1 T 20 where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 3t 2, 0 t 6 where t is the time in hours. a Find the composition NT t and interpret its meaning in context. b Find the bacteria count after 0.5 hour. c Find the time when the bacteria count reaches 1500. 74. COST The weekly cost C of producing x units in a manufacturing process is given by Cx 60x 750. The number of units x produced in t hours is given by xt 50t. a Find and interpret C xt. b Find the cost of the units produced in 4 hours. c Find the time that must elapse in order for the cost to increase to 15,000. 75. SALARY You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3 of your sales over 500,000. Consider the two f x x 500,000 and functions given by g x 0.03x. If x is greater than 500,000, which of the following represents your bonus Explain your reasoning. a f gx b g f x 76. CONSUMER AWARENESS The suggested retail price of a new hybrid car is p dollars. The dealership advertises a factory rebate of 2000 and a 10 discount. a Write a function R in terms of p giving the cost of the hybrid car after receiving the rebate from the factory. b Write a function S in terms of p giving the cost of the hybrid car after receiving the dealership discount. c Form the composite functions R S p and S R p and interpret each. d Find R S20,500 and S R20,500. Which yields the lower cost for the hybrid car Explain. Combinations of Functions: Composite Functions 237 EXPLORATION TRUE OR FALSE In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer. 77. If f x x 1 and gx 6x, then f g x g f x. 78. If you are given two functions f x and gx, you can calculate f gx if and only if the range of g is a subset of the domain of f. In Exercises 79 and 80, three siblings are of three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one half the age of the youngest. 79. a Write a composite function that gives the oldest siblings age in terms of the youngest. Explain how you arrived at your answer. b If the oldest sibling is 16 years old, find the ages of the other two siblings. 80. a Write a composite function that gives the youngest siblings age in terms of the oldest. Explain how you arrived at your answer. b If the youngest sibling is two years old, find the ages of the other two siblings. 81. PROOF Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function. 82. CONJECTURE Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis. 83. PROOF a Given a function f, prove that gx is even and hx is odd, where gx 12 f x f x and hx 12 f x f x. b Use the result of part a to prove that any function can be written as a sum of even and odd functions. Hint: Add the two equations in part a . c Use the result of part b to write each function as a sum of even and odd functions. f x x2 2x 1, kx 1 x1 84. CAPSTONE Consider the functions f x x2 and gx x. a Find fg and its domain. b Find f g and g f. Find the domain of each composite function. Are they the same Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 238 Chapter 2 Functions and Their Graphs 2.7 INVERSE FUNCTIONS What you should learn Find inverse functions informally and verify that two functions are inverse functions of each other. Use graphs of functions to determine whether functions have inverse functions. Use the Horizontal Line Test to determine if functions are one to one. Find inverse functions algebraically. Why you should learn it Inverse functions can be used to model and solve real life problems. For instance, in Exercise 99 on page 246, an inverse function can be used to determine the year in which there was a given dollar amount of sales of LCD televisions in the United States. Inverse Functions Recall from Section 2.2 that a function can be represented by a set of ordered pairs. For instance, the function f x x 4 from the set A 1, 2, 3, 4 to the set B 5, 6, 7, 8 can be written as follows. f x x 4: 1, 5, 2, 6, 3, 7, 4, 8 In this case, by interchanging the first and second coordinates of each of these ordered pairs, you can form the inverse function of f, which is denoted by f 1. It is a function from the set B to the set A, and can be written as follows. f 1x x 4: 5, 1, 6, 2, 7, 3, 8, 4 Note that the domain of f is equal to the range of f 1, and vice versa, as shown in Figure 2.66. Also note that the functions f and f 1 have the effect of undoing each other. In other words, when you form the composition of f with f 1 or the composition of f 1 with f, you obtain the identity function. f f 1x f x 4 x 4 4 x f 1 f x f 1x 4 x 4 4 x Sean GallupGetty Images f x = x + 4 Domain of f Range of f x f x Range of f 1 FIGURE Example 1 f 1 x = x 4 Domain of f 1 2.66 Finding Inverse Functions Informally Find the inverse function of f x 4x. Then verify that both f f 1x and f 1 f x are equal to the identity function. Solution The function f multiplies each input by 4. To undo this function, you need to divide each input by 4. So, the inverse function of f x 4x is x f 1x . 4 You can verify that both f f 1x x and f 1 f x x as follows. f f 1x f 4 4 4 x x x f 1 f x f 14x 4x x 4 Now try Exercise 7. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.7 Inverse Functions 239 Definition of Inverse Function Let f and g be two functions such that f gx x for every x in the domain of g g f x x for every x in the domain of f. and Under these conditions, the function g is the inverse function of the function f. The function g is denoted by f 1 read f inverse . So, f f 1x x f 1 f x x. and The domain of f must be equal to the range of f 1, and the range of f must be equal to the domain of f 1. Do not be confused by the use of 1 to denote the inverse function f 1. In this text, whenever f 1 is written, it always refers to the inverse function of the function f and not to the reciprocal of f x. If the function g is the inverse function of the function f, it must also be true that the function f is the inverse function of the function g. For this reason, you can say that the functions f and g are inverse functions of each other. Example 2 Verifying Inverse Functions Which of the functions is the inverse function of f x gx x2 5 hx 5 x2 5 2 x Solution By forming the composition of f with g, you have f gx f x 5 2 5 25 x2 x 12 2 5 x. Because this composition is not equal to the identity function x, it follows that g is not the inverse function of f. By forming the composition of f with h, you have f hx f x 2 5 5 5 x. 5 5 2 2 x x So, it appears that h is the inverse function of f. You can confirm this by showing that the composition of h with f is also equal to the identity function, as shown below. h f x h x 5 2 5 2x22x 5 x2 Now try Exercise 19. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 240 Chapter 2 Functions and Their Graphs y The Graph of an Inverse Function y=x The graphs of a function f and its inverse function f 1 are related to each other in the following way. If the point a, b lies on the graph of f, then the point b, a must lie on the graph of f 1, and vice versa. This means that the graph of f 1 is a reflection of the graph of f in the line y x, as shown in Figure 2.67. y = f x a, b y=f 1 x Example 3 b, a Sketch the graphs of the inverse functions f x 2x 3 and f 1x 12x 3 on the same rectangular coordinate system and show that the graphs are reflections of each other in the line y x. x FIGURE 2.67 f 1 x = Finding Inverse Functions Graphically Solution 1 x 2 The graphs of f and f 1 are shown in Figure 2.68. It appears that the graphs are reflections of each other in the line y x. You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point a, b is on the graph of f, the point b, a is on the graph of f 1. f x = 2 x 3 + 3 y 6 1, 2 1, 1 Graph of f x 2x 3 Graph of f 1x 12x 3 1, 5 0, 3 1, 1 2, 1 3, 3 5, 1 3, 0 1, 1 1, 2 3, 3 3, 3 2, 1 3, 0 x 6 6 1, 1 5, 1 y=x 0, 3 1, 5 Now try Exercise 25. FIGURE 2.68 Example 4 Finding Inverse Functions Graphically Sketch the graphs of the inverse functions f x x 2 x 0 and f 1x x on the same rectangular coordinate system and show that the graphs are reflections of each other in the line y x. Solution y The graphs of f and f 1 are shown in Figure 2.69. It appears that the graphs are reflections of each other in the line y x. You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point a, b is on the graph of f, the point b, a is on the graph of f 1. 3, 9 9 f x = x 2 8 7 6 5 4 y=x 2, 4 9, 3 3 4, 2 2 1 f 1 x = 1, 1 x x 0, 0 FIGURE 2.69 3 4 5 6 7 8 9 Graph of f x x 2, x 0 Graph of f 1x x 0, 0 1, 1 2, 4 3, 9 0, 0 1, 1 4, 2 9, 3 Try showing that f f 1x x and f 1 f x x. Now try Exercise 27. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.7 Inverse Functions 241 One to One Functions The reflective property of the graphs of inverse functions gives you a nice geometric test for determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions. Horizontal Line Test for Inverse Functions A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point. If no horizontal line intersects the graph of f at more than one point, then no y value is matched with more than one x value. This is the essential characteristic of what are called one to one functions. One to One Functions A function f is one to one if each value of the dependent variable corresponds to exactly one value of the independent variable. A function f has an inverse function if and only if f is one to one. Consider the function given by f x x2. The table on the left is a table of values for f x x2. The table of values on the right is made up by interchanging the columns of the first table. The table on the right does not represent a function because the input x 4 is matched with two different outputs: y 2 and y 2. So, f x x2 is not one to one and does not have an inverse function. y 3 1 x 3 2 1 2 3 f x = x 3 1 2 3 FIGURE 2.70 x f x x2 x y 2 4 4 2 1 1 1 1 0 0 0 0 1 1 1 1 2 4 4 2 3 9 9 3 y Example 5 Applying the Horizontal Line Test 3 2 x 3 2 2 2 3 FIGURE 2.71 3 f x = x 2 1 a. The graph of the function given by f x x 3 1 is shown in Figure 2.70. Because no horizontal line intersects the graph of f at more than one point, you can conclude that f is a one to one function and does have an inverse function. b. The graph of the function given by f x x 2 1 is shown in Figure 2.71. Because it is possible to find a horizontal line that intersects the graph of f at more than one point, you can conclude that f is not a one to one function and does not have an inverse function. Now try Exercise 39. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 242 Chapter 2 Functions and Their Graphs Finding Inverse Functions Algebraically WARNING CAUTION Note what happens when you try to find the inverse function of a function that is not one to one. Original function f x x2 1 y x2 Finding an Inverse Function Replace f x by y. 1 1. Use the Horizontal Line Test to decide whether f has an inverse function. Interchange x and y. x y2 1 y x 1 2. In the equation for f x, replace f x by y. 3. Interchange the roles of x and y, and solve for y. Isolate y term. x 1 y2 For simple functions such as the one in Example 1 , you can find inverse functions by inspection. For more complicated functions, however, it is best to use the following guidelines. The key step in these guidelines is Step 3interchanging the roles of x and y. This step corresponds to the fact that inverse functions have ordered pairs with the coordinates reversed. 4. Replace y by f 1x in the new equation. 5. Verify that f and f 1 are inverse functions of each other by showing that the domain of f is equal to the range of f 1, the range of f is equal to the domain of f 1, and f f 1x x and f 1 f x x. Solve for y. You obtain two y values for each x. Example 6 y 6 Finding an Inverse Function Algebraically Find the inverse function of f x = 5 3x 2 f x 4 5 3x . 2 Solution 6 4 x 2 4 6 The graph of f is a line, as shown in Figure 2.72. This graph passes the Horizontal Line Test. So, you know that f is one to one and has an inverse function. 2 4 6 FIGURE f x 5 3x 2 Write original function. y 5 3x 2 Replace f x by y. x 5 3y 2 Interchange x and y. 2.72 2x 5 3y Multiply each side by 2. 3y 5 2x Isolate the y term. y 5 2x 3 Solve for y. f 1x 5 2x 3 Replace y by f 1x. Note that both f and f 1 have domains and ranges that consist of the entire set of real numbers. Check that f f 1x x and f 1 f x x. Now try Exercise 63. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.7 f 1 x = x2 + 3 ,x0 2 Example 7 y y=x 3 0, 32 x FIGURE 2.73 Solution The graph of f is a curve, as shown in Figure 2.73. Because this graph passes the Horizontal Line Test, you know that f is one to one and has an inverse function. 2 2 Finding an Inverse Function f x 2x 3. 4 1 243 Find the inverse function of 5 2 1 Inverse Functions 32 , 0 2 3 4 f x = 5 2x 3 f x 2x 3 Write original function. y 2x 3 Replace f x by y. x 2y 3 Interchange x and y. x2 2y 3 Square each side. 2y x2 3 Isolate y. y f 1x x2 3 2 Solve for y. x2 3 , x 0 2 Replace y by f 1x. The graph of f 1 in Figure 2.73 is the reflection of the graph of f in the line y x. Note that the range of f is the interval 0, , which implies that the domain of f 1 is the interval 0, . Moreover, the domain of f is the interval 32, , which implies that the range of f 1 is the interval 32, . Verify that f f 1x x and f 1 f x x. Now try Exercise 69. CLASSROOM DISCUSSION The Existence of an Inverse Function Write a short paragraph describing why the following functions do or do not have inverse functions. a. Let x represent the retail price of an item in dollars , and let f x represent the sales tax on the item. Assume that the sales tax is 6 of the retail price and that the sales tax is rounded to the nearest cent. Does this function have an inverse function Hint: Can you undo this function For instance, if you know that the sales tax is 0.12, can you determine exactly what the retail price is b. Let x represent the temperature in degrees Celsius, and let f x represent the temperature in degrees Fahrenheit. Does this function have an inverse function Hint: The formula for converting from degrees Celsius to degrees Fahrenheit is 9 F 5 C 32. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 244 Chapter 2 2.7 Functions and Their Graphs EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. If the composite functions f gx and g f x both equal x, then the function g is the ________ function of f. 2. 3. 4. 5. The inverse function of f is denoted by ________. The domain of f is the ________ of f 1, and the ________ of f 1 is the range of f. The graphs of f and f 1 are reflections of each other in the line ________. A function f is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable. 6. A graphical test for the existence of an inverse function of f is called the _______ Line Test. SKILLS AND APPLICATIONS In Exercises 714, find the inverse function of f informally. Verify that f f 1x x and f 1 f x x. 7. f x 6x 9. f x x 9 8. f x 10. f x x 4 11. f x 3x 1 x1 12. f x 5 13. f x 14. f x y 2 x 1 x5 4 3 2 1 2 3 1 x 1 2 3 4 3 25. 26. y 16. 4 3 2 1 2 1 1 2 2 3 2 15. 24. x 3 2 3 y x3 , 2 gx 4x 9 3 x 5 gx 3 2x gx x 2 f x x 5, gx x 5 x1 f x 7x 1, gx 7 3x f x 3 4x, gx 4 3 x 3 8x f x , gx 8 1 1 f x , gx x x f x x 4, gx x 2 4, x 0 3 1 x f x 1 x 3, gx 2 f x 9 x , x 0, gx 9 x, x 9 23. f x 2x, gx 3 2 1 x x9 , 4 In Exercises 2334, show that f and g are inverse functions a algebraically and b graphically. y d 1 2 3 4 x 1 2 3 4 5 6 4 3 2 1 3 7 2x 6 19. f x x 3, gx 2 7 22. f x 4 y c 3 21. f x x3 5, x 1 2 1 2 In Exercises 1922, verify that f and g are inverse functions. 20. f x 6 5 4 3 2 1 x 3 2 1 y b 3 2 1 3 In Exercises 1518, match the graph of the function with the graph of its inverse function. The graphs of the inverse functions are labeled a , b , c , and d . a y 18. 4 1 3x 3 x y 17. 27. 6 5 4 3 2 1 28. x 1 2 3 4 5 6 29. 30. 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.7 32. f x 1 1x , x 0, gx , 1x x 33. f x x1 , x5 34. f x x3 2x 3 , gx x2 x1 gx 0 x 1 5x 1 x1 36. 38. In Exercises 4348, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one to one and so has an inverse function. 44. 45. x 1 0 1 2 3 4 f x 2 1 2 1 2 6 4x 6 f x 10 hx x 4 x 4 gx x 53 f x 2x16 x2 f x 18x 22 1 x 3 2 1 0 2 3 f x 10 6 4 1 3 10 x 2 1 0 1 2 3 f x 2 0 2 4 6 8 x 3 2 1 0 1 2 f x 10 7 4 1 2 5 46. 47. 48. In Exercises 49 62, a find the inverse function of f, b graph both f and f 1 on the same set of coordinate axes, c describe the relationship between the graphs of f and f 1, and d state the domain and range of f and f 1. In Exercises 37 and 38, use the table of values for y f x to complete a table for y f 1x. 37. f x 2x 3 50. f x 3x 1 f x x 5 2 52. f x x 3 1 f x 4 x 2, 0 x 2 f x x 2 2, x 0 4 2 55. f x 56. f x x x 49. 51. 53. 54. 57. f x In Exercises 39 42, does the function have an inverse function y y 40. 6 x1 x2 58. f x 3 x 1 59. f x 61. f x 39. 6x 4 4x 5 62. f x 2 2 2 4 4 6 2 y 41. 2 x 2 x 2 2 2 2 66. f x 3x 5 68. f x 3x 4 5 x6 3,x, xx 00 x, x 0 72. f x x 3x, x 0 71. f x 2 x 2 x 8 1 x2 69. f x x 32, x 3 70. qx x 52 4 2 64. f x 67. px 4 y 42. 65. gx 4 2 8x 4 2x 6 In Exercises 6376, determine whether the function has an inverse function. If it does, find the inverse function. 63. f x x4 4 x3 x2 60. f x x 35 6 x 245 43. gx In Exercises 35 and 36, does the function have an inverse function 35. Inverse Functions 4 6 2 4 x2 75. f x 2x 3 73. hx www.elsolucionario.net 74. f x x 2, x2 76. f x x 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 246 Chapter 2 Functions and Their Graphs THINK ABOUT IT In Exercises 77 86, restrict the domain of the function f so that the function is one to one and has an inverse function. Then find the inverse function f 1. State the domains and ranges of f and f 1. Explain your results. There are many correct answers. 98. SHOE SIZES The table shows womens shoe sizes in the United States and the corresponding European shoe sizes. Let y gx represent the function that gives the womens European shoe size in terms of x, the womens U.S. size. 77. f x x 22 78. f x 1 x 4 79. f x x 2 80. f x x 5 Womens U.S. shoe size Womens European shoe size 83. f x 2x2 5 84. f x 12 x2 1 85. f x x 4 1 86. f x x 1 2 4 5 6 7 8 9 35 37 38 39 40 42 81. f x x 62 82. f x x 42 In Exercises 87 92, use the functions given by f x 18 x 3 and gx x 3 to find the indicated value or function. 88. g1 f 13 90. g1 g14 92. g1 f 1 87. f 1 g11 89. f 1 f 16 91. f g1 In Exercises 9396, use the functions given by f x x 4 and gx 2x 5 to find the specified function. 93. g1 f 1 95. f g1 94. f 1 g1 96. g f 1 97. SHOE SIZES The table shows mens shoe sizes in the United States and the corresponding European shoe sizes. Let y f x represent the function that gives the mens European shoe size in terms of x, the mens U.S. size. a b c d e Mens U.S. shoe size Mens European shoe size 8 9 10 11 12 13 41 42 43 45 46 47 Is f one to one Explain. Find f 11. Find f 143, if possible. Find f f 141. Find f 1 f 13. a Is g one to one Explain. b Find g6. c Find g142. d Find gg139. e Find g1 g5. 99. LCD TVS The sales S in millions of dollars of LCD televisions in the United States from 2001 through 2007 are shown in the table. The time in years is given by t, with t 1 corresponding to 2001. Source: Consumer Electronics Association Year, t Sales, St 1 2 3 4 5 6 7 62 246 664 1579 3258 8430 14,532 a Does S1 exist b If S1 exists, what does it represent in the context of the problem c If S1 exists, find S18430. d If the table was extended to 2009 and if the sales of LCD televisions for that year was 14,532 million, would S1 exist Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 2.7 100. POPULATION The projected populations P in millions of people in the United States for 2015 through 2040 are shown in the table. The time in years is given by t, with t 15 corresponding to 2015. Source: U.S. Census Bureau Year, t Population, Pt 15 20 25 30 35 40 325.5 341.4 357.5 373.5 389.5 405.7 a Does P1 exist b If P1 exists, what does it represent in the context of the problem c If P1 exists, find P1357.5. d If the table was extended to 2050 and if the projected population of the U.S. for that year was 373.5 million, would P1 exist Explain. 101. HOURLY WAGE Your wage is 10.00 per hour plus 0.75 for each unit produced per hour. So, your hourly wage y in terms of the number of units produced x is y 10 0.75x. a Find the inverse function. What does each variable represent in the inverse function b Determine the number of units produced when your hourly wage is 24.25. 102. DIESEL MECHANICS The function given by y 0.03x 2 245.50, 0 x 100 approximates the exhaust temperature y in degrees Fahrenheit, where x is the percent load for a diesel engine. a Find the inverse function. What does each variable represent in the inverse function b Use a graphing utility to graph the inverse function. c The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval EXPLORATION TRUE OR FALSE In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer. 103. If f is an even function, then f 1 exists. 104. If the inverse function of f exists and the graph of f has a y intercept, then the y intercept of f is an x intercept of f 1. 247 Inverse Functions 105. PROOF Prove that if f and g are one to one functions, then f g1x g1 f 1x. 106. PROOF Prove that if f is a one to one odd function, then f 1 is an odd function. In Exercises 107 and 108, use the graph of the function f to create a table of values for the given points. Then create a second table that can be used to find f 1, and sketch the graph of f 1 if possible. y 107. y 108. 8 f 6 4 f 4 6 4 4 x 2 x 4 2 2 2 8 In Exercises 109112, determine if the situation could be represented by a one to one function. If so, write a statement that describes the inverse function. 109. The number of miles n a marathon runner has completed in terms of the time t in hours 110. The population p of South Carolina in terms of the year t from 1960 through 2008 111. The depth of the tide d at a beach in terms of the time t over a 24 hour period 112. The height h in inches of a human born in the year 2000 in terms of his or her age n in years. 113. THINK ABOUT IT The function given by f x k2 x x 3 has an inverse function, and f 13 2. Find k. 114. THINK ABOUT IT Consider the functions given by f x x 2 and f 1x x 2. Evaluate f f 1x and f 1 f x for the indicated values of x. What can you conclude about the functions x 10 0 7 45 f f 1x f 1 f x 115. THINK ABOUT IT Restrict the domain of f x x2 1 to x 0. Use a graphing utility to graph the function. Does the restricted function have an inverse function Explain. 116. CAPSTONE Describe and correct the error. 1 Given f x x 6, then f 1x . x 6 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 248 Chapter 2 Functions and Their Graphs Section 2.4 Section 2.3 Section 2.2 Section 2.1 2 CHAPTER SUMMARY What Did You Learn ExplanationExamples Review Exercises Use slope to graph linear equations in two variables p. 170 . The Slope Intercept Form of the Equation of a Line Find the slope of a line given two points on the line p. 172 . The slope m of the nonvertical line through x1, y1 and x2, y2 is m y2 y1x2 x1, where x1 x2. 912 Write linear equations in two variables p. 174 . Point Slope Form of the Equation of a Line 1320 Use slope to identify parallel and perpendicular lines p. 175 . Parallel lines: Slopes are equal. Use slope and linear equations in two variables to model and solve real life problems p. 176 . A linear equation in two variables can be used to describe the book value of exercise equipment in a given year. See Example 7. 23, 24 Determine whether relations between two variables are functions p. 185 . A function f from a set A domain to a set B range is a relation that assigns to each element x in the set A exactly one element y in the set B. 25 28 Use function notation, evaluate functions, and find domains p. 187 . Equation: f x 5 x2 29 36 Use functions to model and solve real life problems p. 191 . A function can be used to model the number of alternative fueled vehicles in the United States. See Example 10. 37, 38 Evaluate difference quotients p. 192 . Difference quotient: f x h f xh, h 39, 40 Use the Vertical Line Test for functions p. 201 . A graph represents a function if and only if no vertical line intersects the graph at more than one point. 41 44 Find the zeros of functions p. 202 . Zeros of f x: x values for which f x 0 45 50 Determine intervals on which functions are increasing or decreasing p. 203 , find relative minimum and maximum values p. 204 , and find the average rate of change of a function p. 205 . To determine whether a function is increasing, decreasing, or constant on an interval, evaluate the function for several values of x. The points at which the behavior of a function changes can help determine the relative minimum or relative maximum. 51 60 Identify even and odd functions p. 206 . Even: For each x in the domain of f, f x f x. Identify and graph linear p. 212 and squaring functions p. 213 . Linear: f x ax b 1 8 The graph of the equation y mx b is a line whose slope is m and whose y intercept is 0, b. The equation of the line with slope m passing through the point x1, y1 is y y1 mx x1. 21, 22 Perpendicular lines: Slopes are negative reciprocals of each other. Domain of f x 5 x2 : f 2: f 2 5 22 1 All real numbers 0 The average rate of change between any two points is the slope of the line secant line through the two points. 61 64 Odd: For each x in the domain of f, f x f x. Squaring: f x x2 y y 5 5 4 f x = x + 4 4 3 3 2 2 1 1 1 65 68 f x = x 2 1 x 1 2 3 4 5 3 2 1 1 www.elsolucionario.net x 1 2 3 0, 0 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Summary What Did You Learn ExplanationExamples Identify and graph cubic, square root, reciprocal p. 214 , step, and other piecewise defined functions p. 215 . Cubic: f x x3 Square Root: f x x 6978 y 3 4 2 3 f x = x 3 0, 0 Section 2.4 Review Exercises y 3 2 249 f x = x 2 4 2 x 1 1 2 0, 0 3 2 1 1 3 2 Reciprocal: f x 1x x 1 5 Step: f x x y y 3 f x = 2 3 1 x 2 1 1 1 3 x 1 2 3 3 2 1 x 1 2 3 f x = x Section 2.7 Section 2.6 Section 2.5 3 Recognize graphs of parent functions p. 216 . Eight of the most commonly used functions in algebra are shown in Figure 2.48. 79, 80 Use vertical and horizontal shifts p. 219 , reflections p. 221 , and nonrigid transformations p. 223 to sketch graphs of functions. Vertical shifts: hx f x c or hx f x c 8194 Horizontal shifts: hx f x c or hx f x c Reflection in x axis: hx f x Reflection in y axis: hx f x Nonrigid transformations: hx cf x or hx f cx f gx f x gx fgx f xgx, gx Add, subtract, multiply, and divide functions p. 229 . f gx f x gx fgx f x gx Find the composition of one function with another function p. 231 . The composition of the function f with the function g is f gx f gx. 97102 Use combinations and compositions of functions to model and solve real life problems p. 233 . A composite function can be used to represent the number of bacteria in food as a function of the amount of time the food has been out of refrigeration. See Example 8. 103, 104 Find inverse functions informally and verify that two functions are inverse functions of each other p. 238 . Let f and g be two functions such that f gx x for every x in the domain of g and g f x x for every x in the domain of f. Under these conditions, the function g is the inverse function of the function f. 105108 Use graphs of functions to determine whether functions have inverse functions p. 240 . If the point a, b lies on the graph of f, then the point b, a must lie on the graph of f 1, and vice versa. In short, f 1 is a reflection of f in the line y x. 109, 110 Use the Horizontal Line Test to determine if functions are one to one p. 241 . Horizontal Line Test for Inverse Functions 111114 Find inverse functions algebraically p. 242 . To find inverse functions, replace f x by y, interchange the roles of x and y, and solve for y. Replace y by f 1x. 95, 96 0 A function f has an inverse function if and only if no horizontal line intersects f at more than one point. www.elsolucionario.net 115120 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 250 Chapter 2 Functions and Their Graphs 2 REVIEW EXERCISES 2.1 In Exercises 1 8, find the slope and y intercept if possible of the equation of the line. Sketch the line. 1. 3. 5. 7. y 2x 7 y6 y 52 x 1 3x y 13 2. 4. 6. 8. y 4x 3 x 3 y 56 x 5 10x 2y 9 9. 6, 4, 3, 4 11. 4.5, 6, 2.1, 3 10. 1, 5, 5 2 13. 14. 15. 16. 3, 0 8, 5 10, 3 12, 6 m 23 m0 m 12 m is undefined. 18. 2, 1, 4, 1 20. 11, 2, 6, 1 Line 5x 4y 8 2x 3y 5 RATE OF CHANGE In Exercises 23 and 24, you are given the dollar value of a product in 2010 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. Let t 10 represent 2010. 2010 Value 23. 12,500 24. 72.95 d gx x 1 x 1 c h0 d h2 c f t d f 0 4 32. f x 2 x 1 a f 1 b f 5 33. f x 25 x 2 In Exercises 21 and 22, write the slope intercept forms of the equations of the lines through the given point a parallel to the given line and b perpendicular to the given line. Point 21. 3, 2 22. 8, 3 2 d f t 1 In Exercises 3336, find the domain of the function. Verify your result with a graph. In Exercises 1720, find the slope intercept form of the equation of the line passing through the points. 17. 0, 0, 0, 10 19. 1, 0, 6, 2 2xx 2,1, a h2 b h1 In Exercises 1316, find the slope intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Slope 29. f x x 2 1 a f 2 b f 4 c f t 2 43 30. gx x a g8 b gt 1 c g27 31. hx 12. 3, 2, 8, 2 Point 26. 2x y 3 0 28. y x 2 In Exercises 2932, evaluate the function at each specified value of the independent variable and simplify. In Exercises 912, plot the points and find the slope of the line passing through the pair of points. 3 2, 25. 16x y 4 0 27. y 1 x 35. h x x x2 x 6 34. gs 5s 5 3s 9 36. h t t 1 37. PHYSICS The velocity of a ball projected upward from ground level is given by v t 32t 48, where t is the time in seconds and v is the velocity in feet per second. a Find the velocity when t 1. b Find the time when the ball reaches its maximum height. Hint: Find the time when v t 0. c Find the velocity when t 2. 38. MIXTURE PROBLEM From a full 50 liter container of a 40 concentration of acid, x liters is removed and replaced with 100 acid. a Write the amount of acid in the final mixture as a function of x. b Determine the domain and range of the function. c Determine x if the final mixture is 50 acid. In Exercises 39 and 40, find the difference quotient and simplify your answer. Rate 850 decrease per year 5.15 increase per year 2.2 In Exercises 2528, determine whether the equation represents y as a function of x. 39. f x 2x2 3x 1, f x h f x , h h 0 40. f x x3 5x2 x, f x h f x , h h 0 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 251 Review Exercises 2.3 In Exercises 4144, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 41. y x 3 42. y 2 35x 3 y 2x 1 62. f x x 4 20x 2 5 6x 2 64. f x 2.4 In Exercises 65 and 66, write the linear function f such that it has the indicated function values. Then sketch the graph of the function. 1 3 2 1 x 3 2 1 1 2 3 2 3 x 1 2 3 4 5 44. x 4 y 43. x 4 y 2 y 67. f x x2 5 69. gx 3x3 71. f x x y 8 2 x 2 4 73. gx 4 8 2 x 4 8 4 2 2 In Exercises 4550, find the zeros of the function algebraically. 45. f x x 2 4x 21 47. f x 8x 3 11 x 65. f 2 6, f 1 3 66. f 0 5, f 4 8 In Exercises 6778, graph the function. 10 4 2 61. f x x 5 4x 7 63. f x 2xx 2 3 y 5 4 1 In Exercises 6164, determine whether the function is even, odd, or neither. 46. f x 5x 2 4x 1 48. f x 2x 1 49. f x x3 x2 50. f x x3 x 2 25x 25 68. f x 3 x2 70. hx x3 2 72. f x x 1 3 x 74. gx 75. f x x 2 77. f x 76. gx x 4 5x4x3, 5, x 1 x 1 x 2 2, x 2 78. f x 5, 2 x 0 8x 5, x 0 In Exercises 79 and 80, the figure shows the graph of a transformed parent function. Identify the parent function. y 79. 53. f x x2 2x 1 55. f x x3 6x 4 54. f x x 4 4x 2 2 56. f x x 3 4x2 1 In Exercises 5760, find the average rate of change of the function from x1 to x2. Function 57. 58. 59. 60. f x 8x 4 f x x 3 12x 2 f x 2 x 1 f x 1 x 3 x 2 x1 x1 x1 x1 x Values 0, x 2 0, x 2 3, x 2 1, x 2 4 4 7 6 6 6 4 4 2 2 8 In Exercises 5356, use a graphing utility to graph the function and approximate any relative minimum or relative maximum values. 8 8 52. f x x2 42 4 2 y 80. 10 In Exercises 51 and 52, use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. 51. f x x x 1 1 x5 x 2 2 2 x 2 4 6 8 2.5 In Exercises 8194, h is related to one of the parent functions described in this chapter. a Identify the parent function f. b Describe the sequence of transformations from f to h. c Sketch the graph of h. d Use function notation to write h in terms of f. 81. hx x 2 9 83. hx x 4 85. hx x 22 3 87. hx x 6 89. hx x 4 6 90. hx x 12 3 91. hx 5x 9 93. hx 2x 4 www.elsolucionario.net 82. hx x 23 2 84. hx x 3 5 86. hx 12x 12 2 88. hx x 1 9 92. hx 13 x 3 94. hx 12x 1 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 252 Chapter 2 Functions and Their Graphs 2.6 In Exercises 95 and 96, find a f gx, b f gx, c fgx, and d fgx. What is the domain of fg In Exercises 109 and 110, determine whether the function has an inverse function. y 109. 95. f x x 3, gx 2x 1 96. f x x2 4, gx 3 x y 110. 4 2 2 2 In Exercises 97100, find a f g and b g f. Find the domain of each function and each composite function. x 2 2 4 97. f x 13 x 3, gx 3x 1 1 98. f x , gx 2x 3 x 4 x 2 2 4 4 6 In Exercises 111114, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one to one and so has an inverse function. 3 99. f x x3 4, gx x7 100. f x x 1, gx x2 111. f x 4 13 x 2 t3 112. f x x 12 In Exercises 101 and 102, find two functions f and g such that f gx hx. There are many correct answers. 113. ht 101. hx 1 2x3 In Exercises 115118, a find the inverse function of f, b graph both f and f 1 on the same set of coordinate axes, c describe the relationship between the graphs of f and f 1, and d state the domains and ranges of f and f 1. 3 102. hx x2 103. PHONE EXPENDITURES The average annual expenditures in dollars for residential r t and cellular c t phone services from 2001 through 2006 can be approximated by the functions rt 27.5t 705 and c t 151.3t 151, where t represents the year, with t 1 corresponding to 2001. Source: Bureau of Labor Statistics a Find and interpret r ct. b Use a graphing utility to graph r t, ct, and r ct in the same viewing window. c Find r c13. Use the graph in part b to verify your result. 104. BACTERIA COUNT The number N of bacteria in a refrigerated food is given by NT 25T 2 50T 300, 2 T 20 where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by T t 2t 1, 0 t 9 where t is the time in hours. a Find the composition NT t and interpret its meaning in context, and b find the time when the bacteria count reaches 750. 2.7 In Exercises 105108, find the inverse function of f informally. Verify that f f 1x x and f 1 f x x. x4 5 105. f x 3x 8 106. f x 107. f x x3 1 3 x 108. f x 2 115. f x 12x 3 117. f x x 1 114. gx x 6 116. f x 5x 7 118. f x x3 2 In Exercises 119 and 120, restrict the domain of the function f to an interval over which the function is increasing and determine f 1 over that interval. 119. f x 2x 42 120. f x x 2 EXPLORATION TRUE OR FALSE In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. Relative to the graph of f x x, the function given by hx x 9 13 is shifted 9 units to the left and 13 units downward, then reflected in the x axis. 122. If f and g are two inverse functions, then the domain of g is equal to the range of f. 123. WRITING Explain how to tell whether a relation between two variables is a function. 124. WRITING Explain the difference between the Vertical Line Test and the Horizontal Line Test. 125. WRITING Describe the basic characteristics of the cubic function. Describe the basic characteristics of f x x3 1. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Test 2 CHAPTER TEST 253 See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1 and 2, find the slope intercept form of the equation of the line passing through the points. Then sketch the line. 1. 4, 5, 2, 7 2. 3, 0.8, 7, 6 3. Find equations of the lines that pass through the point 0, 4 and are a parallel to and b perpendicular to the line 5x 2y 3. In Exercises 4 and 5, evaluate the function at each specified value of the independent variable and simplify. 4. f x x 2 15 a f 8 b f 14 x 9 5. f x 2 x 81 a f 7 b f 5 c f x 6 c f x 9 In Exercises 6 and 7, find the domain of the function. 6. f x x 6 2 7. f x 10 3 x In Exercises 810, a use a graphing utility to graph the function, b approximate the intervals over which the function is increasing, decreasing, or constant, and c determine whether the function is even, odd, or neither. 8. f x 2x 6 5x 4 x 2 9. f x 4x3 x 10. f x x 5 11. Use a graphing utility to approximate any relative minimum or maximum values of f x x 3 2x 1. 12. Find the average rate of change of f x 2x 2 5x 3 from x1 1 to x2 3. 13. Sketch the graph of f x 3x4x 7,1, 2 x 3 . x 3 In Exercises 1416, a identify the parent function in the transformation, b describe the sequence of transformations from f to h, and c sketch the graph of h. 14. hx 3x 15. hx x 5 8 16. hx 2x 53 3 In Exercises 17 and 18, find a f gx, b f gx, c fgx, d fgx, e f gx, and f g f x. 17. f x 3x2 7, gx x2 4x 5 1 18. f x , gx 2x x In Exercises 1921, determine whether the function has an inverse function, and if so, find the inverse function. 19. f x x 3 8 20. f x x 2 3 6 21. f x 3xx 22. It costs a company 58 to produce 6 units of a product and 78 to produce 10 units. How much does it cost to produce 25 units, assuming that the cost function is linear www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 254 Chapter 2 Functions and Their Graphs 2 CUMULATIVE TEST FOR CHAPTERS P2 See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1 and 2, simplify the expression. 1. 8x 2 y3 30x1y 2 2. 18x 3y 4 In Exercises 35, perform the operation and simplify the result. 3. 4x 2x 32 x 4. x 2x 2 x 3 5. 2 1 s3 s1 In Exercises 6 8, factor the expression completely. 6. 25 x 22 7. x 5x 2 6x 3 8. 54x3 16 In Exercises 9 and 10, write an expression for the area of the region. 9. x1 10. 2x + 4 x 3x x+5 x+4 2 x + 1 In Exercises 1113, graph the equation without using a graphing utility. 11. x 3y 12 0 12. y x 2 9 13. y 4 x In Exercises 1416, solve the equation and check your solution. 14. 3x 5 6x 8 15. x 3 14x 6 16. 1 10 x 2 4x 3 In Exercises 1722, solve the equation using any convenient method and check your solutions. State the method you used. 17. x 2 4x 3 0 2 19. 3 x2 24 21. 3x 2 9x 1 0 18. 2x 2 8x 12 0 20. 3x 2 5x 6 0 1 22. 2 x 2 7 25 In Exercises 2328, solve the equation if possible . 23. x 4 12x 3 4x 2 48x 0 25. x 23 13 17 27. 3x 4 27 24. 8x 3 48x 2 72x 0 26. x 10 x 2 28. x 12 2 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Cumulative Test for Chapters P2 255 In Exercises 29 and 30, determine whether each value of x is a solution of the inequality. 29. 4x 2 7 a x 1 c x 32 y b x 12 d x 2 In Exercises 3134, solve the inequality and sketch the solution on the real number line. 4 2 x 2 30. 5x 1 4 a x 1 c x 1 b x 12 d x 2 2 4 4 FIGURE FOR 36 31. x 1 6 33. 5x 2 12x 7 0 32. 5 6x 3 34. x 2 x 4 0 35. Find the slope intercept form of the equation of the line passing through 12, 1 and 3, 8. 36. Explain why the graph at the left does not represent y as a function of x. x 37. Evaluate if possible the function given by f x for each value. x2 a f 6 b f 2 c f s 2 In Exercises 3840, determine whether the function is even, odd, or neither. 38. f x 5 4 x 39. f x x 5 x 3 2 40. f x 2x 4 4 3 x. 41. Compare the graph of each function with the graph of y Note: It is not necessary to sketch the graphs. 13 3 x 2 3 x 2 x a r x 2 b h x c gx In Exercises 42 and 43, find a f gx, b f gx, c fgx, and d fgx. What is the domain of fg 42. f x x 4, gx 3x 1 43. f x x 1, gx x 2 1 In Exercises 44 and 45, find a f g and b g f. Find the domain of each composite function. 44. f x 2x 2, gx x 6 45. f x x 2, gx x 46. Determine whether hx 3x 4 has an inverse function. If so, find the inverse function. 47. A group of n people decide to buy a 36,000 minibus. Each person will pay an equal share of the cost. If three additional people join the group, the cost per person will decrease by 1000. Find n. 48. For groups of 60 or more, a charter bus company determines the rate per person according to the formula Rate 10.00 0.05n 60, n 60. a Write the revenue R as a function of n. b Use a graphing utility to graph the revenue function. Move the cursor along the function to estimate the number of passengers that will maximize the revenue. 49. The height of an object thrown vertically upward from a height of 8 feet at a velocity of 36 feet per second can be modeled by st 16t 2 36t 8, where s is the height in feet and t is the time in seconds . Find the average rate of change of the function from t1 0 to t2 2. Interpret your answer in the context of the problem. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS Biconditional Statements Recall from the Proofs in Mathematics in Chapter 1 that a conditional statement is a statement of the form if p, then q. A statement of the form p if and only if q is called a biconditional statement. A biconditional statement, denoted by pq Biconditional statement is the conjunction of the conditional statement p q and its converse q p. A biconditional statement can be either true or false. To be true, both the conditional statement and its converse must be true. Example 1 Analyzing a Biconditional Statement Consider the statement x 3 if and only if x2 9. a. Is the statement a biconditional statement b. Is the statement true Solution a. The statement is a biconditional statement because it is of the form p if and only if q. b. The statement can be rewritten as the following conditional statement and its converse. Conditional statement: If x 3, then x2 9. Converse: If x2 9, then x 3. The first of these statements is true, but the second is false because x could also equal 3. So, the biconditional statement is false. Knowing how to use biconditional statements is an important tool for reasoning in mathematics. Example 2 Analyzing a Biconditional Statement Determine whether the biconditional statement is true or false. If it is false, provide a counterexample. A number is divisible by 5 if and only if it ends in 0. Solution The biconditional statement can be rewritten as the following conditional statement and its converse. Conditional statement: If a number is divisible by 5, then it ends in 0. Converse: If a number ends in 0, then it is divisible by 5. The conditional statement is false. A counterexample is the number 15, which is divisible by 5 but does not end in 0. 256 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROBLEM SOLVING This collection of thought provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of 2000, plus a commission of 7 of sales. You are offered a new job at 2300 per month, plus a commission of 5 of sales. a Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. b Write a linear equation for the monthly wage W2 of your new job offer in terms of the monthly sales S. c Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify d You think you can sell 20,000 per month. Should you change jobs Explain. 2. For the numbers 2 through 9 on a telephone keypad see figure , create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions Explain. 3. What can be said about the sum and difference of each of the following a Two even functions b Two odd functions c An odd function and an even function 4. The two functions given by f x x and gx x are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a general formula for a family of linear functions that are their own inverse functions. 5. Prove that a function of the following form is even. y a2n x2n a2n2x2n2 . . . a2 x2 a0 6. A miniature golf professional is trying to make a hole inone on the miniature golf green shown. A coordinate plane is placed over the golf green. The golf ball is at the point 2.5, 2 and the hole is at the point 9.5, 2. The professional wants to bank the ball off the side wall of the green at the point x, y. Find the coordinates of the point x, y. Then write an equation for the path of the ball. y x, y 8 ft x 12 ft FIGURE FOR 6 7. At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400 mile trip. a What was the total duration of the voyage in hours b What was the average speed in miles per hour c Write a function relating the distance of the Titanic from New York City and the number of hours traveled. Find the domain and range of the function. d Graph the function from part c . 8. Consider the function given by f x x 2 4x 3. Find the average rate of change of the function from x1 to x2. a x1 1, x2 2 b x1 1, x2 1.5 c x1 1, x2 1.25 d x1 1, x2 1.125 e x1 1, x2 1.0625 f Does the average rate of change seem to be approaching one value If so, what value g Find the equations of the secant lines through the points x1, f x1 and x2, f x2 for parts a e . h Find the equation of the line through the point 1, f 1 using your answer from part f as the slope of the line. 9. Consider the functions given by f x 4x and gx x 6. a Find f gx. b Find f g1x. c Find f 1x and g1x. d Find g1 f 1x and compare the result with that of part b . e Repeat parts a through d for f x x3 1 and gx 2x. f Write two one to one functions f and g, and repeat parts a through d for these functions. g Make a conjecture about f g1x and g1 f 1x. 257 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 10. You are in a boat 2 miles from the nearest point on the coast. You are to travel to a point Q, 3 miles down the coast and 1 mile inland see figure . You can row at 2 miles per hour and you can walk at 4 miles per hour. 3x 1 mi Q 3 mi 1,0, y Not drawn to scale. a Write the total time T of the trip as a function of x. b Determine the domain of the function. c Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. d Use the zoom and trace features to find the value of x that minimizes T. e Write a brief paragraph interpreting these values. 11. The Heaviside function Hx is widely used in engineering applications. See figure. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Hx f g hx f g hx. 14. Consider the graph of the function f shown in the figure. Use this graph to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. a f x 1 b f x 1 c 2f x d f x e f x f f x g f x 2 mi x 13. Show that the Associative Property holds for compositions of functionsthat is, 4 2 4 x 2 2 4 2 4 15. Use the graphs of f and f 1 to complete each table of function values. y x 0 x 0 y 4 4 2 2 x 2 Sketch the graph of each function by hand. a Hx 2 b Hx 2 c Hx d Hx e 12 Hx f Hx 2 2 2 a b 1 x 1 2 f x 2 2 4 2 4 f 1 4 x f f 3 2 2 4 4 y 3 2 1 2 0 4 x 1 3 x 2 0 1 f f 1x 3 2 c 3 3 x 2 0 1 f f 1x 1 . 1x a What are the domain and range of f 12. Let f x d b Find f f x. What is the domain of this function c Find f f f x. Is the graph a line Why or why not 4 x f 3 0 4 x 1 258 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Polynomial Functions 3.1 Quadratic Functions and Models 3.2 Polynomial Functions of Higher Degree 3.3 Polynomial and Synthetic Division 3.4 Zeros of Polynomial Functions 3.5 Mathematical Modeling and Variation 3 In Mathematics Functions defined by polynomial expressions are called polynomial functions. Polynomial functions are used to model real life situations, such as a companys revenue, the design of a propane tank, or the height of a thrown baseball. For instance, you can model the per capita cigarette consumption in the United States with a polynomial function. You can use the model to determine whether the addition of cigarette warnings affected consumption. See Exercise 85, page 268. Michael NewmanPhotoEdit In Real Life IN CAREERS There are many careers that use polynomial functions. Several are listed below. Architect Exercise 84, page 268 Ecologist Exercises 75 and 76, page 318 Forester Exercise 103, page 282 Oceanographer Exercise 83, page 318 259 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 260 Chapter 3 Polynomial Functions 3.1 QUADRATIC FUNCTIONS AND MODELS What you should learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and maximum values of quadratic functions in real life applications. Why you should learn it Quadratic functions can be used to model data to analyze consumer behavior. For instance, in Exercise 79 on page 268, you will use a quadratic function to model the revenue earned from manufacturing handheld video games. The Graph of a Quadratic Function In this and the next section, you will study the graphs of polynomial functions. In Section 2.4, you were introduced to the following basic functions. f x ax b Linear function f x c Constant function f x x Squaring function 2 These functions are examples of polynomial functions. Definition of Polynomial Function Let n be a nonnegative integer and let an, an1, . . . , a2, a1, a0 be real numbers with an 0. The function given by f x an x n an1 x n1 . . . a 2 x 2 a1 x a 0 is called a polynomial function of x with degree n. Polynomial functions are classified by degree. For instance, a constant function f x c with c 0 has degree 0, and a linear function f x ax b with a 0 has degree 1. In this section, you will study second degree polynomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f x x 2 6x 2 John HenleyCorbis gx 2x 12 3 hx 9 14 x 2 kx 3x 2 4 mx x 2x 1 Note that the squaring function is a simple quadratic function that has degree 2. Definition of Quadratic Function Let a, b, and c be real numbers with a f x ax 2 bx c 0. The function given by Quadratic function is called a quadratic function. The graph of a quadratic function is a special type of U shaped curve called a parabola. Parabolas occur in many real life applicationsespecially those involving reflective properties of satellite dishes and flashlight reflectors. You will study these properties in Section 4.3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.1 261 Quadratic Functions and Models All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola, as shown in Figure 3.1. If the leading coefficient is positive, the graph of f x ax 2 bx c is a parabola that opens upward. If the leading coefficient is negative, the graph of f x ax 2 bx c is a parabola that opens downward. y y Opens upward f x = ax 2 + bx + c, a 0 Vertex is highest point Axis Axis Vertex is lowest point f x = ax 2 + bx + c, a 0 x x Opens downward Leading coefficient is positive. FIGURE 3.1 Leading coefficient is negative. The simplest type of quadratic function is f x ax 2. Its graph is a parabola whose vertex is 0, 0. If a 0, the vertex is the point with the minimum y value on the graph, and if a 0, the vertex is the point with the maximum y value on the graph, as shown in Figure 3.2. y y 3 3 2 2 1 3 2 x 1 1 1 1 f x = ax 2, a 0 2 3 Minimum: 0, 0 3 2 x 1 1 1 2 2 3 3 Leading coefficient is positive. FIGURE 3.2 Maximum: 0, 0 2 3 f x = ax 2, a 0 Leading coefficient is negative. When sketching the graph of f x ax 2, it is helpful to use the graph of y x 2 as a reference, as discussed in Section 2.5. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 262 Chapter 3 Polynomial Functions Example 1 Sketching Graphs of Quadratic Functions a. Compare the graphs of y x 2 and f x 13x 2. b. Compare the graphs of y x 2 and gx 2x 2. Solution You can review the techniques for shifting, reflecting, and stretching graphs in Section 2.5. a. Compared with y x 2, each output of f x 13x 2 shrinks by a factor of 13, creating the broader parabola shown in Figure 3.3. b. Compared with y x 2, each output of gx 2x 2 stretches by a factor of 2, creating the narrower parabola shown in Figure 3.4. y y = x2 g x = 2 x 2 y 4 4 3 3 f x = 13 x 2 2 2 1 1 y = x2 2 FIGURE x 1 1 2 3.3 2 FIGURE x 1 1 2 3.4 Now try Exercise 13. In Example 1, note that the coefficient a determines how widely the parabola given by f x ax 2 opens. If a is small, the parabola opens more widely than if a is large. Recall from Section 2.5 that the graphs of y f x c, y f x c, y f x, and y f x are rigid transformations of the graph of y f x. For instance, in Figure 3.5, notice how the graph of y x 2 can be transformed to produce the graphs of f x x 2 1 and gx x 22 3. y 2 g x = x + 2 3 y 2 3 0, 1 y = x2 2 f x = x 2 + 1 2 y = x2 1 x 2 1 4 3 2, 3 www.elsolucionario.net 1 2 2 2 Reflection in x axis followed by an upward shift of one unit FIGURE 3.5 x 1 3 Left shift of two units followed by a downward shift of three units http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.1 Quadratic Functions and Models 263 The Standard Form of a Quadratic Function The standard form of a quadratic function identifies four basic transformations of the graph of y x 2. a. The factor a produces a vertical stretch or shrink. b. If a 0, the graph is reflected in the x axis. c. The factor x h2 represents a horizontal shift of h units. d. The term k represents a vertical shift of k units. The standard form of a quadratic function is f x ax h 2 k. This form is especially convenient for sketching a parabola because it identifies the vertex of the parabola as h, k. Standard Form of a Quadratic Function The quadratic function given by f x ax h 2 k, a 0 is in standard form. The graph of f is a parabola whose axis is the vertical line x h and whose vertex is the point h, k. If a 0, the parabola opens upward, and if a 0, the parabola opens downward. To graph a parabola, it is helpful to begin by writing the quadratic function in standard form using the process of completing the square, as illustrated in Example 2. In this example, notice that when completing the square, you add and subtract the square of half the coefficient of x within the parentheses instead of adding the value to each side of the equation as is done in Section 1.4. Example 2 Graphing a Parabola in Standard Form Sketch the graph of f x 2x 2 8x 7 and identify the vertex and the axis of the parabola. Solution Begin by writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out any coefficient of x2 that is not 1. f x 2x 2 8x 7 You can review the techniques for completing the square in Section 1.4. Write original function. 2x 2 4x 7 Factor 2 out of x terms. 2x 2 4x 4 4 7 Add and subtract 4 within parentheses. 422 f x = 2 x + 2 2 1 After adding and subtracting 4 within the parentheses, you must now regroup the terms to form a perfect square trinomial. The 4 can be removed from inside the parentheses; however, because of the 2 outside of the parentheses, you must multiply 4 by 2, as shown below. y 4 1 2, 1 FIGURE 3.6 x = 2 Regroup terms. 2x 2 4x 4 8 7 Simplify. 2 2x 2 1 Write in standard form. 1 3 f x 2x 2 4x 4 24 7 3 2 y = 2x 2 x 1 From this form, you can see that the graph of f is a parabola that opens upward and has its vertex at 2, 1. This corresponds to a left shift of two units and a downward shift of one unit relative to the graph of y 2x 2, as shown in Figure 3.6. In the figure, you can see that the axis of the parabola is the vertical line through the vertex, x 2. Now try Exercise 19. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 264 Chapter 3 Polynomial Functions To find the x intercepts of the graph of f x ax 2 bx c, you must solve the equation ax 2 bx c 0. If ax 2 bx c does not factor, you can use the Quadratic Formula to find the x intercepts. Remember, however, that a parabola may not have x intercepts. You can review the techniques for using the Quadratic Formula in Section 1.4. Example 3 Finding the Vertex and x Intercepts of a Parabola Sketch the graph of f x x 2 6x 8 and identify the vertex and x intercepts. Solution f x x 2 6x 8 Write original function. x 2 6x 8 Factor 1 out of x terms. x 6x 9 9 8 Add and subtract 9 within parentheses. 2 622 y f x = x 3 2 + 1 4, 0 x 1 3 x 3 1 Write in standard form. From this form, you can see that f is a parabola that opens downward with vertex 3, 1. The x intercepts of the graph are determined as follows. 3, 1 1 1 Regroup terms. 2 2 2, 0 x 2 6x 9 9 8 5 1 x 2 6x 8 0 x 2x 4 0 2 3 y= x 2 4 FIGURE Factor out 1. Factor. x20 x2 Set 1st factor equal to 0. x40 x4 Set 2nd factor equal to 0. So, the x intercepts are 2, 0 and 4, 0, as shown in Figure 3.7. Now try Exercise 25. 3.7 Example 4 Writing the Equation of a Parabola Write the standard form of the equation of the parabola whose vertex is 1, 2 and that passes through the point 3, 6. Solution Because the vertex of the parabola is at h, k 1, 2, the equation has the form f x ax 12 2. y 2 4 2 Substitute for h and k in standard form. Because the parabola passes through the point 3, 6, it follows that f 3 6. So, 1, 2 x 4 6 y = f x 3, 6 f x ax 12 2 Write in standard form. 6 a3 1 2 Substitute 3 for x and 6 for f x. 6 4a 2 Simplify. 8 4a Subtract 2 from each side. 2 a. Divide each side by 4. 2 The equation in standard form is f x 2x 12 2. The graph of f is shown in Figure 3.8. FIGURE 3.8 Now try Exercise 47. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.1 265 Quadratic Functions and Models Finding Minimum and Maximum Values Many applications involve finding the maximum or minimum value of a quadratic function. By completing the square of the quadratic function f x ax2 bx c, you can rewrite the function in standard form see Exercise 95 . f x a x b 2a c 4ab 2 2 So, the vertex of the graph of f is Standard form b b ,f 2a 2a , which implies the following. Minimum and Maximum Values of Quadratic Functions Consider the function f x ax 2 bx c with vertex 1. If a 0, f has a minimum at x . b b . The minimum value is f . 2a 2a 2. If a 0, f has a maximum at x Example 5 b b , f 2a 2a b b . The maximum value is f . 2a 2a The Maximum Height of a Baseball A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f x 0.0032x 2 x 3, where f x is the height of the baseball in feet and x is the horizontal distance from home plate in feet . What is the maximum height reached by the baseball Algebraic Solution Graphical Solution For this quadratic function, you have Use a graphing utility to graph f x ax bx c y 0.0032x2 x 3 2 0.0032x2 x 3 which implies that a 0.0032 and b 1. Because a 0, the function has a maximum when x b2a. So, you can conclude that the baseball reaches its maximum height when it is x feet from home plate, where x is b x 2a so that you can see the important features of the parabola. Use the maximum feature see Figure 3.9 or the zoom and trace features see Figure 3.10 of the graphing utility to approximate the maximum height on the graph to be y 81.125 feet at x 156.25. 100 y = 0.0032x 2 + x + 3 81.3 1 20.0032 156.25 feet. 0 400 At this distance, the maximum height is FIGURE 152.26 159.51 81 0 3.9 FIGURE 3.10 f 156.25 0.0032156.252 156.25 3 81.125 feet. Now try Exercise 75. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 266 Chapter 3 3.1 Polynomial Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. Linear, constant, and squaring functions are examples of ________ functions. 2. A polynomial function of degree n and leading coefficient an is a function of the form f x an x n an1 x n1 . . . a1x a0 an 0 where n is a ________ ________ and an, an1, . . . , a1, a0 are ________ numbers. 3. A ________ function is a second degree polynomial function, and its graph is called a ________. 4. The graph of a quadratic function is symmetric about its ________. 5. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________. 6. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________. SKILLS AND APPLICATIONS In Exercises 712, match the quadratic function with its graph. The graphs are labeled a , b , c , d , e , and f . y a y b 6 6 4 4 2 2 x 4 4 2 1, 2 2 0, 2 y c x 2 4 y d 4, 0 6 x 4, 0 4 2 6 4 2 4 6 8 6 2 y f 2, 4 4 6 2 4 2 2 2 2, 0 x 2 6 x 2 4 b d b d hx 3 f x x 12 2 hx 13 x 3 f x 12x 22 1 2 gx 12x 1 3 hx 12x 22 1 kx 2x 1 2 4 x2 gx x 2 1 kx x 2 3 gx 3x2 1 kx x 32 f x 1 x2 f x x 2 7 f x 12x 2 4 f x x 42 3 hx x 2 8x 16 27. f x x 2 x 54 29. f x x 2 2x 5 31. hx 4x 2 4x 21 33. f x 14x 2 2x 12 18. 20. 22. 24. 26. gx x2 8 hx 12 x 2 f x 16 14 x 2 f x x 62 8 gx x 2 2x 1 28. f x x 2 3x 14 30. f x x 2 4x 1 32. f x 2x 2 x 1 34. f x 13x2 3x 6 6 7. f x x 22 9. f x x 2 2 11. f x 4 x 22 8. f x x 42 10. f x x 1 2 2 12. f x x 42 In Exercises 1316, graph each function. Compare the graph of each function with the graph of y x2. 13. a f x 12 x 2 c hx 32 x 2 f x x 2 1 In Exercises 1734, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x intercept s . 17. 19. 21. 23. 25. 4 x y e 2 2 14. a c 15. a c 16. a b c d b gx 18 x 2 d kx 3x 2 In Exercises 3542, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x intercepts. Then check your results algebraically by writing the quadratic function in standard form. 35. 37. 39. 40. 41. f x x 2 2x 3 gx x2 8x 11 36. f x x 2 x 30 38. f x x 2 10x 14 f x 2x 16x 31 f x 4x 2 24x 41 1 3 gx 2x 2 4x 2 42. f x 5x 2 6x 5 2 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.1 In Exercises 4346, write an equation for the parabola in standard form. y 43. 1, 4 3, 0 y 44. 6 2 4 x 2 2 2 2 y 2, 2 3, 0 2 2, 1 45. In Exercises 6570, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x intercepts. There are many correct answers. y 46. 6 x 6 4 2 2, 0 4 3, 2 2 1, 0 6 2 x 2 4 6 In Exercises 4756, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. Vertex: 2, 5; point: 0, 9 Vertex: 4, 1; point: 2, 3 Vertex: 1, 2; point: 1, 14 Vertex: 2, 3; point: 0, 2 Vertex: 5, 12; point: 7, 15 Vertex: 2, 2; point: 1, 0 1 3 Vertex: 4, 2 ; point: 2, 0 5 3 Vertex: 2, 4 ; point: 2, 4 5 7 16 Vertex: 2, 0; point: 2, 3 61 3 Vertex: 6, 6; point: 10, 2 y 2 8 4 8 75. PATH OF A DIVER y y 4 71. The sum is 110. 72. The sum is S. 73. The sum of the first and twice the second is 24. 74. The sum of the first and three times the second is 42. The path of a diver is given by where y is the height in feet and x is the horizontal distance from the end of the diving board in feet . What is the maximum height of the diver 76. HEIGHT OF A BALL The height y in feet of a punted football is given by 58. y 2x 2 5x 3 x In Exercises 71 74, find two positive real numbers whose product is a maximum. 4 24 y x 2 x 12 9 9 GRAPHICAL REASONING In Exercises 57 and 58, determine the x intercept s of the graph visually. Then find the x intercept s algebraically to confirm your results. 57. y x 2 4x 5 66. 5, 0, 5, 0 68. 4, 0, 8, 0 5 70. 2, 0, 2, 0 65. 1, 0, 3, 0 67. 0, 0, 10, 0 1 69. 3, 0, 2, 0 8 2 60. f x 2x 2 10x 62. f x x 2 8x 20 7 64. f x 10x 2 12x 45 x 6 4 4 267 In Exercises 5964, use a graphing utility to graph the quadratic function. Find the x intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when f x 0. 59. f x x 2 4x 61. f x x 2 9x 18 63. f x 2x 2 7x 30 0, 3 1, 0 Quadratic Functions and Models x 6 4 2 2 4 16 2 9 x x 1.5 2025 5 where x is the horizontal distance in feet from the point at which the ball is punted. a How high is the ball when it is punted b What is the maximum height of the punt c How long is the punt 77. MINIMUM COST A manufacturer of lighting fixtures has daily production costs of C 800 10x 0.25x 2, where C is the total cost in dollars and x is the number of units produced. How many fixtures should be produced each day to yield a minimum cost 78. MAXIMUM PROFIT The profit P in hundreds of dollars that a company makes depends on the amount x in hundreds of dollars the company spends on advertising according to the model P 230 20x 0.5x 2. What expenditure for advertising will yield a maximum profit www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 268 Chapter 3 Polynomial Functions 79. MAXIMUM REVENUE The total revenue R earned in thousands of dollars from manufacturing handheld video games is given by R p 25p2 1200p where p is the price per unit in dollars . a Find the revenues when the price per unit is 20, 25, and 30. b Find the unit price that will yield a maximum revenue. What is the maximum revenue Explain your results. 80. MAXIMUM REVENUE The total revenue R earned per day in dollars from a pet sitting service is given by R p 12p2 150p, where p is the price charged per pet in dollars . a Find the revenues when the price per pet is 4, 6, and 8. b Find the price that will yield a maximum revenue. What is the maximum revenue Explain your results. 81. NUMERICAL, GRAPHICAL, AND ANALYTICAL ANALYSIS A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals see figure . b Determine the radius of each semicircular end of the room. Determine the distance, in terms of y, around the inside edge of each semicircular part of the track. c Use the result of part b to write an equation, in terms of x and y, for the distance traveled in one lap around the track. Solve for y. d Use the result of part c to write the area A of the rectangular region as a function of x. What dimensions will produce a rectangle of maximum area 83. MAXIMUM REVENUE A small theater has a seating capacity of 2000. When the ticket price is 20, attendance is 1500. For each 1 decrease in price, attendance increases by 100. a Write the revenue R of the theater as a function of ticket price x. b What ticket price will yield a maximum revenue What is the maximum revenue 84. MAXIMUM AREA A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window see figure . The perimeter of the window is 16 feet. x 2 y x x y x a Write the area A of the corrals as a function of x. b Create a table showing possible values of x and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area. c Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area. d Write the area function in standard form to find analytically the dimensions that will produce the maximum area. e Compare your results from parts b , c , and d . 82. GEOMETRY An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200 meter singlelane running track. a Draw a diagram that illustrates the problem. Let x and y represent the length and width of the rectangular region, respectively. a Write the area A of the window as a function of x. b What dimensions will produce a window of maximum area 85. GRAPHICAL ANALYSIS From 1950 through 2005, the per capita consumption C of cigarettes by Americans age 18 and older can be modeled by C 3565.0 60.30t 1.783t 2, 0 t 55, where t is the year, with t 0 corresponding to 1950. Source: Tobacco Outlook Report a Use a graphing utility to graph the model. b Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect Explain. c In 2005, the U.S. population age 18 and over was 296,329,000. Of those, about 59,858,458 were smokers. What was the average annual cigarette consumption per smoker in 2005 What was the average daily cigarette consumption per smoker www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.1 86. DATA ANALYSIS: SALES The sales y in billions of dollars for Harley Davidson from 2000 through 2007 are shown in the table. Source: U.S. HarleyDavidson, Inc. Quadratic Functions and Models 269 92. f x x2 bx 16; Maximum value: 48 93. f x x2 bx 26; Minimum value: 10 94. f x x2 bx 25; Minimum value: 50 95. Write the quadratic function Year Sales, y 2000 2001 2002 2003 2004 2005 2006 2007 2.91 3.36 4.09 4.62 5.02 5.34 5.80 5.73 f x ax 2 bx c in standard form to verify that the vertex occurs at 2ab , f 2ab . 96. CAPSTONE The profit P in millions of dollars for a recreational vehicle retailer is modeled by a quadratic function of the form P at 2 bt c a Use a graphing utility to create a scatter plot of the data. Let x represent the year, with x 0 corresponding to 2000. b Use the regression feature of the graphing utility to find a quadratic model for the data. c Use the graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data d Use the trace feature of the graphing utility to approximate the year in which the sales for HarleyDavidson were the greatest. e Verify your answer to part d algebraically. f Use the model to predict the sales for HarleyDavidson in 2010. EXPLORATION TRUE OR FALSE In Exercises 8790, determine whether the statement is true or false. Justify your answer. 87. The function given by f x 12x 2 1 has no x intercepts. 88. The graphs of f x 4x 2 10x 7 and gx 12x 2 30x 1 have the same axis of symmetry. 89. The graph of a quadratic function with a negative leading coefficient will have a maximum value at its vertex. 90. The graph of a quadratic function with a positive leading coefficient will have a minimum value at its vertex. THINK ABOUT IT In Exercises 9194, find the values of b such that the function has the given maximum or minimum value. where t represents the year. If you were president of the company, which of the models below would you prefer Explain your reasoning. a a is positive and b2a t. b a is positive and t b2a. c a is negative and b2a t. d a is negative and t b2a. 97. GRAPHICAL ANALYSIS a Graph y ax2 for a 2, 1, 0.5, 0.5, 1 and 2. How does changing the value of a affect the graph b Graph y x h2 for h 4, 2, 2, and 4. How does changing the value of h affect the graph c Graph y x2 k for k 4, 2, 2, and 4. How does changing the value of k affect the graph 98. Describe the sequence of transformation from f to g given that f x x2 and gx ax h2 k. Assume a, h, and k are positive. 99. Is it possible for a quadratic equation to have only one x intercept Explain. 100. Assume that the function given by f x ax 2 bx c, a 0 has two real zeros. Show that the x coordinate of the vertex of the graph is the average of the zeros of f. Hint: Use the Quadratic Formula. PROJECT: HEIGHT OF A BASKETBALL To work an extended application analyzing the height of a basketball after it has been dropped, visit this texts website at academic.cengage.com. 91. f x x2 bx 75; Maximum value: 25 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 270 Chapter 3 Polynomial Functions 3.2 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE What you should learn Use transformations to sketch graphs of polynomial functions. Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. Find and use zeros of polynomial functions as sketching aids. Use the Intermediate Value Theorem to help locate zeros of polynomial functions. Graphs of Polynomial Functions In this section, you will study basic features of the graphs of polynomial functions. The first feature is that the graph of a polynomial function is continuous. Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps, as shown in Figure 3.11 a . The graph shown in Figure 3.11 b is an example of a piecewisedefined function that is not continuous. y y Why you should learn it You can use polynomial functions to analyze business situations such as how revenue is related to advertising expenses, as discussed in Exercise 104 on page 282. a Polynomial functions have continuous graphs. FIGURE Bill AronPhotoEdit, Inc. x x b Functions with graphs that are not continuous are not polynomial functions. 3.11 The second feature is that the graph of a polynomial function has only smooth, rounded turns, as shown in Figure 3.12. A polynomial function cannot have a sharp turn. For instance, the function given by f x x, which has a sharp turn at the point 0, 0, as shown in Figure 3.13, is not a polynomial function. y y 6 5 4 3 2 x Polynomial functions have graphs with smooth, rounded turns. FIGURE 3.12 4 3 2 1 2 f x = x x 1 2 3 4 0, 0 Graphs of polynomial functions cannot have sharp turns. FIGURE 3.13 The graphs of polynomial functions of degree greater than 2 are more difficult to analyze than the graphs of polynomials of degree 0, 1, or 2. However, using the features presented in this section, coupled with your knowledge of point plotting, intercepts, and symmetry, you should be able to make reasonably accurate sketches by hand. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 For power functions given by f x x n, if n is even, then the graph of the function is symmetric with respect to the y axis, and if n is odd, then the graph of the function is symmetric with respect to the origin. 271 Polynomial Functions of Higher Degree The polynomial functions that have the simplest graphs are monomials of the form f x x n, where n is an integer greater than zero. From Figure 3.14, you can see that when n is even, the graph is similar to the graph of f x x 2, and when n is odd, the graph is similar to the graph of f x x 3. Moreover, the greater the value of n, the flatter the graph near the origin. Polynomial functions of the form f x x n are often referred to as power functions. y y y = x4 2 1, 1 1 y = x3 y = x2 1, 1 1 x 1 1, 1 1, 1 1 a If n is even, the graph of y x n touches the axis at the x intercept. 1 1 x 1 FIGURE y = x5 b If n is odd, the graph of y x n crosses the axis at the x intercept. 3.14 Example 1 Sketching Transformations of Polynomial Functions Sketch the graph of each function. a. f x x 5 b. hx x 14 Solution a. Because the degree of f x x 5 is odd, its graph is similar to the graph of y x 3. In Figure 3.15, note that the negative coefficient has the effect of reflecting the graph in the x axis. b. The graph of hx x 14, as shown in Figure 3.16, is a left shift by one unit of the graph of y x 4. y 1, 1 You can review the techniques for shifting, reflecting, and stretching graphs in Section 2.5. 3 1 f x = x 5 2 x 1 1 1 FIGURE y h x = x + 1 4 1, 1 3.15 2, 1 1 0, 1 1, 0 2 FIGURE 1 x 1 3.16 Now try Exercise 17. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 272 Chapter 3 Polynomial Functions The Leading Coefficient Test In Example 1, note that both graphs eventually rise or fall without bound as x moves to the right. Whether the graph of a polynomial function eventually rises or falls can be determined by the functions degree even or odd and by its leading coefficient, as indicated in the Leading Coefficient Test. Leading Coefficient Test As x moves without bound to the left or to the right, the graph of the polynomial function f x a n x n . . . a1x a0 eventually rises or falls in the following manner. 1. When n is odd: y y f x as x f x as x f x as x f x as x x If the leading coefficient is positive an 0, the graph falls to the left and rises to the right. x If the leading coefficient is negative an 0, the graph rises to the left and falls to the right. 2. When n is even: y The notation f x as x indicates that the graph falls to the left. The notation f x as x indicates that the graph rises to the right. y f x as x f x as x f x as x x If the leading coefficient is positive an 0, the graph rises to the left and right. f x as x x If the leading coefficient is negative an 0, the graph falls to the left and right. The dashed portions of the graphs indicate that the test determines only the right hand and left hand behavior of the graph. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 Example 2 273 Polynomial Functions of Higher Degree Applying the Leading Coefficient Test Describe the right hand and left hand behavior of the graph of each function. a. f x x3 4x b. f x x 4 5x 2 4 c. f x x 5 x Solution a. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right, as shown in Figure 3.17. b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure 3.18. c. Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, as shown in Figure 3.19. f x = x 3 + 4x f x = x 5 x f x = x 4 5x 2 + 4 y y y 3 6 2 4 1 2 1 3 1 x 1 2 3 x 4 FIGURE 3.17 FIGURE 4 3.18 x 2 1 2 FIGURE 3.19 Now try Exercise 23. In Example 2, note that the Leading Coefficient Test tells you only whether the graph eventually rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maximum points, must be determined by other tests. Zeros of Polynomial Functions It can be shown that for a polynomial function f of degree n, the following statements are true. Remember that the zeros of a function of x are the x values for which the function is zero. 1. The function f has, at most, n real zeros. You will study this result in detail in the discussion of the Fundamental Theorem of Algebra in Section 3.4. 2. The graph of f has, at most, n 1 turning points. Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa. Finding the zeros of polynomial functions is one of the most important problems in algebra. There is a strong interplay between graphical and algebraic approaches to this problem. Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph. Finding zeros of polynomial functions is closely related to factoring and finding x intercepts. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 274 Chapter 3 Polynomial Functions Real Zeros of Polynomial Functions To do Example 3 algebraically, you need to be able to completely factor polynomials. You can review the techniques for factoring in Section P.4. If f is a polynomial function and a is a real number, the following statements are equivalent. 1. x a is a zero of the function f. 2. x a is a solution of the polynomial equation f x 0. 3. x a is a factor of the polynomial f x. 4. a, 0 is an x intercept of the graph of f. Example 3 Finding the Zeros of a Polynomial Function Find all real zeros of f x 2x4 2x 2. Then determine the number of turning points of the graph of the function. Algebraic Solution Graphical Solution To find the real zeros of the function, set f x equal to zero and solve for x. Use a graphing utility to graph y 2x 4 2x2. In Figure 3.20, the graph appears to have zeros at 0, 0, 1, 0, and 1, 0. Use the zero or root feature, or the zoom and trace features, of the graphing utility to verify these zeros. So, the real zeros are x 0, x 1, and x 1. From the figure, you can see that the graph has three turning points. This is consistent with the fact that a fourth degree polynomial can have at most three turning points. 2x 4 2x2 0 2x2x2 1 0 Set f x equal to 0. Remove common monomial factor. 2x2x 1x 1 0 Factor completely. So, the real zeros are x 0, x 1, and x 1. Because the function is a fourth degree polynomial, the graph of f can have at most 4 1 3 turning points. 2 y = 2x 4 + 2x 2 3 3 2 FIGURE 3.20 Now try Exercise 35. In Example 3, note that because the exponent is greater than 1, the factor 2x2 yields the repeated zero x 0. Because the exponent is even, the graph touches the x axis at x 0, as shown in Figure 3.20. Repeated Zeros A factor x ak, k 1, yields a repeated zero x a of multiplicity k. 1. If k is odd, the graph crosses the x axis at x a. 2. If k is even, the graph touches the x axis but does not cross the x axis at x a. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 T E C H N O LO G Y Example 4 uses an algebraic approach to describe the graph of the function. A graphing utility is a complement to this approach. Remember that an important aspect of using a graphing utility is to find a viewing window that shows all significant features of the graph. For instance, the viewing window in part a illustrates all of the significant features of the function in Example 4 while the viewing window in part b does not. a. 3 4 5 275 Polynomial Functions of Higher Degree A polynomial function is written in standard form if its terms are written in descending order of exponents from left to right. Before applying the Leading Coefficient Test to a polynomial function, it is a good idea to check that the polynomial function is written in standard form. Example 4 Sketching the Graph of a Polynomial Function Sketch the graph of f x 3x 4 4x 3. Solution 1. Apply the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, you know that the graph eventually rises to the left and to the right see Figure 3.21 . 2. Find the Zeros of the Polynomial. By factoring f x 3x 4 4x 3 x33x 4 Remove common factor. you can see that the zeros of f are x 0 and x 43 both of odd multiplicity . So, the x intercepts occur at 0, 0 and 43, 0. Add these points to your graph, as shown in Figure 3.21. 3. Plot a Few Additional Points. To sketch the graph by hand, find a few additional points, as shown in the table. Then plot the points see Figure 3.22 . x 3 1 0.5 1 1.5 7 0.3125 1 1.6875 f x 0.5 b. 2 2 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 3.22. Because both zeros are of odd multiplicity, you know that the graph should cross the x axis at x 0 and x 43. If you are unsure of the shape of that portion of the graph, plot some additional points. 0.5 y y 7 7 6 6 5 Up to left 4 f x = 3x 4 4x 3 5 Up to right 4 3 3 2 0, 0 4 3 2 1 1 FIGURE 43 , 0 x 1 2 3 4 3.21 4 3 2 1 1 FIGURE x 2 3 4 3.22 Now try Exercise 75. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 276 Chapter 3 Polynomial Functions Example 5 Sketching the Graph of a Polynomial Function Sketch the graph of f x 2x 3 6x 2 92x. Solution 1. Apply the Leading Coefficient Test. Because the leading coefficient is negative and the degree is odd, you know that the graph eventually rises to the left and falls to the right see Figure 3.23 . 2. Find the Zeros of the Polynomial. By factoring f x 2x3 6x2 92 x 12 x 4x2 12x 9 12 x 2x 32 Observe in Example 5 that the sign of f x is positive to the left of and negative to the right of the zero x 0. Similarly, the sign of f x is negative to the left and to the right of the zero x 32. This suggests that if the zero of a polynomial function is of odd multiplicity, then the sign of f x changes from one side to the other side of the zero. If the zero is of even multiplicity, then the sign of f x does not change from one side of the zero to the other side. The following table helps to illustrate this concept. 3. Plot a Few Additional Points. To sketch the graph by hand, find a few additional points, as shown in the table. Then plot the points see Figure 3.24 . x 0 0.5 f x 4 0 1 Sign 0.5 1 2 4 1 0.5 1 f x 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 3.24. As indicated by the multiplicities of the zeros, the graph crosses the x axis at 0, 0 but does not cross the x axis at 32, 0. y y 6 f x = 2x 3 + 6x 2 92 x 5 Up to left 3 0, 0 1 3 2 2 4 3 2 1 1 f x 0.5 0 1 Sign Down to right 2 x 0.5 4 0.5 x you can see that the zeros of f are x 0 odd multiplicity and x 32 even multiplicity . So, the x intercepts occur at 0, 0 and 32, 0. Add these points to your graph, as shown in Figure 3.23. 32 , 0 1 2 1 x 3 4 3 2 1 1 4 2 FIGURE 3.23 x 3 4 2 FIGURE 3.24 Now try Exercise 77. This sign analysis may be helpful in graphing polynomial functions. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 Polynomial Functions of Higher Degree 277 The Intermediate Value Theorem The next theorem, called the Intermediate Value Theorem, illustrates the existence of real zeros of polynomial functions. This theorem implies that if a, f a and b, f b are two points on the graph of a polynomial function such that f a f b, then for any number d between f a and f b there must be a number c between a and b such that f c d. See Figure 3.25. y f b f c = d f a a FIGURE x cb 3.25 Intermediate Value Theorem Let a and b be real numbers such that a b. If f is a polynomial function such that f a f b, then, in the interval a, b, f takes on every value between f a and f b. The Intermediate Value Theorem helps you locate the real zeros of a polynomial function in the following way. If you can find a value x a at which a polynomial function is positive, and another value x b at which it is negative, you can conclude that the function has at least one real zero between these two values. For example, the function given by f x x 3 x 2 1 is negative when x 2 and positive when x 1. Therefore, it follows from the Intermediate Value Theorem that f must have a real zero somewhere between 2 and 1, as shown in Figure 3.26. y f x = x 3 + x 2 + 1 1, 1 f 1 = 1 2 2, 3 FIGURE x 1 2 f has a zero 1 between 2 and 1. 2 3 f 2 = 3 3.26 By continuing this line of reasoning, you can approximate any real zeros of a polynomial function to any desired accuracy. This concept is further demonstrated in Example 6. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 278 Chapter 3 Polynomial Functions Example 6 Approximating a Zero of a Polynomial Function Use the Intermediate Value Theorem to approximate the real zero of f x x 3 x 2 1. Solution Begin by computing a few function values, as follows. y f x = x 3 x 2 + 1 0, 1 1, 1 1 1, 1 FIGURE f 0.8 0.152 x 1 f x 2 11 1 1 0 1 1 1 Because f 1 is negative and f 0 is positive, you can apply the Intermediate Value Theorem to conclude that the function has a zero between 1 and 0. To pinpoint this zero more closely, divide the interval 1, 0 into tenths and evaluate the function at each point. When you do this, you will find that 2 1 x 2 f has a zero between 0.8 and 0.7. 3.27 and f 0.7 0.167. So, f must have a zero between 0.8 and 0.7, as shown in Figure 3.27. For a more accurate approximation, compute function values between f 0.8 and f 0.7 and apply the Intermediate Value Theorem again. By continuing this process, you can approximate this zero to any desired accuracy. Now try Exercise 93. T E C H N O LO G Y You can use the table feature of a graphing utility to approximate the zeros of a polynomial function. For instance, for the function given by f x 2x3 3x2 3 create a table that shows the function values for 20 x 20, as shown in the first table at the right. Scroll through the table looking for consecutive function values that differ in sign. From the table, you can see that f 0 and f 1 differ in sign. So, you can conclude from the Intermediate Value Theorem that the function has a zero between 0 and 1. You can adjust your table to show function values for 0 x 1 using increments of 0.1, as shown in the second table at the right. By scrolling through the table you can see that f 0.8 and f 0.9 differ in sign. So, the function has a zero between 0.8 and 0.9. If you repeat this process several times, you should obtain x y 0.806 as the zero of the function. Use the zero or root feature of a graphing utility to confirm this result. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 3.2 EXERCISES 279 Polynomial Functions of Higher Degree See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps. 2. The ________ ________ ________ is used to determine the left hand and right hand behavior of the graph of a polynomial function. 3. Polynomial functions of the form f x ________ are often referred to as power functions. 4. A polynomial function of degree n has at most ________ real zeros and at most ________ turning points. 5. If x a is a zero of a polynomial function f, then the following three statements are true. a x a is a ________ of the polynomial equation f x 0. b ________ is a factor of the polynomial f x. c a, 0 is an ________ of the graph of f. 6. If a real zero of a polynomial function is of even multiplicity, then the graph of f ________ the x axis at x a, and if it is of odd multiplicity, then the graph of f ________ the x axis at x a. 7. A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right. 8. The ________ ________ Theorem states that if f is a polynomial function such that f a f b, then, in the interval a, b, f takes on every value between f a and f b. SKILLS AND APPLICATIONS In Exercises 916, match the polynomial function with its graph. The graphs are labeled a , b , c , d , e , f , g , and h . y a x 8 8 4 y 8 4 x 4 4 8 8 6 4 4 x 2 y e x 4 8 2 y 4 8 8 4 x 4 4 8 4 2 f 8 4 x 2 2 4 6 2 4 4 x 2 2 4 10. f x x 2 4x 12. f x 2x 3 3x 1 1 4 14. f x 3x 3 x 2 3 1 5 9 16. f x 5x 2x 3 5x In Exercises 1720, sketch the graph of y x n and each transformation. 8 4 x 2 9. f x 2x 3 11. f x 2x 2 5x 1 13. f x 4x 4 3x 2 15. f x x 4 2x 3 y d 4 4 2 8 c y h y b 8 8 y g 4 17. y x 3 a f x x 43 1 c f x 4x 3 18. y x 5 a f x x 15 1 c f x 1 2x 5 19. y x 4 a f x x 34 c f x 4 x 4 e f x 2x4 1 www.elsolucionario.net b f x x 3 4 d f x x 43 4 b f x x 5 1 1 d f x 2x 15 b f x x 4 3 1 d f x 2x 14 1 4 f f x 2 x 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 280 Chapter 3 Polynomial Functions 20. y x 6 a f x 18x 6 c f x x 6 5 6 e f x 14 x 2 b f x x 2 4 d f x 14x 6 1 f f x 2x6 1 In Exercises 2130, describe the right hand and left hand behavior of the graph of the polynomial function. 21. 23. 25. 26. 27. 28. 29. 30. f x 15x 3 4x 22. f x 2x 2 3x 1 7 2 g x 5 2x 3x 24. h x 1 x 6 f x 2.1x 5 4x 3 2 f x 4x 5 7x 6.5 f x 6 2x 4x 2 5x 3 f x 3x 4 2x 54 h t 34t 2 3t 6 f s 78s 3 5s 2 7s 1 GRAPHICAL ANALYSIS In Exercises 3134, use a graphing utility to graph the functions f and g in the same viewing window. Zoom out sufficiently far to show that the right hand and left hand behaviors of f and g appear identical. 31. 32. 33. 34. f x 3x 3 9x 1, gx 3x 3 f x 13x 3 3x 2, gx 13x 3 f x x 4 4x 3 16x, gx x 4 f x 3x 4 6x 2, gx 3x 4 In Exercises 35 50, a find all the real zeros of the polynomial function, b determine the multiplicity of each zero and the number of turning points of the graph of the function, and c use a graphing utility to graph the function and verify your answers. 35. f x x 2 36 36. f x 81 x 2 37. h t t 2 6t 9 38. f x x 2 10x 25 1 2 1 2 1 5 3 39. f x 3 x 3 x 3 40. f x 2x 2 2x 2 41. f x 3x3 12x2 3x 42. gx 5xx 2 2x 1 43. f t t 3 8t 2 16t 44. f x x 4 x 3 30x 2 45. gt t 5 6t 3 9t 46. f x x 5 x 3 6x 4 2 47. f x 3x 9x 6 48. f x 2x 4 2x 2 40 49. gx x3 3x 2 4x 12 50. f x x 3 4x 2 25x 100 GRAPHICAL ANALYSIS In Exercises 5154, a use a graphing utility to graph the function, b use the graph to approximate any x intercepts of the graph, c set y 0 and solve the resulting equation, and d compare the results of part c with any x intercepts of the graph. 51. y 4x 3 20x 2 25x 52. y 4x 3 4x 2 8x 8 53. y x 5 5x 3 4x 1 54. y 4x 3x 2 9 6 In Exercises 55 64, find a polynomial function that has the given zeros. There are many correct answers. 55. 57. 59. 61. 63. 0, 8 2, 6 0, 4, 5 4, 3, 3, 0 1 3, 1 3 56. 58. 60. 62. 64. 0, 7 4, 5 0, 1, 10 2, 1, 0, 1, 2 2, 4 5, 4 5 In Exercises 6574, find a polynomial of degree n that has the given zero s . There are many correct answers. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. Zero s x 3 x 12, 6 x 5, 0, 1 x 2, 4, 7 x 0, 3, 3 x9 x 5, 1, 2 x 4, 1, 3, 6 x 0, 4 x 1, 4, 7, 8 Degree n2 n2 n3 n3 n3 n3 n4 n4 n5 n5 In Exercises 7588, sketch the graph of the function by a applying the Leading Coefficient Test, b finding the zeros of the polynomial, c plotting sufficient solution points, and d drawing a continuous curve through the points. 75. 77. 78. 79. 81. 82. 83. f x x 3 25x 1 f t 4t 2 2t 15 gx x 2 10x 16 76. gx x 4 9x 2 f x x 3 2x 2 80. f x 8 x 3 f x 3x3 15x 2 18x f x 4x 3 4x 2 15x f x 5x2 x3 84. f x 48x 2 3x 4 1 2 85. f x x x 4 86. hx 3x 3x 42 1 87. gt 4t 22t 22 1 88. gx 10x 12x 33 In Exercises 8992, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. 1 89. f x x 3 16x 90. f x 4x 4 2x 2 1 91. gx 5x 12x 32x 9 1 92. hx 5x 223x 52 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 In Exercises 9396, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. 93. 94. 95. 96. f x x 3 3x 2 3 f x 0.11x 3 2.07x 2 9.81x 6.88 gx 3x 4 4x 3 3 h x x 4 10x 2 3 97. NUMERICAL AND GRAPHICAL ANALYSIS An open box is to be made from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides see figure . x 36 2x x x a Write a function Vx that represents the volume of the box. b Determine the domain of the function. c Use a graphing utility to create a table that shows box heights x and the corresponding volumes V. Use the table to estimate the dimensions that will produce a maximum volume. d Use a graphing utility to graph V and use the graph to estimate the value of x for which Vx is maximum. Compare your result with that of part c . 98. MAXIMUM VOLUME An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure. 24 in. x 281 c Sketch a graph of the function and estimate the value of x for which Vx is maximum. 99. CONSTRUCTION A roofing contractor is fabricating gutters from 12 inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls see figure . x 12 2x x a Let x represent the height of the sidewall of the gutter. Write a function A that represents the cross sectional area of the gutter. b The length of the aluminum sheeting is 16 feet. Write a function V that represents the volume of one run of gutter in terms of x. c Determine the domain of the function in part b . d Use a graphing utility to create a table that shows the sidewall heights x and the corresponding volumes V. Use the table to estimate the dimensions that will produce a maximum volume. e Use a graphing utility to graph V. Use the graph to estimate the value of x for which Vx is a maximum. Compare your result with that of part d . f Would the value of x change if the aluminum sheeting were of different lengths Explain. 100. CONSTRUCTION An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components see figure . 4r r xx 24 in. xx x Polynomial Functions of Higher Degree a Write a function Vx that represents the volume of the box. b Determine the domain of the function V. a Write a function that represents the total volume V of the tank in terms of r. b Find the domain of the function. c Use a graphing utility to graph the function. d The total volume of the tank is to be 120 cubic feet. Use the graph from part c to estimate the radius and length of the cylindrical portion of the tank. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 282 Chapter 3 Polynomial Functions 101. REVENUE The total revenues R in millions of dollars for Krispy Kreme from 2000 through 2007 are shown in the table. Year Revenue, R 2000 2001 2002 2003 2004 2005 2006 2007 300.7 394.4 491.5 665.6 707.8 543.4 461.2 429.3 A model that represents these data is given by R 3.0711t 4 42.803t3 160.59t2 62.6t 307, 0 t 7, where t represents the year, with t 0 corresponding to 2000. Source: Krispy Kreme a Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. b How well does the model fit the data c Use a graphing utility to approximate any relative extrema of the model over its domain. d Use a graphing utility to approximate the intervals over which the revenue for Krispy Kreme was increasing and decreasing over its domain. e Use the results of parts c and d to write a short paragraph about Krispy Kremes revenue during this time period. 102. REVENUE The total revenues R in millions of dollars for Papa Johns International from 2000 through 2007 are shown in the table. Year Revenue, R 2000 2001 2002 2003 2004 2005 2006 2007 944.7 971.2 946.2 917.4 942.4 968.8 1001.6 1063.6 a Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. b How well does the model fit the data c Use a graphing utility to approximate any relative extrema of the model over its domain. d Use a graphing utility to approximate the intervals over which the revenue for Papa Johns International was increasing and decreasing over its domain. e Use the results of parts c and d to write a short paragraph about the revenue for Papa Johns International during this time period. 103. TREE GROWTH The growth of a red oak tree is approximated by the function G 0.003t 3 0.137t 2 0.458t 0.839 where G is the height of the tree in feet and t 2 t 34 is its age in years . a Use a graphing utility to graph the function. Hint: Use a viewing window in which 10 x 45 and 5 y 60. b Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year. c Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by y 0.009t 2 0.274t 0.458. Find the vertex of this parabola. d Compare your results from parts b and c . 104. REVENUE The total revenue R in millions of dollars for a company is related to its advertising expense by the function R 1 x 3 600x2, 0 x 400 100,000 where x is the amount spent on advertising in tens of thousands of dollars . Use the graph of this function, shown in the figure on the next page, to estimate the point on the graph at which the function is increasing most rapidly. This point is called the point of diminishing returns because any expense above this amount will yield less return per dollar invested in advertising. A model that represents these data is given by R 0.5635t 4 9.019t 3 40.20t2 49.0t 947, 0 t 7, where t represents the year, with t 0 corresponding to 2000. Source: Papa Johns International www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.2 Revenue in millions of dollars R 350 300 250 200 150 100 50 x 100 200 300 400 Advertising expense in tens of thousands of dollars FIGURE FOR 104 EXPLORATION TRUE OR FALSE In Exercises 105107, determine whether the statement is true or false. Justify your answer. 105. A fifth degree polynomial can have five turning points in its graph. 106. It is possible for a sixth degree polynomial to have only one solution. 107. The graph of the function given by f x 2 x x 2 x3 x 4 x5 x 6 x7 rises to the left and falls to the right. 108. CAPSTONE For each graph, describe a polynomial function that could represent the graph. Indicate the degree of the function and the sign of its leading coefficient. y y a b x Polynomial Functions of Higher Degree 283 109. GRAPHICAL REASONING Sketch a graph of the function given by f x x 4. Explain how the graph of each function g differs if it does from the graph of each function f. Determine whether g is odd, even, or neither. a gx f x 2 b gx f x 2 c gx f x d gx f x 1 1 e gx f 2x f gx 2 f x g gx f x34 h gx f f x 110. THINK ABOUT IT For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right hand and lefthand behavior of the graph of the function. a f x x3 2x2 x 1 b f x 2x5 2x2 5x 1 c f x 2x5 x2 5x 3 d f x x3 5x 2 e f x 2x2 3x 4 f f x x 4 3x2 2x 1 g f x x2 3x 2 111. THINK ABOUT IT Sketch the graph of each polynomial function. Then count the number of zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe a f x x3 9x b f x x 4 10x2 9 c f x x5 16x 112. Explore the transformations of the form gx ax h5 k. x c y d x y x a Use a graphing utility to graph the functions y1 13x 25 1 and y2 35x 25 3. Determine whether the graphs are increasing or decreasing. Explain. b Will the graph of g always be increasing or decreasing If so, is this behavior determined by a, h, or k Explain. c Use a graphing utility to graph the function given by Hx x 5 3x 3 2x 1. Use the graph and the result of part b to determine whether H can be written in the form Hx ax h5 k. Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 284 Chapter 3 Polynomial Functions 3.3 POLYNOMIAL AND SYNTHETIC DIVISION What you should learn Use long division to divide polynomials by other polynomials. Use synthetic division to divide polynomials by binomials of the form x k. Use the Remainder Theorem and the Factor Theorem. Why you should learn it Synthetic division can help you evaluate polynomial functions. For instance, in Exercise 85 on page 291, you will use synthetic division to determine the amount donated to support higher education in the United States in 2010. Long Division of Polynomials In this section, you will study two procedures for dividing polynomials. These procedures are especially valuable in factoring and finding the zeros of polynomial functions. To begin, suppose you are given the graph of f x 6x 3 19x 2 16x 4. Notice that a zero of f occurs at x 2, as shown in Figure 3.28. Because x 2 is a zero of f, you know that x 2 is a factor of f x. This means that there exists a second degree polynomial qx such that f x x 2 qx. To find qx, you can use long division, as illustrated in Example 1. Example 1 Long Division of Polynomials Divide 6x 3 19x 2 16x 4 by x 2, and use the result to factor the polynomial completely. Solution 6x 3 6x 2. x 7x 2 Think 7x. x 2x Think 2. x MBIAlamy Think 6x 2 7x 2 x 2 6x3 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x 2x 4 2x 4 0 Subtract. Multiply: 7x x 2. Subtract. Multiply: 2x 2. Subtract. From this division, you can conclude that y 1 Multiply: 6x2x 2. 12 , 0 23 , 0 1 6x 3 19x 2 16x 4 x 26x 2 7x 2 and by factoring the quadratic 6x 2 7x 2, you have 2, 0 x 3 Note that this factorization agrees with the graph shown in Figure 3.28 in that the three x intercepts occur at x 2, x 12, and x 23. 1 2 3 FIGURE 6x 3 19x 2 16x 4 x 22x 13x 2. Now try Exercise 11. f x = 6x 3 19x 2 + 16x 4 3.28 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.3 Polynomial and Synthetic Division 285 In Example 1, x 2 is a factor of the polynomial 6x 3 19x 2 16x 4, and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if you divide x 2 3x 5 by x 1, you obtain the following. x2 x 1 3x 5 x2 x 2x 5 2x 2 3 x2 Divisor Quotient Dividend Remainder In fractional form, you can write this result as follows. Remainder Dividend Quotient x 2 3x 5 3 x2 x1 x1 Divisor Divisor This implies that x 2 3x 5 x 1 x 2 3 Multiply each side by x 1. which illustrates the following theorem, called the Division Algorithm. The Division Algorithm If f x and dx are polynomials such that dx 0, and the degree of dx is less than or equal to the degree of f x, there exist unique polynomials qx and rx such that f x dxqx rx Dividend Quotient Divisor Remainder where r x 0 or the degree of rx is less than the degree of dx. If the remainder rx is zero, dx divides evenly into f x. The Division Algorithm can also be written as f x r x qx . dx dx In the Division Algorithm, the rational expression f xdx is improper because the degree of f x is greater than or equal to the degree of dx. On the other hand, the rational expression r xdx is proper because the degree of r x is less than the degree of dx. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 286 Chapter 3 Polynomial Functions Before you apply the Division Algorithm, follow these steps. 1. Write the dividend and divisor in descending powers of the variable. 2. Insert placeholders with zero coefficients for missing powers of the variable. Example 2 Long Division of Polynomials Divide x3 1 by x 1. Solution Because there is no x 2 term or x term in the dividend, you need to line up the subtraction by using zero coefficients or leaving spaces for the missing terms. x2 x 1 x 1 0x 2 0x 1 x 3 x2 x 2 0x x2 x x1 x1 0 x3 So, x 1 divides evenly into x 3 1, and you can write x3 1 x 2 x 1, x x1 1. Now try Exercise 17. You can check the result of Example 2 by multiplying. x 1x 2 x 1 x 3 x2 x x2 x 1 x3 1 You can check a long division problem by multiplying. You can review the techniques for multiplying polynomials in Section P.3. Example 3 Long Division of Polynomials Divide 5x2 2 3x 2x 4 4x3 by 2x 3 x2. Solution Begin by writing the dividend and divisor in descending powers of x. 2x 2 1 x 2 2x 3 2x 4 4x 3 5x 2 3x 2 2x 4 4x 3 6x 2 x 2 3x 2 x 2 2x 3 x1 Note that the first subtraction eliminated two terms from the dividend. When this happens, the quotient skips a term. You can write the result as x1 2x4 4x 3 5x 2 3x 2 2x 2 1 2 . x 2 2x 3 x 2x 3 Now try Exercise 23. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.3 Polynomial and Synthetic Division 287 Synthetic Division There is a nice shortcut for long division of polynomials by divisors of the form x k. This shortcut is called synthetic division. The pattern for synthetic division of a cubic polynomial is summarized as follows. The pattern for higher degree polynomials is similar. Synthetic Division for a Cubic Polynomial To divide ax3 bx 2 cx d by x k, use the following pattern. k a b c d Coefficients of dividend ka Vertical pattern: Add terms. Diagonal pattern: Multiply by k. a r Remainder Coefficients of quotient This algorithm for synthetic division works only for divisors of the form x k. Remember that x k x k. Example 4 Using Synthetic Division Use synthetic division to divide x 4 10x 2 2x 4 by x 3. Solution You should set up the array as follows. Note that a zero is included for the missing x3 term in the dividend. 3 0 10 2 1 4 Then, use the synthetic division pattern by adding terms in columns and multiplying the results by 3. Divisor: x 3 3 Dividend: x 4 10x 2 2x 4 1 0 3 10 9 2 3 4 3 1 3 1 1 1 Remainder: 1 Quotient: x3 3x2 x 1 So, you have x4 10x 2 2x 4 1 x 3 3x 2 x 1 . x3 x3 Now try Exercise 27. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 288 Chapter 3 Polynomial Functions The Remainder and Factor Theorems The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem If a polynomial f x is divided by x k, the remainder is r f k. For a proof of the Remainder Theorem, see Proofs in Mathematics on page 327. The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f x when x k, divide f x by x k. The remainder will be f k, as illustrated in Example 5. Example 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x 2. f x 3x3 8x 2 5x 7 Solution Using synthetic division, you obtain the following. 2 3 8 6 5 4 7 2 3 2 1 9 Because the remainder is r 9, you can conclude that f 2 9. r f k This means that 2, 9 is a point on the graph of f. You can check this by substituting x 2 in the original function. Check f 2 323 822 52 7 38 84 10 7 9 Now try Exercise 55. Another important theorem is the Factor Theorem, stated below. This theorem states that you can test to see whether a polynomial has x k as a factor by evaluating the polynomial at x k. If the result is 0, x k is a factor. The Factor Theorem A polynomial f x has a factor x k if and only if f k 0. For a proof of the Factor Theorem, see Proofs in Mathematics on page 327. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.3 Example 6 289 Polynomial and Synthetic Division Factoring a Polynomial: Repeated Division Show that x 2 and x 3 are factors of f x 2x 4 7x 3 4x 2 27x 18. Then find the remaining factors of f x. Algebraic Solution Using synthetic division with the factor x 2, you obtain the following. 2 2 7 4 4 22 27 36 18 18 2 11 18 9 0 0 remainder, so f 2 0 and x 2 is a factor. Take the result of this division and perform synthetic division again using the factor x 3. 3 2 2 11 6 18 15 5 3 Graphical Solution From the graph of f x 2x 4 7x3 4x2 27x 18, you can see that there are four x intercepts see Figure 3.29 . These occur at x 3, x 32, x 1, and x 2. Check this algebraically. This implies that x 3, x 32 , x 1, and x 2 are factors of f x. Note that x 32 and 2x 3 are equivalent factors because they both yield the same zero, x 32. f x = 2x 4 + 7x 3 4x 2 27x 18 y 9 9 0 40 0 remainder, so f 3 0 and x 3 is a factor. 30 32 , 0 2010 2x2 5x 3 Because the resulting quadratic expression factors as 2x 2 5x 3 2x 3x 1 4 1 2, 0 1 3 x 4 1, 0 20 3, 0 the complete factorization of f x is 30 f x x 2x 32x 3x 1. 40 FIGURE 3.29 Now try Exercise 67. Note in Example 6 that the complete factorization of f x implies that f has four real zeros: x 2, x 3, x 32, and x 1. This is confirmed by the graph of f, which is shown in the Figure 3.29. Uses of the Remainder in Synthetic Division The remainder r, obtained in the synthetic division of f x by x k, provides the following information. 1. The remainder r gives the value of f at x k. That is, r f k. 2. If r 0, x k is a factor of f x. 3. If r 0, k, 0 is an x intercept of the graph of f. Throughout this text, the importance of developing several problem solving strategies is emphasized. In the exercises for this section, try using more than one strategy to solve several of the exercises. For instance, if you find that x k divides evenly into f x with no remainder , try sketching the graph of f. You should find that k, 0 is an x intercept of the graph. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 290 3.3 Chapter 3 Polynomial Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY 1. Two forms of the Division Algorithm are shown below. Identify and label each term or function. f x dxqx r x f x r x qx dx dx In Exercises 26, fill in the blanks. 2. The rational expression pxqx is called ________ if the degree of the numerator is greater than or equal to that of the denominator, and is called ________ if the degree of the numerator is less than that of the denominator. 3. In the Division Algorithm, the rational expression f xdx is ________ because the degree of f x is greater than or equal to the degree of dx. 4. An alternative method to long division of polynomials is called ________ ________, in which the divisor must be of the form x k. 5. The ________ Theorem states that a polynomial f x has a factor x k if and only if f k 0. 6. The ________ Theorem states that if a polynomial f x is divided by x k, the remainder is r f k. SKILLS AND APPLICATIONS ANALYTICAL ANALYSIS In Exercises 7 and 8, use long division to verify that y1 y2. x2 4 , y2 x 2 x2 x2 x4 3x 2 1 39 8. y1 , y2 x 2 8 2 x2 5 x 5 7. y1 In Exercises 27 46, use synthetic division to divide. GRAPHICAL ANALYSIS In Exercises 9 and 10, a use a graphing utility to graph the two equations in the same viewing window, b use the graphs to verify that the expressions are equivalent, and c use long division to verify the results algebraically. x2 2x 1 2 , y2 x 1 x3 x3 x 4 x2 1 1 10. y1 , y2 x2 2 x2 1 x 1 9. y1 In Exercises 1126, use long division to divide. 11. 12. 13. 14. 15. 16. 17. 19. 21. 23. 24. 5x3 16 20x x 4 x2 x 3 x4 2x3 4x 2 15x 5 25. 26. 3 x 1 x 12 2x 2 10x 12 x 3 5x 2 17x 12 x 4 4x3 7x 2 11x 5 4x 5 6x3 16x 2 17x 6 3x 2 x 4 5x 3 6x 2 x 2 x 2 x3 4x 2 3x 12 x 3 x3 27 x 3 18. x3 125 x 5 7x 3 x 2 20. 8x 5 2x 1 3 2 x 9 x 1 22. x 5 7 x 3 1 3x 2x3 9 8x2 x2 1 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 39. 41. 43. 45. 46. 3x3 17x 2 15x 25 x 5 5x3 18x 2 7x 6 x 3 6x3 7x2 x 26 x 3 2x3 14x2 20x 7 x 6 4x3 9x 8x 2 18 x 2 9x3 16x 18x 2 32 x 2 x3 75x 250 x 10 3x3 16x 2 72 x 6 5x3 6x 2 8 x 4 5x3 6x 8 x 2 10x 4 50x3 800 x 5 13x 4 120x 80 38. x6 x3 3 x3 512 x 729 40. x9 x8 3x 4 3x 4 42. x2 x2 4 180x x 5 3x 2x 2 x3 44. x6 x1 3 2 4x 16x 23x 15 1 x2 3 2 3x 4x 5 x 32 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.3 In Exercises 47 54, write the function in the form f x x kqx r for the given value of k, and demonstrate that f k r. 47. 48. 49. 50. 51. 52. 53. 54. f x x3 x 2 14x 11, k 4 f x x3 5x 2 11x 8, k 2 f x 15x 4 10x3 6x 2 14, k 23 f x 10x3 22x 2 3x 4, k 15 f x x3 3x 2 2x 14, k 2 f x x 3 2x 2 5x 4, k 5 f x 4x3 6x 2 12x 4, k 1 3 f x 3x3 8x 2 10x 8, k 2 2 In Exercises 5558, use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. 55. f x 2x3 7x 3 a f 1 b f 2 c f 12 6 4 2 56. gx 2x 3x x 3 a g2 b g1 c g3 57. hx x3 5x 2 7x 4 a h3 b h2 c h2 58. f x 4x4 16x3 7x 2 20 a f 1 b f 2 c f 5 d f 2 d g1 d h5 d f 10 Polynomial and Synthetic Division Function 70. f x 71. 72. 73. 74. Factors 10x 24 f x 6x3 41x 2 9x 14 f x 10x3 11x 2 72x 45 f x 2x3 x 2 10x 5 f x x3 3x 2 48x 144 8x 4 14x3 291 71x 2 x 2, x 4 2x 1, 3x 2 2x 5, 5x 3 2x 1, x 5 x 43 , x 3 GRAPHICAL ANALYSIS In Exercises 7580, a use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, b determine one of the exact zeros, and c use synthetic division to verify your result from part b , and then factor the polynomial completely. 75. 76. 77. 78. 79. 80. f x x3 2x 2 5x 10 gx x3 4x 2 2x 8 ht t 3 2t 2 7t 2 f s s3 12s 2 40s 24 hx x5 7x 4 10x3 14x2 24x gx 6x 4 11x3 51x2 99x 27 In Exercises 8184, simplify the rational expression by using long division or synthetic division. 4x 3 8x 2 x 3 x 3 x 2 64x 64 82. 2x 3 x8 x 4 6x3 11x 2 6x 83. x 2 3x 2 4 x 9x 3 5x 2 36x 4 84. x2 4 81. In Exercises 5966, use synthetic division to show that x is a solution of the third degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. 59. 60. 61. 62. 63. 64. 65. 66. x3 7x 6 0, x 2 x3 28x 48 0, x 4 2x3 15x 2 27x 10 0, x 12 48x3 80x 2 41x 6 0, x 23 x3 2x 2 3x 6 0, x 3 x3 2x 2 2x 4 0, x 2 x3 3x 2 2 0, x 1 3 x3 x 2 13x 3 0, x 2 5 85. DATA ANALYSIS: HIGHER EDUCATION The amounts A in billions of dollars donated to support higher education in the United States from 2000 through 2007 are shown in the table, where t represents the year, with t 0 corresponding to 2000. In Exercises 6774, a verify the given factors of the function f, b find the remaining factor s of f, c use your results to write the complete factorization of f, d list all real zeros of f, and e confirm your results by using a graphing utility to graph the function. Function x 2 5x 2 67. f x 3 68. f x 3x 2x 2 19x 6 69. f x x 4 4x3 15x 2 58x 40 2x 3 Factors x 2, x 1 x 3, x 2 x 5, x 4 www.elsolucionario.net Year, t Amount, A 0 1 2 3 4 5 6 7 23.2 24.2 23.9 23.9 24.4 25.6 28.0 29.8 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 292 Chapter 3 Polynomial Functions a Use a graphing utility to create a scatter plot of the data. b Use the regression feature of the graphing utility to find a cubic model for the data. Graph the model in the same viewing window as the scatter plot. c Use the model to create a table of estimated values of A. Compare the model with the original data. d Use synthetic division to evaluate the model for the year 2010. Even though the model is relatively accurate for estimating the given data, would you use this model to predict the amount donated to higher education in the future Explain. 86. DATA ANALYSIS: HEALTH CARE The amounts A in billions of dollars of national health care expenditures in the United States from 2000 through 2007 are shown in the table, where t represents the year, with t 0 corresponding to 2000. Year, t Amount, A 0 1 2 3 4 5 6 7 30.5 32.2 34.2 38.0 42.7 47.9 52.7 57.6 x3 2x 2 13x 10 x 2 4x 12 is improper. 90. Use the form f x x kqx r to create a cubic function that a passes through the point 2, 5 and rises to the right, and b passes through the point 3, 1 and falls to the right. There are many correct answers. THINK ABOUT IT In Exercises 91 and 92, perform the division by assuming that n is a positive integer. 91. x 3n 3x 2n 5x n 6 x 3n 9x 2n 27x n 27 92. n x 3 xn 2 93. WRITING Briefly explain what it means for a divisor to divide evenly into a dividend. 94. WRITING Briefly explain how to check polynomial division, and justify your reasoning. Give an example. EXPLORATION In Exercises 95 and 96, find the constant c such that the denominator will divide evenly into the numerator. 95. x 3 4x 2 3x c x5 96. x 5 2x 2 x c x2 97. THINK ABOUT IT Find the x 4 is a factor of x3 kx2 98. THINK ABOUT IT Find the x 3 is a factor of x3 kx2 a Use a graphing utility to create a scatter plot of the data. b Use the regression feature of the graphing utility to find a cubic model for the data. Graph the model in the same viewing window as the scatter plot. c Use the model to create a table of estimated values of A. Compare the model with the original data. d Use synthetic division to evaluate the model for the year 2010. EXPLORATION 87. If 7x 4 is a factor of some polynomial function f, then 47 is a zero of f. 88. 2x 1 is a factor of the polynomial value of k such that 2kx 8. value of k such that 2kx 12. 99. WRITING Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division xn 1x 1. Create a numerical example to test your formula. a x2 1 x1 c x4 1 x1 100. CAPSTONE TRUE OR FALSE In Exercises 8789, determine whether the statement is true or false. Justify your answer. 6x 6 x 5 92x 4 45x 3 184x 2 4x 48. 89. The rational expression b x3 1 x1 Consider the division f x x k where f x x 3 2x 3x 13. a What is the remainder when k 3 Explain. b If it is necessary to find f 2, it is easier to evaluate the function directly or to use synthetic division Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 Zeros of Polynomial Functions 293 3.4 ZEROS OF POLYNOMIAL FUNCTIONS What you should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find conjugate pairs of complex zeros. Find zeros of polynomials by factoring. Use Descartess Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. Why you should learn it Finding zeros of polynomial functions is an important part of solving real life problems. For instance, in Exercise 120 on page 306, the zeros of a polynomial function can help you analyze the attendance at womens college basketball games. The Fundamental Theorem of Algebra You know that an nth degree polynomial can have at most n real zeros. In the complex number system, this statement can be improved. That is, in the complex number system, every nth degree polynomial function has precisely n zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved by the German mathematician Carl Friedrich Gauss 17771855 . The Fundamental Theorem of Algebra If f x is a polynomial of degree n, where n 0, then f has at least one zero in the complex number system. Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, you obtain the Linear Factorization Theorem. Linear Factorization Theorem If f x is a polynomial of degree n, where n 0, then f has precisely n linear factors f x anx c1x c2 . . . x cn where c1, c2, . . . , cn are complex numbers. Recall that in order to find the zeros of a function f x, set f x equal to 0 and solve the resulting equation for x. For instance, the function in Example 1 a has a zero at x 2 because x20 x 2. For a proof of the Linear Factorization Theorem, see Proofs in Mathematics on page 328. Note that the Fundamental Theorem of Algebra and the Linear Factorization Theorem tell you only that the zeros or factors of a polynomial exist, not how to find them. Such theorems are called existence theorems. Remember that the n zeros of a polynomial function can be real or complex, and they may be repeated. Example 1 Zeros of Polynomial Functions a. The first degree polynomial f x x 2 has exactly one zero: x 2. b. Counting multiplicity, the second degree polynomial function f x x 2 6x 9 x 3x 3 has exactly two zeros: x 3 and x 3. This is called a repeated zero. c. The third degree polynomial function f x x 3 4x xx 2 4 xx 2ix 2i Examples 1 b , 1 c , and 1 d involve factoring polynomials. You can review the techniques for factoring polynomials in Section P.4. has exactly three zeros: x 0, x 2i, and x 2i. d. The fourth degree polynomial function f x x 4 1 x 1x 1x i x i has exactly four zeros: x 1, x 1, x i, and x i. Now try Exercise 9. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 294 Chapter 3 Polynomial Functions The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polynomial having integer coefficients to the leading coefficient and to the constant term of the polynomial. HISTORICAL NOTE The Rational Zero Test Fogg Art MuseumHarvard University If the polynomial f x an x n an1 x n1 . . . a 2 x 2 a1x a0 has integer coefficients, every rational zero of f has the form Rational zero p q where p and q have no common factors other than 1, and p a factor of the constant term a0 Although they were not contemporaries, Jean Le Rond dAlembert 17171783 worked independently of Carl Gauss in trying to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequently known as the Theorem of dAlembert. q a factor of the leading coefficient an. To use the Rational Zero Test, you should first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Possible rational zeros factors of constant term factors of leading coefficient Having formed this list of possible rational zeros, use a trial and error method to determine which, if any, are actual zeros of the polynomial. Note that when the leading coefficient is 1, the possible rational zeros are simply the factors of the constant term. Example 2 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x 3 x 1. Solution f x = x 3 + x + 1 y 3 f 1 13 1 1 2 3 1 3 2 x 1 1 2 3 FIGURE 3.30 Because the leading coefficient is 1, the possible rational zeros are 1, the factors of the constant term. By testing these possible zeros, you can see that neither works. 2 3 f 1 13 1 1 1 So, you can conclude that the given polynomial has no rational zeros. Note from the graph of f in Figure 3.30 that f does have one real zero between 1 and 0. However, by the Rational Zero Test, you know that this real zero is not a rational number. Now try Exercise 15. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 Example 3 When the list of possible rational zeros is small, as in Example 2, it may be quicker to test the zeros by evaluating the function. When the list of possible rational zeros is large, as in Example 3, it may be quicker to use a different approach to test the zeros, such as using synthetic division or sketching a graph. 295 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x 4 x 3 x 2 3x 6. Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Possible rational zeros: 1, 2, 3, 6 By applying synthetic division successively, you can determine that x 1 and x 2 are the only two rational zeros. 1 2 You can review the techniques for synthetic division in Section 3.3. Zeros of Polynomial Functions 1 1 1 1 2 3 3 6 6 1 2 3 6 0 1 2 2 3 0 6 6 1 0 3 0 0 remainder, so x 1 is a zero. 0 remainder, so x 2 is a zero. So, f x factors as f x x 1x 2x 2 3. Because the factor x 2 3 produces no real zeros, you can conclude that x 1 and x 2 are the only real zeros of f, which is verified in Figure 3.31. y 8 6 f x = x 4 x 3 + x 2 3 x 6 1, 0 8 6 4 2 2, 0 x 4 6 8 6 8 FIGURE 3.31 Now try Exercise 19. If the leading coefficient of a polynomial is not 1, the list of possible rational zeros can increase dramatically. In such cases, the search can be shortened in several ways: 1 a programmable calculator can be used to speed up the calculations; 2 a graph, drawn either by hand or with a graphing utility, can give a good estimate of the locations of the zeros; 3 the Intermediate Value Theorem along with a table generated by a graphing utility can give approximations of zeros; and 4 synthetic division can be used to test the possible rational zeros. Finding the first zero is often the most difficult part. After that, the search is simplified by working with the lower degree polynomial obtained in synthetic division, as shown in Example 3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 296 Chapter 3 Polynomial Functions Example 4 Using the Rational Zero Test Find the rational zeros of f x 2x 3 3x 2 8x 3. Solution Remember that when you try to find the rational zeros of a polynomial function with many possible rational zeros, as in Example 4, you must use trial and error. There is no quick algebraic method to determine which of the possibilities is an actual zero; however, sketching a graph may be helpful. The leading coefficient is 2 and the constant term is 3. Possible rational zeros: Factors of 3 1, 3 1 3 1, 3, , Factors of 2 1, 2 2 2 By synthetic division, you can determine that x 1 is a rational zero. 1 2 3 2 8 5 3 3 2 5 3 0 So, f x factors as f x x 12x 2 5x 3 x 12x 1x 3 and you can conclude that the rational zeros of f are x 1, x 12, and x 3. Now try Exercise 25. Recall from Section 3.2 that if x a is a zero of the polynomial function f, then x a is a solution of the polynomial equation f x 0. y 15 10 Example 5 Solving a Polynomial Equation 5 x Find all the real solutions of 10x3 15x2 16x 12 0. 1 5 10 Solution The leading coefficient is 10 and the constant term is 12. Possible rational solutions: f x = 10x 3 + 15x 2 + 16x 12 FIGURE 3.32 Factors of 12 1, 2, 3, 4, 6, 12 Factors of 10 1, 2, 5, 10 With so many possibilities 32, in fact , it is worth your time to stop and sketch a graph. From Figure 3.32, it looks like three reasonable solutions would be x 65, x 12, and x 2. Testing these by synthetic division shows that x 2 is the only rational solution. So, you have x 210x2 5x 6 0. Using the Quadratic Formula for the second factor, you find that the two additional solutions are irrational numbers. x 5 265 1.0639 20 x 5 265 0.5639 20 and You can review the techniques for using the Quadratic Formula in Section 1.4. Now try Exercise 31. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 Zeros of Polynomial Functions 297 Conjugate Pairs In Examples 1 c and 1 d , note that the pairs of complex zeros are conjugates. That is, they are of the form a bi and a bi. Complex Zeros Occur in Conjugate Pairs Let f x be a polynomial function that has real coefficients. If a bi, where b 0, is a zero of the function, the conjugate a bi is also a zero of the function. Be sure you see that this result is true only if the polynomial function has real coefficients. For instance, the result applies to the function given by f x x 2 1 but not to the function given by gx x i. Example 6 Finding a Polynomial with Given Zeros Find a fourth degree polynomial function with real coefficients that has 1, 1, and 3i as zeros. Solution Because 3i is a zero and the polynomial is stated to have real coefficients, you know that the conjugate 3i must also be a zero. So, from the Linear Factorization Theorem, f x can be written as f x ax 1x 1x 3ix 3i. For simplicity, let a 1 to obtain f x x 2 2x 1x 2 9 x 4 2x 3 10x 2 18x 9. Now try Exercise 45. Factoring a Polynomial The Linear Factorization Theorem shows that you can write any nth degree polynomial as the product of n linear factors. f x anx c1x c2x c3 . . . x cn However, this result includes the possibility that some of the values of ci are complex. The following theorem says that even if you do not want to get involved with complex factors, you can still write f x as the product of linear andor quadratic factors. For a proof of this theorem, see Proofs in Mathematics on page 328. Factors of a Polynomial Every polynomial of degree n 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 298 Chapter 3 Polynomial Functions A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure you see that this is not the same as being irreducible over the rationals. For example, the quadratic x 2 1 x i x i is irreducible over the reals and therefore over the rationals . On the other hand, the quadratic x 2 2 x 2 x 2 is irreducible over the rationals but reducible over the reals. Example 7 Finding the Zeros of a Polynomial Function Find all the zeros of f x x 4 3x 3 6x 2 2x 60 given that 1 3i is a zero of f. Algebraic Solution Graphical Solution Because complex zeros occur in conjugate pairs, you know that 1 3i is also a zero of f. This means that both Because complex zeros always occur in conjugate pairs, you know that 1 3i is also a zero of f. Because the polynomial is a fourth degree polynomial, you know that there are two other zeros of the function. Use a graphing utility to graph x 1 3i and x 1 3i are factors of f. Multiplying these two factors produces x 1 3i x 1 3i x 1 3ix 1 3i x 12 9i 2 y x 4 3x3 6x2 2x 60 as shown in Figure 3.33. x 2 2x 10. y = x4 3x3 + 6x2 + 2x 60 Using long division, you can divide x 2 2x 10 into f to obtain the following. x2 x2 2x 10 6x 2 x 4 2x 3 10x 2 x 3 4x 2 x3 2x 2 6x 2 6x 2 x4 3x 3 80 x 6 2x 60 4 2x 10x 12x 60 12x 60 0 5 80 FIGURE 3.33 You can see that 2 and 3 appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the graphing utility to confirm that x 2 and x 3 are zeros of the graph. So, you can conclude that the zeros of f are x 1 3i, x 1 3i, x 3, and x 2. So, you have f x x 2 2x 10x 2 x 6 x 2 2x 10x 3x 2 and you can conclude that the zeros of f are x 1 3i, x 1 3i, x 3, and x 2. Now try Exercise 55. You can review the techniques for polynomial long division in Section 3.3. In Example 7, if you were not told that 1 3i is a zero of f, you could still find all zeros of the function by using synthetic division to find the real zeros 2 and 3. Then you could factor the polynomial as x 2x 3x 2 2x 10. Finally, by using the Quadratic Formula, you could determine that the zeros are x 2, x 3, x 1 3i, and x 1 3i. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 Zeros of Polynomial Functions 299 Example 8 shows how to find all the zeros of a polynomial function, including complex zeros. In Example 8, the fifth degree polynomial function has three real zeros. In such cases, you can use the zoom and trace features or the zero or root feature of a graphing utility to approximate the real zeros. You can then use these real zeros to determine the complex zeros algebraically. Example 8 Finding the Zeros of a Polynomial Function Write f x x 5 x 3 2x 2 12x 8 as the product of linear factors, and list all of its zeros. Solution The possible rational zeros are 1, 2, 4, and 8. Synthetic division produces the following. 1 1 0 1 1 1 2 12 2 4 8 8 1 1 2 4 8 0 2 1 1 1 2 4 8 2 2 8 8 1 4 4 0 1 is a zero. 2 is a zero. So, you have f x x 5 x 3 2x 2 12x 8 x 1x 2x3 x2 4x 4. f x = x5 + x3 + 2x2 12x You can factor x3 x2 4x 4 as x 1x2 4, and by factoring x 2 4 as +8 x 2 4 x 4 x 4 y x 2ix 2i you obtain f x x 1x 1x 2x 2ix 2i 10 which gives the following five zeros of f. x 1, x 1, x 2, x 2i, and 5 2, 0 x 4 FIGURE 1, 0 2 3.34 4 x 2i From the graph of f shown in Figure 3.34, you can see that the real zeros are the only ones that appear as x intercepts. Note that x 1 is a repeated zero. Now try Exercise 77. T E C H N O LO G Y You can use the table feature of a graphing utility to help you determine which of the possible rational zeros are zeros of the polynomial in Example 8. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When you do this, you will see that there are two rational zeros, 2 and 1, as shown at the right. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 300 Chapter 3 Polynomial Functions Other Tests for Zeros of Polynomials You know that an nth degree polynomial function can have at most n real zeros. Of course, many nth degree polynomials do not have that many real zeros. For instance, f x x 2 1 has no real zeros, and f x x 3 1 has only one real zero. The following theorem, called Descartess Rule of Signs, sheds more light on the number of real zeros of a polynomial. Descartess Rule of Signs Let f x an x n an1x n1 . . . a2x2 a1x a0 be a polynomial with real coefficients and a0 0. 1. The number of positive real zeros of f is either equal to the number of variations in sign of f x or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f x or less than that number by an even integer. A variation in sign means that two consecutive coefficients have opposite signs. When using Descartess Rule of Signs, a zero of multiplicity k should be counted as k zeros. For instance, the polynomial x 3 3x 2 has two variations in sign, and so has either two positive or no positive real zeros. Because x3 3x 2 x 1x 1x 2 you can see that the two positive real zeros are x 1 of multiplicity 2. Example 9 Using Descartess Rule of Signs Describe the possible real zeros of f x 3x 3 5x 2 6x 4. Solution The original polynomial has three variations in sign. to f x = 3x 3 5x 2 + 6x 4 to f x 3x3 5x2 6x 4 y to 3 The polynomial 2 f x 3x3 5x2 6x 4 1 3 2 1 x 2 1 2 3 FIGURE 3.35 3x 3 5x 2 6x 4 3 has no variations in sign. So, from Descartess Rule of Signs, the polynomial f x 3x 3 5x 2 6x 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. From the graph in Figure 3.35, you can see that the function has only one real zero, at x 1. Now try Exercise 87. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 Zeros of Polynomial Functions 301 Another test for zeros of a polynomial function is related to the sign pattern in the last row of the synthetic division array. This test can give you an upper or lower bound of the real zeros of f. A real number b is an upper bound for the real zeros of f if no zeros are greater than b. Similarly, b is a lower bound if no real zeros of f are less than b. Upper and Lower Bound Rules Let f x be a polynomial with real coefficients and a positive leading coefficient. Suppose f x is divided by x c, using synthetic division. 1. If c 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c 0 and the numbers in the last row are alternately positive and negative zero entries count as positive or negative , c is a lower bound for the real zeros of f. Example 10 Finding the Zeros of a Polynomial Function Find the real zeros of f x 6x 3 4x 2 3x 2. Solution The possible real zeros are as follows. Factors of 2 1, 2 1 1 1 2 1, , , , , 2 Factors of 6 1, 2, 3, 6 2 3 6 3 The original polynomial f x has three variations in sign. The polynomial f x 6x3 4x2 3x 2 6x3 4x2 3x 2 has no variations in sign. As a result of these two findings, you can apply Descartess Rule of Signs to conclude that there are three positive real zeros or one positive real zero, and no negative zeros. Trying x 1 produces the following. 1 6 4 6 3 2 2 5 6 2 5 3 So, x 1 is not a zero, but because the last row has all positive entries, you know that x 1 is an upper bound for the real zeros. So, you can restrict the search to zeros between 0 and 1. By trial and error, you can determine that x 23 is a zero. So, f x x 2 6x2 3. 3 Because 6x 2 3 has no real zeros, it follows that x 23 is the only real zero. Now try Exercise 95. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 302 Chapter 3 Polynomial Functions Before concluding this section, here are two additional hints that can help you find the real zeros of a polynomial. 1. If the terms of f x have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing f x x 4 5x 3 3x 2 x xx 3 5x 2 3x 1 you can see that x 0 is a zero of f and that the remaining zeros can be obtained by analyzing the cubic factor. 2. If you are able to find all but two zeros of f x, you can always use the Quadratic Formula on the remaining quadratic factor. For instance, if you succeeded in writing f x x 4 5x 3 3x 2 x xx 1x 2 4x 1 you can apply the Quadratic Formula to x 2 4x 1 to conclude that the two remaining zeros are x 2 5 and x 2 5. Example 11 Using a Polynomial Model You are designing candle making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candles square base. What should the dimensions of your candle mold be Solution The volume of a pyramid is V 13 Bh, where B is the area of the base and h is the height. The area of the base is x 2 and the height is x 2. So, the volume of the pyramid is V 13 x 2x 2. Substituting 25 for the volume yields the following. 1 25 x 2x 2 3 Substitute 25 for V. 75 x3 2x 2 Multiply each side by 3. 0 x3 2x 2 75 Write in general form. The possible rational solutions are x 1, 3, 5, 15, 25, 75. Use synthetic division to test some of the possible solutions. Note that in this case, it makes sense to test only positive x values. Using synthetic division, you can determine that x 5 is a solution. 5 1 2 0 75 1 5 3 15 15 75 0 The other two solutions, which satisfy x 2 3x 15 0, are imaginary and can be discarded. You can conclude that the base of the candle mold should be 5 inches by 5 inches and the height of the mold should be 5 2 3 inches. Now try Exercise 115. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 3.4 EXERCISES Zeros of Polynomial Functions 303 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. The ________ ________ of ________ states that if f x is a polynomial of degree n n 0, then f has at least one zero in the complex number system. 2. The ________ ________ ________ states that if f x is a polynomial of degree n n 0, then f has precisely n linear factors, f x anx c1x c2 . . . x cn, where c1, c2, . . . , cn are complex numbers. 3. The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test. 4. If a bi is a complex zero of a polynomial with real coefficients, then so is its ________, a bi. 5. Every polynomial of degree n 0 with real coefficients can be written as the product of ________ and ________ factors with real coefficients, where the ________ factors have no real zeros. 6. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be ________ over the ________. 7. The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________. 8. A real number b is a n ________ bound for the real zeros of f if no real zeros are less than b, and is a n ________ bound if no real zeros are greater than b. SKILLS AND APPLICATIONS In Exercises 914, find all the zeros of the function. 9. 10. 11. 12. 13. 14. 17. f x 2x4 17x 3 35x 2 9x 45 y f x xx 62 f x x 2x 3x 2 1 g x x 2x 43 f x x 5x 82 f x x 6x ix i ht t 3t 2t 3i t 3i x 2 4 6 40 48 In Exercises 15 18, use the Rational Zero Test to list all possible rational zeros of f. Verify that the zeros of f shown on the graph are contained in the list. 18. f x 4x 5 8x4 5x3 10x 2 x 2 y 4 2 15. f x x 3 2x 2 x 2 x 2 y 6 3 6 4 2 x 1 1 4 16. f x x 3 4x 2 4x 16 y 18 9 6 3 1 6 x 1 3 In Exercises 1928, find all the rational zeros of the function. 2 5 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. f x x 3 6x 2 11x 6 f x x 3 7x 6 gx x 3 4x 2 x 4 hx x 3 9x 2 20x 12 ht t 3 8t 2 13t 6 px x 3 9x 2 27x 27 Cx 2x 3 3x 2 1 f x 3x 3 19x 2 33x 9 f x 9x 4 9x 3 58x 2 4x 24 f x 2x4 15x 3 23x 2 15x 25 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 304 Chapter 3 Polynomial Functions In Exercises 2932, find all real solutions of the polynomial equation. 29. 30. 31. 32. z 4 z 3 z2 3z 6 0 x 4 13x 2 12x 0 2y 4 3y 3 16y 2 15y 4 0 x 5 x4 3x 3 5x 2 2x 0 In Exercises 55 62, use the given zero to find all the zeros of the function. In Exercises 3336, a list the possible rational zeros of f, b sketch the graph of f so that some of the possible zeros in part a can be disregarded, and then c determine all real zeros of f. 33. 34. 35. 36. f x x 3 x 2 4x 4 f x 3x 3 20x 2 36x 16 f x 4x 3 15x 2 8x 3 f x 4x 3 12x 2 x 15 In Exercises 37 40, a list the possible rational zeros of f, b use a graphing utility to graph f so that some of the possible zeros in part a can be disregarded, and then c determine all real zeros of f. 37. 38. 39. 40. f x 2x4 13x 3 21x 2 2x 8 f x 4x 4 17x 2 4 f x 32x 3 52x 2 17x 3 f x 4x 3 7x 2 11x 18 GRAPHICAL ANALYSIS In Exercises 41 44, a use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, b determine one of the exact zeros use synthetic division to verify your result , and c factor the polynomial completely. 41. f x x 4 3x 2 2 42. Pt t 4 7t 2 12 5 4 3 43. hx x 7x 10x 14x 2 24x 44. gx 6x 4 11x 3 51x 2 99x 27 In Exercises 4550, find a polynomial function with real coefficients that has the given zeros. There are many correct answers. 45. 1, 5i 47. 2, 5 i 2 49. 3, 1, 3 2 i 46. 4, 3i 48. 5, 3 2i 50. 5, 5, 1 3 i In Exercises 5154, write the polynomial a as the product of factors that are irreducible over the rationals, b as the product of linear and quadratic factors that are irreducible over the reals, and c in completely factored form. 51. f x x 4 6x 2 27 52. f x x 4 2x 3 3x 2 12x 18 Hint: One factor is x 2 6. 53. f x x 4 4x 3 5x 2 2x 6 Hint: One factor is x 2 2x 2. 54. f x x 4 3x 3 x 2 12x 20 Hint: One factor is x 2 4. Function 55. 56. 57. 58. 59. 60. 61. 62. Zero f x x x 4x 4 f x 2x 3 3x 2 18x 27 f x 2x 4 x 3 49x 2 25x 25 g x x 3 7x 2 x 87 g x 4x 3 23x 2 34x 10 2i 3i 5i 5 2i 3 i h x 3x 3 4x 2 8x 8 f x x 4 3x 3 5x 2 21x 22 f x x 3 4x 2 14x 20 1 3i 3 2i 1 3i 3 2 In Exercises 6380, find all the zeros of the function and write the polynomial as a product of linear factors. 64. f x x 2 x 56 f x x 2 36 2 66. gx x2 10x 17 hx x 2x 17 68. f y y 4 256 f x x 4 16 2 f z z 2z 2 h x x 3 3x 2 4x 2 g x x 3 3x 2 x 5 f x x 3 x 2 x 39 h x x 3 x 6 h x x 3 9x 2 27x 35 f x 5x 3 9x 2 28x 6 g x 2x 3 x 2 8x 21 g x x 4 4x 3 8x 2 16x 16 78. h x x 4 6x 3 10x 2 6x 9 79. f x x 4 10x 2 9 80. f x x 4 29x 2 100 63. 65. 67. 69. 70. 71. 72. 73. 74. 75. 76. 77. In Exercises 8186, find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to discard any rational zeros that are obviously not zeros of the function. 81. 82. 83. 84. 85. 86. f x x 3 24x 2 214x 740 f s 2s 3 5s 2 12s 5 f x 16x 3 20x 2 4x 15 f x 9x 3 15x 2 11x 5 f x 2x 4 5x 3 4x 2 5x 2 g x x 5 8x 4 28x 3 56x 2 64x 32 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 In Exercises 8794, use Descartess Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 87. 89. 91. 92. 93. 94. gx 2x 3 3x 2 3 88. hx 4x 2 8x 3 hx 2x3 3x 2 1 90. hx 2x 4 3x 2 5 gx 5x 10x f x 4x 3 3x 2 2x 1 f x 5x 3 x 2 x 5 f x 3x 3 2x 2 x 3 In Exercises 9598, use synthetic division to verify the upper and lower bounds of the real zeros of f. 95. f x x3 3x2 2x 1 a Upper: x 1 b Lower: 96. f x x 3 4x 2 1 a Upper: x 4 b Lower: 97. f x x 4 4x 3 16x 16 a Upper: x 5 b Lower: 98. f x 2x 4 8x 3 a Upper: x 3 b Lower: x 4 Zeros of Polynomial Functions 305 a Let x represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. b Use the diagram to write the volume V of the box as a function of x. Determine the domain of the function. c Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. d Find values of x such that V 56. Which of these values is a physical impossibility in the construction of the box Explain. 112. GEOMETRY A rectangular package to be sent by a delivery service see figure can have a maximum combined length and girth perimeter of a cross section of 120 inches. x 1 x x x 3 x 4 y In Exercises 99102, find all the real zeros of the function. 99. 100. 101. 102. f x 4x 3 3x 1 f z 12z 3 4z 2 27z 9 f y 4y 3 3y 2 8y 6 g x 3x 3 2x 2 15x 10 In Exercises 103106, find all the rational zeros of the polynomial function. 25 1 103. Px x 4 4 x 2 9 44x 4 25x 2 36 3 23 104. f x x 3 2 x 2 2 x 6 122x 3 3x 2 23x 12 1 1 1 105. f x x3 4 x 2 x 4 44x3 x 2 4x 1 11 2 1 1 1 3 106. f z z 6 z 2 z 3 66z3 11z2 3z 2 In Exercises 107110, match the cubic function with the numbers of rational and irrational zeros. a Rational zeros: 0; irrational zeros: 1 b Rational zeros: 3; irrational zeros: 0 c Rational zeros: 1; irrational zeros: 2 d Rational zeros: 1; irrational zeros: 0 107. f x x 3 1 108. f x x 3 2 109. f x x 3 x 110. f x x 3 2x 111. GEOMETRY An open box is to be made from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. a Write a function Vx that represents the volume of the package. b Use a graphing utility to graph the function and approximate the dimensions of the package that will yield a maximum volume. c Find values of x such that V 13,500. Which of these values is a physical impossibility in the construction of the package Explain. 113. ADVERTISING COST A company that produces MP3 players estimates that the profit P in dollars for selling a particular model is given by P 76x 3 4830x 2 320,000, 0 x 60 where x is the advertising expense in tens of thousands of dollars . Using this model, find the smaller of two advertising amounts that will yield a profit of 2,500,000. 114. ADVERTISING COST A company that manufactures bicycles estimates that the profit P in dollars for selling a particular model is given by P 45x 3 2500x 2 275,000, 0 x 50 where x is the advertising expense in tens of thousands of dollars . Using this model, find the smaller of two advertising amounts that will yield a profit of 800,000. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 306 Chapter 3 Polynomial Functions 115. GEOMETRY A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size to hold five times as much food as the current bin. Assume each dimension is increased by the same amount. a Write a function that represents the volume V of the new bin. b Find the dimensions of the new bin. 116. GEOMETRY A manufacturer wants to enlarge an existing manufacturing facility such that the total floor area is 1.5 times that of the current facility. The floor area of the current facility is rectangular and measures 250 feet by 160 feet. The manufacturer wants to increase each dimension by the same amount. a Write a function that represents the new floor area A. b Find the dimensions of the new floor. c Another alternative is to increase the current floors length by an amount that is twice an increase in the floors width. The total floor area is 1.5 times that of the current facility. Repeat parts a and b using these criteria. 117. COST The ordering and transportation cost C in thousands of dollars for the components used in manufacturing a product is given by C 100 x 200 2 x , x 1 x 30 where x is the order size in hundreds . In calculus, it can be shown that the cost is a minimum when 3x 3 40x 2 2400x 36,000 0. Use a calculator to approximate the optimal order size to the nearest hundred units. 118. HEIGHT OF A BASEBALL A baseball is thrown upward from a height of 6 feet with an initial velocity of 48 feet per second, and its height h in feet is ht 16t 2 48t 6, 0 t 3 where t is the time in seconds . You are told the ball reaches a height of 64 feet. Is this possible 119. PROFIT The demand equation for a certain product is p 140 0.0001x, where p is the unit price in dollars of the product and x is the number of units produced and sold. The cost equation for the product is C 80x 150,000, where C is the total cost in dollars and x is the number of units produced. The total profit obtained by producing and selling x units is P R C xp C. You are working in the marketing department of the company that produces this product, and you are asked to determine a price p that will yield a profit of 9 million dollars. Is this possible Explain. 120. ATHLETICS The attendance A in millions at NCAA womens college basketball games for the years 2000 through 2007 is shown in the table. Source: National Collegiate Athletic Association, Indianapolis, IN Year Attendance, A 2000 2001 2002 2003 2004 2005 2006 2007 8.7 8.8 9.5 10.2 10.0 9.9 9.9 10.9 a Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t 0 corresponding to 2000. b Use the regression feature of the graphing utility to find a quartic model for the data. c Graph the model and the scatter plot in the same viewing window. How well does the model fit the data d According to the model in part b , in what year s was the attendance at least 10 million e According to the model, will the attendance continue to increase Explain. EXPLORATION TRUE OR FALSE In Exercises 121 and 122, decide whether the statement is true or false. Justify your answer. 121. It is possible for a third degree polynomial function with integer coefficients to have no real zeros. 122. If x i is a zero of the function given by f x x 3 ix2 ix 1 then x i must also be a zero of f. THINK ABOUT IT In Exercises 123128, determine if possible the zeros of the function g if the function f has zeros at x r1, x r2, and x r3. 123. gx f x 125. gx f x 5 127. gx 3 f x www.elsolucionario.net 124. gx 3f x 126. gx f 2x 128. gx f x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.4 129. THINK ABOUT IT A third degree polynomial function f has real zeros 2, 12, and 3, and its leading coefficient is negative. Write an equation for f. Sketch the graph of f. How many different polynomial functions are possible for f 130. CAPSTONE Use a graphing utility to graph the function given by f x x 4 4x 2 k for different values of k. Find values of k such that the zeros of f satisfy the specified characteristics. Some parts do not have unique answers. a Four real zeros b Two real zeros, each of multiplicity 2 c Two real zeros and two complex zeros d Four complex zeros e Will the answers to parts a through d change for the function g, where gx f x 2 f Will the answers to parts a through d change for the function g, where gx f 2x 131. THINK ABOUT IT Sketch the graph of a fifth degree polynomial function whose leading coefficient is positive and that has a zero at x 3 of multiplicity 2. 132. WRITING Compile a list of all the various techniques for factoring a polynomial that have been covered so far in the text. Give an example illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate. 133. THINK ABOUT IT Let y f x be a quartic polynomial with leading coefficient a 1 and f i f 2i 0. Write an equation for f. 134. THINK ABOUT IT Let y f x be a cubic polynomial with leading coefficient a 1 and f 2 f i 0. Write an equation for f. In Exercises 135 and 136, the graph of a cubic polynomial function y f x is shown. It is known that one of the zeros is 1 i. Write an equation for f. y 135. 136. , 2 Positive 2, 1 Negative 1, 4 Negative 4, Positive a What are the three real zeros of the polynomial function f b What can be said about the behavior of the graph of f at x 1 c What is the least possible degree of f Explain. Can the degree of f ever be odd Explain. d Is the leading coefficient of f positive or negative Explain. e Write an equation for f. There are many correct answers. f Sketch a graph of the equation you wrote in part e . 138. a Find a quadratic function f with integer coefficients that has bi as zeros. Assume that b is a positive integer. b Find a quadratic function f with integer coefficients that has a bi as zeros. Assume that b is a positive integer. 139. GRAPHICAL REASONING The graph of one of the following functions is shown below. Identify the function shown in the graph. Explain why each of the others is not the correct function. Use a graphing utility to verify your result. a f x x 2x 2 x 3.5 b g x x 2 x 3.5 c h x x 2 x 3.5x 2 1 d k x x 1 x 2x 3.5 y x 2 1 2 4 x 2 1 2 3 20 30 40 2 3 Value of f x 10 1 1 1 Interval 1 x 307 137. Use the information in the table to answer each question. y 2 Zeros of Polynomial Functions 3 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 308 Chapter 3 Polynomial Functions 3.5 MATHEMATICAL MODELING AND VARIATION What you should learn Introduction Use mathematical models to approximate sets of data points. Use the regression feature of a graphing utility to find the equation of a least squares regression line. Write mathematical models for direct variation. Write mathematical models for direct variation as an nth power. Write mathematical models for inverse variation. Write mathematical models for joint variation. You have already studied some techniques for fitting models to data. For instance, in Section 2.1, you learned how to find the equation of a line that passes through two points. In this section, you will study other techniques for fitting models to data: least squares regression and direct and inverse variation. The resulting models are either polynomial functions or rational functions. Rational functions will be studied in Chapter 4. Example 1 A Mathematical Model The populations y in millions of the United States from 2000 through 2007 are shown in the table. Source: U.S. Census Bureau Why you should learn it You can use functions as models to represent a wide variety of real life data sets. For instance, in Exercise 83 on page 318, a variation model can be used to model the water temperatures of the ocean at various depths. Year Population, y 2000 2001 2002 2003 2004 2005 2006 2007 282.4 285.3 288.2 290.9 293.6 296.3 299.2 302.0 A linear model that approximates the data is y 2.78t 282.5 for 0 t 7, where t is the year, with t 0 corresponding to 2000. Plot the actual data and the model on the same graph. How closely does the model represent the data Solution The actual data are plotted in Figure 3.36, along with the graph of the linear model. From the graph, it appears that the model is a good fit for the actual data. You can see how well the model fits by comparing the actual values of y with the values of y given by the model. The values given by the model are labeled y* in the table below. U.S. Population Population in millions y t 0 1 2 3 4 5 6 7 300 y 282.4 285.3 288.2 290.9 293.6 296.3 299.2 302.0 295 y* 282.5 285.3 288.1 290.8 293.6 296.4 299.2 302.0 305 290 285 Now try Exercise 11. y = 2.78t + 282.5 280 t 1 2 3 4 5 6 Year 0 2000 FIGURE 3.36 7 Note in Example 1 that you could have chosen any two points to find a line that fits the data. However, the given linear model was found using the regression feature of a graphing utility and is the line that best fits the data. This concept of a best fitting line is discussed on the next page. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 Mathematical Modeling and Variation 309 Least Squares Regression and Graphing Utilities So far in this text, you have worked with many different types of mathematical models that approximate real life data. In some instances the model was given as in Example 1 , whereas in other instances you were asked to find the model using simple algebraic techniques or a graphing utility. To find a model that approximates the data most accurately, statisticians use a measure called the sum of square differences, which is the sum of the squares of the differences between actual data values and model values. The best fitting linear model, called the least squares regression line, is the one with the least sum of square differences. Recall that you can approximate this line visually by plotting the data points and drawing the line that appears to fit bestor you can enter the data points into a calculator or computer and use the linear regression feature of the calculator or computer. When you use the regression feature of a graphing calculator or computer program, you will notice that the program may also output an r value. This r value is the correlation coefficient of the data and gives a measure of how well the model fits the data. The closer the value of r is to 1, the better the fit. Example 2 Debt in trillions of dollars The data in the table show the outstanding household credit market debt D in trillions of dollars from 2000 through 2007. Construct a scatter plot that represents the data and find the least squares regression line for the data. Source: Board of Governors of the Federal Reserve System Household Credit Market Debt D Finding a Least Squares Regression Line 14 13 12 11 10 9 8 7 6 t 1 2 3 4 5 6 7 Year 0 2000 FIGURE 3.37 t D D* 0 1 2 3 4 5 6 7 7.0 7.7 8.5 9.5 10.6 11.8 12.9 13.8 6.7 7.7 8.7 9.7 10.7 11.8 12.8 13.8 Year Household credit market debt, D 2000 2001 2002 2003 2004 2005 2006 2007 7.0 7.7 8.5 9.5 10.6 11.8 12.9 13.8 Solution Let t 0 represent 2000. The scatter plot for the points is shown in Figure 3.37. Using the regression feature of a graphing utility, you can determine that the equation of the least squares regression line is D 1.01t 6.7. To check this model, compare the actual D values with the D values given by the model, which are labeled D* in the table at the left. The correlation coefficient for this model is r 0.997, which implies that the model is a good fit. Now try Exercise 17. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 310 Chapter 3 Polynomial Functions Direct Variation There are two basic types of linear models. The more general model has a y intercept that is nonzero. y mx b, b 0 The simpler model y kx has a y intercept that is zero. In the simpler model, y is said to vary directly as x, or to be directly proportional to x. Direct Variation The following statements are equivalent. 1. y varies directly as x. 2. y is directly proportional to x. 3. y kx for some nonzero constant k. k is the constant of variation or the constant of proportionality. Example 3 Direct Variation In Pennsylvania, the state income tax is directly proportional to gross income. You are working in Pennsylvania and your state income tax deduction is 46.05 for a gross monthly income of 1500. Find a mathematical model that gives the Pennsylvania state income tax in terms of gross income. Solution Pennsylvania Taxes State income tax in dollars State income tax k Labels: State income tax y Gross income x Income tax rate k Equation: y kx 100 y kx y = 0.0307x 80 46.05 k1500 60 0.0307 k 1500, 46.05 40 Gross income dollars dollars percent in decimal form To solve for k, substitute the given information into the equation y kx, and then solve for k. y Write direct variation model. Substitute y 46.05 and x 1500. Simplify. So, the equation or model for state income tax in Pennsylvania is 20 y 0.0307x. x 1000 2000 3000 4000 Gross income in dollars FIGURE Verbal Model: 3.38 In other words, Pennsylvania has a state income tax rate of 3.07 of gross income. The graph of this equation is shown in Figure 3.38. Now try Exercise 43. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 Mathematical Modeling and Variation 311 Direct Variation as an nth Power Another type of direct variation relates one variable to a power of another variable. For example, in the formula for the area of a circle A r2 the area A is directly proportional to the square of the radius r. Note that for this formula, is the constant of proportionality. Direct Variation as an nth Power Note that the direct variation model y kx is a special case of y kx n with n 1. The following statements are equivalent. 1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3. y kx n for some constant k. Example 4 The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. See Figure 3.39. t = 0 sec t = 1 sec 10 FIGURE 20 3.39 30 Direct Variation as nth Power 40 t = 3 sec 50 60 70 a. Write an equation relating the distance traveled to the time. b. How far will the ball roll during the first 3 seconds Solution a. Letting d be the distance in feet the ball rolls and letting t be the time in seconds , you have d kt 2. Now, because d 8 when t 1, you can see that k 8, as follows. d kt 2 8 k12 8k So, the equation relating distance to time is d 8t 2. b. When t 3, the distance traveled is d 83 2 89 72 feet. Now try Exercise 75. In Examples 3 and 4, the direct variations are such that an increase in one variable corresponds to an increase in the other variable. This is also true in the model 1 d 5F, F 0, where an increase in F results in an increase in d. You should not, however, assume that this always occurs with direct variation. For example, in the model y 3x, an increase in x results in a decrease in y, and yet y is said to vary directly as x. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 312 Chapter 3 Polynomial Functions Inverse Variation Inverse Variation The following statements are equivalent. 1. y varies inversely as x. 3. y 2. y is inversely proportional to x. k for some constant k. x If x and y are related by an equation of the form y kx n, then y varies inversely as the nth power of x or y is inversely proportional to the nth power of x . Some applications of variation involve problems with both direct and inverse variation in the same model. These types of models are said to have combined variation. Example 5 P1 P2 V1 V2 P2 P1 then V2 V1 3.40 If the temperature is held constant and pressure increases, volume decreases. FIGURE Direct and Inverse Variation A gas law states that the volume of an enclosed gas varies directly as the temperature and inversely as the pressure, as shown in Figure 3.40. The pressure of a gas is 0.75 kilogram per square centimeter when the temperature is 294 K and the volume is 8000 cubic centimeters. a Write an equation relating pressure, temperature, and volume. b Find the pressure when the temperature is 300 K and the volume is 7000 cubic centimeters. Solution a. Let V be volume in cubic centimeters , let P be pressure in kilograms per square centimeter , and let T be temperature in Kelvin . Because V varies directly as T and inversely as P, you have V kT . P Now, because P 0.75 when T 294 and V 8000, you have 8000 k k294 0.75 6000 1000 . 294 49 So, the equation relating pressure, temperature, and volume is V 1000 T . 49 P b. When T 300 and V 7000, the pressure is P 1000 300 300 0.87 kilogram per square centimeter. 49 7000 343 Now try Exercise 77. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 Mathematical Modeling and Variation 313 Joint Variation In Example 5, note that when a direct variation and an inverse variation occur in the same statement, they are coupled with the word and. To describe two different direct variations in the same statement, the word jointly is used. Joint Variation The following statements are equivalent. 1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. z kxy for some constant k. If x, y, and z are related by an equation of the form z kx ny m then z varies jointly as the nth power of x and the mth power of y. Example 6 Joint Variation The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter 3 months , the interest on a principal of 5000 is 43.75. a. Write an equation relating the interest, principal, and time. b. Find the interest after three quarters. Solution a. Let I interest in dollars , P principal in dollars , and t time in years . Because I is jointly proportional to P and t, you have I kPt. For I 43.75, P 5000, and t 14, you have 43.75 k5000 4 1 which implies that k 443.755000 0.035. So, the equation relating interest, principal, and time is I 0.035Pt which is the familiar equation for simple interest where the constant of proportionality, 0.035, represents an annual interest rate of 3.5. b. When P 5000 and t 34, the interest is I 0.0355000 4 3 131.25. Now try Exercise 79. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 314 Chapter 3 3.5 Polynomial Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. Two techniques for fitting models to data are called direct ________ and least squares ________. 2. Statisticians use a measure called ________ of________ ________ to find a model that approximates a set of data most accurately. 3. The linear model with the least sum of square differences is called the ________ ________ ________ line. 4. An r value of a set of data, also called a ________ ________, gives a measure of how well a model fits a set of data. 5. Direct variation models can be described as y varies directly as x, or y is ________ ________ to x. 6. In direct variation models of the form y kx, k is called the ________ of ________. 7. The direct variation model y kx n can be described as y varies directly as the nth power of x, or y is ________ ________ to the nth power of x. 8. The mathematical model y k is an example of ________ variation. x 9. Mathematical models that involve both direct and inverse variation are said to have ________ variation. 10. The joint variation model z kxy can be described as z varies jointly as x and y, or z is ________ ________ to x and y. SKILLS AND APPLICATIONS 11. EMPLOYMENT The total numbers of people in thousands in the U.S. civilian labor force from 1992 through 2007 are given by the following ordered pairs. 2000, 142,583 1992, 128,105 2001, 143,734 1993, 129,200 2002, 144,863 1994, 131,056 2003, 146,510 1995, 132,304 2004, 147,401 1996, 133,943 2005, 149,320 1997, 136,297 2006, 151,428 1998, 137,673 1999, 139,368 2007, 153,124 A linear model that approximates the data is y 1695.9t 124,320, where y represents the number of employees in thousands and t 2 represents 1992. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data Source: U.S. Bureau of Labor Statistics 12. SPORTS The winning times in minutes in the womens 400 meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. 1996, 4.12 1948, 5.30 1972, 4.32 2000, 4.10 1952, 5.20 1976, 4.16 2004, 4.09 1956, 4.91 1980, 4.15 2008, 4.05 1960, 4.84 1984, 4.12 1988, 4.06 1964, 4.72 1968, 4.53 A linear model that approximates the data is y 0.020t 5.00, where y represents the winning time in minutes and t 0 represents 1950. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data Does it appear that another type of model may be a better fit Explain. Source: International Olympic Committee In Exercises 1316, sketch the line that you think best approximates the data in the scatter plot. Then find an equation of the line. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. y 13. y 14. 5 5 4 4 3 2 3 2 1 1 x 1 2 3 4 y 15. x 5 2 3 4 5 1 2 3 4 5 y 16. 5 5 4 4 3 2 3 2 1 1 1 x 1 2 3 4 5 x 1992, 4.12 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 17. SPORTS The lengths in feet of the winning mens discus throws in the Olympics from 1920 through 2008 are listed below. Source: International Olympic Committee 1920 146.6 1956 184.9 1984 218.5 1924 151.3 1960 194.2 1988 225.8 1928 155.3 1964 200.1 1992 213.7 1932 162.3 1968 212.5 1996 227.7 1936 165.6 1972 211.3 2000 227.3 1948 173.2 1976 221.5 2004 229.3 1952 180.5 1980 218.7 2008 225.8 a Sketch a scatter plot of the data. Let y represent the length of the winning discus throw in feet and let t 20 represent 1920. b Use a straightedge to sketch the best fitting line through the points and find an equation of the line. c Use the regression feature of a graphing utility to find the least squares regression line that fits the data. d Compare the linear model you found in part b with the linear model given by the graphing utility in part c . e Use the models from parts b and c to estimate the winning mens discus throw in the year 2012. 18. SALES The total sales in billions of dollars for CocaCola Enterprises from 2000 through 2007 are listed below. Source: Coca Cola Enterprises, Inc. 2000 14.750 2004 18.185 2001 15.700 2005 18.706 2002 16.899 2006 19.804 2003 17.330 2007 20.936 a Sketch a scatter plot of the data. Let y represent the total revenue in billions of dollars and let t 0 represent 2000. b Use a straightedge to sketch the best fitting line through the points and find an equation of the line. c Use the regression feature of a graphing utility to find the least squares regression line that fits the data. d Compare the linear model you found in part b with the linear model given by the graphing utility in part c . e Use the models from parts b and c to estimate the sales of Coca Cola Enterprises in 2008. f Use your schools library, the Internet, or some other reference source to analyze the accuracy of the estimate in part e . Mathematical Modeling and Variation 315 19. DATA ANALYSIS: BROADWAY SHOWS The table shows the annual gross ticket sales S in millions of dollars for Broadway shows in New York City from 1995 through 2006. Source: The League of American Theatres and Producers, Inc. Year Sales, S 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 406 436 499 558 588 603 666 643 721 771 769 862 a Use a graphing utility to create a scatter plot of the data. Let t 5 represent 1995. b Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. c Use the graphing utility to graph the scatter plot you created in part a and the model you found in part b in the same viewing window. How closely does the model represent the data d Use the model to estimate the annual gross ticket sales in 2007 and 2009. e Interpret the meaning of the slope of the linear model in the context of the problem. 20. DATA ANALYSIS: TELEVISION SETS The table shows the numbers N in millions of television sets in U.S. households from 2000 through 2006. Source: Television Bureau of Advertising, Inc. www.elsolucionario.net Year Television sets, N 2000 2001 2002 2003 2004 2005 2006 245 248 254 260 268 287 301 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 316 Chapter 3 Polynomial Functions a Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. Let t 0 represent 2000. b Use the graphing utility to create a scatter plot of the data. Then graph the model you found in part a and the scatter plot in the same viewing window. How closely does the model represent the data c Use the model to estimate the number of television sets in U.S. households in 2008. d Use your schools library, the Internet, or some other reference source to analyze the accuracy of the estimate in part c . THINK ABOUT IT In Exercises 21 and 22, use the graph to determine whether y varies directly as some power of x or inversely as some power of x. Explain. y 21. y 22. 8 4 6 2 x x 4 2 4 6 8 In Exercises 2326, use the given value of k to complete the table for the direct variation model y kx 2. Plot the points on a rectangular coordinate system. 2 x 4 6 8 10 y kx2 23. k 1 1 25. k 2 24. k 2 1 26. k 4 In Exercises 2730, use the given value of k to complete the table for the inverse variation model y k . x2 Plot the points on a rectangular coordinate system. 2 x y 27. k 2 29. k 10 4 6 31. 32. 33. 34. x 5 10 15 20 25 y 1 1 2 1 3 1 4 1 5 x 5 10 15 20 25 y 2 4 6 8 10 x 5 10 15 20 25 y 3.5 7 10.5 14 17.5 x 5 10 15 20 25 y 24 12 8 6 24 5 DIRECT VARIATION In Exercises 3538, assume that y is directly proportional to x. Use the given x value and y value to find a linear model that relates y and x. 4 2 2 In Exercises 3134, determine whether the variation model is of the form y kx or y kx, and find k. Then write a model that relates y and x. 8 k x2 28. k 5 30. k 20 10 35. x 5, y 12 37. x 10, y 2050 36. x 2, y 14 38. x 6, y 580 39. SIMPLE INTEREST The simple interest on an investment is directly proportional to the amount of the investment. By investing 3250 in a certain bond issue, you obtained an interest payment of 113.75 after 1 year. Find a mathematical model that gives the interest I for this bond issue after 1 year in terms of the amount invested P. 40. SIMPLE INTEREST The simple interest on an investment is directly proportional to the amount of the investment. By investing 6500 in a municipal bond, you obtained an interest payment of 211.25 after 1 year. Find a mathematical model that gives the interest I for this municipal bond after 1 year in terms of the amount invested P. 41. MEASUREMENT On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters y to inches x. Then use the model to find the numbers of centimeters in 10 inches and 20 inches. 42. MEASUREMENT When buying gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Use this information to find a linear model that relates liters y to gallons x. Then use the model to find the numbers of liters in 5 gallons and 25 gallons. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 43. TAXES Property tax is based on the assessed value of a property. A house that has an assessed value of 150,000 has a property tax of 5520. Find a mathematical model that gives the amount of property tax y in terms of the assessed value x of the property. Use the model to find the property tax on a house that has an assessed value of 225,000. 44. TAXES State sales tax is based on retail price. An item that sells for 189.99 has a sales tax of 11.40. Find a mathematical model that gives the amount of sales tax y in terms of the retail price x. Use the model to find the sales tax on a 639.99 purchase. HOOKES LAW In Exercises 4548, use Hookes Law for springs, which states that the distance a spring is stretched or compressed varies directly as the force on the spring. 45. A force of 265 newtons stretches a spring 0.15 meter see figure . 8 ft FIGURE FOR 48 In Exercises 4958, find a mathematical model for the verbal statement. 49. 50. 51. 52. 53. 54. 55. Equilibrium 0.15 meter 56. 265 newtons a How far will a force of 90 newtons stretch the spring b What force is required to stretch the spring 0.1 meter 46. A force of 220 newtons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter 47. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25 pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy 48. An overhead garage door has two springs, one on each side of the door see figure . A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed. 317 Mathematical Modeling and Variation 57. 58. A varies directly as the square of r. V varies directly as the cube of e. y varies inversely as the square of x. h varies inversely as the square root of s. F varies directly as g and inversely as r 2. z is jointly proportional to the square of x and the cube of y. BOYLES LAW: For a constant temperature, the pressure P of a gas is inversely proportional to the volume V of the gas. NEWTONS LAW OF COOLING: The rate of change R of the temperature of an object is proportional to the difference between the temperature T of the object and the temperature Te of the environment in which the object is placed. NEWTONS LAW OF UNIVERSAL GRAVITATION: The gravitational attraction F between two objects of masses m1 and m2 is proportional to the product of the masses and inversely proportional to the square of the distance r between the objects. LOGISTIC GROWTH: The rate of growth R of a population is jointly proportional to the size S of the population and the difference between S and the maximum population size L that the environment can support. In Exercises 59 66, write a sentence using the variation terminology of this section to describe the formula. 59. Area of a triangle: A 12bh 60. Area of a rectangle: A lw 61. Area of an equilateral triangle: A 3s 24 62. Surface area of a sphere: S 4 r 2 63. Volume of a sphere: V 43 r 3 64. Volume of a right circular cylinder: V r 2h 65. Average speed: r dt 66. Free vibrations: kgW www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 318 Chapter 3 Polynomial Functions In Exercises 6774, find a mathematical model representing the statement. In each case, determine the constant of proportionality. 67. 68. 69. 70. 71. 72. 73. 74. A varies directly as r 2. A 9 when r 3. y varies inversely as x. y 3 when x 25. y is inversely proportional to x. y 7 when x 4. z varies jointly as x and y. z 64 when x 4 and y 8. F is jointly proportional to r and the third power of s. F 4158 when r 11 and s 3. P varies directly as x and inversely as the square of y. P 283 when x 42 and y 9. z varies directly as the square of x and inversely as y. z 6 when x 6 and y 4. v varies jointly as p and q and inversely as the square of s. v 1.5 when p 4.1, q 6.3, and s 1.2. ECOLOGY In Exercises 75 and 76, use the fact that the diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. 1 75. A stream with a velocity of 4 mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter. 76. A stream of velocity v can move particles of diameter d or less. By what factor does d increase when the velocity is doubled 80. MUSIC The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 440 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.2 times as long. 81. FLUID FLOW The velocity v of a fluid flowing in a conduit is inversely proportional to the cross sectional area of the conduit. Assume that the volume of the flow per unit of time is held constant. Determine the change in the velocity of water flowing from a hose when a person places a finger over the end of the hose to decrease its cross sectional area by 25. 82. BEAM LOAD The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth, and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. a The width and length of the beam are doubled. b The width and depth of the beam are doubled. c All three of the dimensions are doubled. d The depth of the beam is halved. 83. DATA ANALYSIS: OCEAN TEMPERATURES An oceanographer took readings of the water temperatures C in degrees Celsius at several depths d in meters . The data collected are shown in the table. Depth, d Temperature, C 1000 2000 3000 4000 5000 4.2 1.9 1.4 1.2 0.9 RESISTANCE In Exercises 77 and 78, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross sectional area. 77. If 28 copper wire which has a diameter of 0.0126 inch has a resistance of 66.17 ohms per thousand feet, what length of 28 copper wire will produce a resistance of 33.5 ohms 78. A 14 foot piece of copper wire produces a resistance of 0.05 ohm. Use the constant of proportionality from Exercise 77 to find the diameter of the wire. 79. WORK The work W in joules done when lifting an object varies jointly with the mass m in kilograms of the object and the height h in meters that the object is lifted. The work done when a 120 kilogram object is lifted 1.8 meters is 2116.8 joules. How much work is done when lifting a 100 kilogram object 1.5 meters a Sketch a scatter plot of the data. b Does it appear that the data can be modeled by the inverse variation model C kd If so, find k for each pair of coordinates. c Determine the mean value of k from part b to find the inverse variation model C kd. d Use a graphing utility to plot the data points and the inverse model from part c . e Use the model to approximate the depth at which the water temperature is 3C. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 3.5 84. DATA ANALYSIS: PHYSICS EXPERIMENT An experiment in a physics lab requires a student to measure the compressed lengths y in centimeters of a spring when various forces of F pounds are applied. The data are shown in the table. Force, F Length, y 0 2 4 6 8 10 12 0 1.15 2.3 3.45 4.6 5.75 6.9 89. Discuss how well the data shown in each scatter plot can be approximated by a linear model. y a 5 5 4 4 3 2 3 2 1 1 x x 1 2 3 4 5 y c a Sketch a scatter plot of the data. b Does it appear that the data can be modeled by Hookes Law If so, estimate k. See Exercises 45 48. c Use the model in part b to approximate the force required to compress the spring 9 centimeters. 85. DATA ANALYSIS: LIGHT INTENSITY A light probe is located x centimeters from a light source, and the intensity y in microwatts per square centimeter of the light is measured. The results are shown as ordered pairs x, y. 34, 0.1543 46, 0.0775 y b 38, 0.1172 50, 0.0645 A model for the data is y 262.76x 2.12. a Use a graphing utility to plot the data points and the model in the same viewing window. b Use the model to approximate the light intensity 25 centimeters from the light source. 86. ILLUMINATION The illumination from a light source varies inversely as the square of the distance from the light source. When the distance from a light source is doubled, how does the illumination change Discuss this model in terms of the data given in Exercise 85. Give a possible explanation of the difference. EXPLORATION TRUE OR FALSE In Exercises 87 and 88, decide whether the statement is true or false. Justify your answer. 87. In the equation for kinetic energy, E 12 mv 2, the amount of kinetic energy E is directly proportional to the mass m of an object and the square of its velocity v. 88. If the correlation coefficient for a least squares regression line is close to 1, the regression line cannot be used to describe the data. 1 2 3 4 5 1 2 3 4 5 y d 5 5 4 4 3 2 3 2 1 30, 0.1881 42, 0.0998 319 Mathematical Modeling and Variation 1 x 1 2 3 4 5 x 90. WRITING A linear model for predicting prize winnings at a race is based on data for 3 years. Write a paragraph discussing the potential accuracy or inaccuracy of such a model. 91. WRITING Suppose the constant of proportionality is positive and y varies directly as x. When one of the variables increases, how will the other change Explain your reasoning. 92. WRITING Suppose the constant of proportionality is positive and y varies inversely as x. When one of the variables increases, how will the other change Explain your reasoning. 93. WRITING a Given that y varies inversely as the square of x and x is doubled, how will y change Explain. b Given that y varies directly as the square of x and x is doubled, how will y change Explain. 94. CAPSTONE The prices of three sizes of pizza at a pizza shop are as follows. 9 inch: 8.78, 12 inch: 11.78, 15 inch: 14.18 You would expect that the price of a certain size of pizza would be directly proportional to its surface area. Is that the case for this pizza shop If not, which size of pizza is the best buy PROJECT: FRAUD AND IDENTITY THEFT To work an extended application analyzing the numbers of fraud complaints and identity theft victims in the United States in 2007, visit this texts website at academic.cengage.com. Data Source: U.S. Census Bureau www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 320 Chapter 3 Polynomial Functions Section 3.3 Section 3.2 Section 3.1 3 CHAPTER SUMMARY What Did You Learn ExplanationExamples Analyze graphs of quadratic functions p. 260 . Let a, b, and c be real numbers with a 0. The function given by f x ax2 bx c is called a quadratic function. Its graph is a U shaped curve called a parabola. All parabolas are symmetric with respect to a line called the axis of symmetry. The point where the axis of symmetry intersects the parabola is the vertex. 1, 2 Write quadratic functions in standard form and use the results to sketch graphs of functions p. 263 . The quadratic function f x ax h2 k, a 0, is in standard form. The graph of f is a parabola whose axis is the vertical line x h and whose vertex is h, k. If a 0, the parabola opens upward. If a 0, the parabola opens downward. 320 Find minimum and maximum values of quadratic functions in real life applications p. 265 . b b ,f 2a 2a If a 0, then f has a minimum when x b2a. If a 0, then f has a maximum when x b2a. . 2126 Use transformations to sketch graphs of polynomial functions p. 270 . The graph of a polynomial function is continuous no breaks, holes, or gaps and has only smooth, rounded turns. 2732 Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions p. 272 . Consider the graph of f x an x n . . . a1x a0. When n is odd: If an 0, the graph falls to the left and rises to the right. If an 0, the graph rises to the left and falls to the right. When n is even: If an 0, the graph rises to the left and right. If an 0, the graph falls to the left and right. 33 36 Find and use zeros of polynomial functions as sketching aids p. 273 . If f is a polynomial function and a is a real number, the following are equivalent: 1 x a is a zero of f, 2 x a is a solution of the equation f x 0, 3 x a is a factor of f x, and 4 a, 0 is an x intercept of the graph of f. 37 46 Use the Intermediate Value Theorem to help locate zeros of polynomial functions p. 277 . Let a and b be real numbers such that a b. If f is a polynomial function such that f a f b, then, in a, b, f takes on every value between f a and f b. 47 50 Use long division to divide polynomials by other polynomials p. 284 . Dividend 5156 Use synthetic division to divide polynomials by binomials of the form x k p. 287 . Divisor: x 3 Consider f x ax2 bx c with vertex Divisor Quotient Remainder x2 3x 5 3 x2 x1 x1 3 Divisor Dividend: x 4 10x2 2x 4 1 1 Quotient: Use the Remainder Theorem and the Factor Theorem p. 288 . Review Exercises 0 3 10 9 2 3 4 3 1 1 1 3 x3 3x2 57 60 Remainder: 1 x1 The Remainder Theorem: If a polynomial f x is divided by x k, the remainder is r f k. The Factor Theorem: A polynomial f x has a factor x k if and only if f k 0. www.elsolucionario.net 61 68 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Summary What Did You Learn ExplanationExamples Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions p. 293 . The Fundamental Theorem of Algebra If f x is a polynomial of degree n, where n 0, then f has at least one zero in the complex number system. 321 Review Exercises 6974 Section 3.5 Section 3.4 Linear Factorization Theorem If f x is a polynomial of degree n, where n 0, then f has precisely n linear factors f x anx c1x c2 . . . x cn where c1, c2, . . ., cn are complex numbers. Find rational zeros of polynomial functions p. 294 . The Rational Zero Test relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term of the polynomial. 7582 Find conjugate pairs of complex zeros p. 297 . Complex Zeros Occur in Conjugate Pairs Let f x be a polynomial function that has real coefficients. If a bi b 0 is a zero of the function, the conjugate a bi is also a zero of the function. 83, 84 Find zeros of polynomials by factoring p. 297 . Every polynomial of degree n 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. 8596 Use Descartess Rule of Signs p. 300 and the Upper and Lower Bound Rules p. 301 to find zeros of polynomials. Descartess Rule of Signs Let f x an x n an1x n1 . . . a2 x2 a1x a0 be a polynomial with real coefficients and a0 0. 1. The number of positive real zeros of f is either equal to the number of variations in sign of f x or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f x or less than that number by an even integer. 97100 Use mathematical models to approximate sets of data points p. 308 . To see how well a model fits a set of data, compare the actual values and model values of y. see Example 1. 101 Use the regression feature of a graphing utility to find the equation of a least squares regression line p. 309 . The sum of square differences is the sum of the squares of the differences between actual data values and model values. The least squares regression line is the linear model with the least sum of square differences. The regression feature of a graphing utility can be used to find the least squares regression line. The correlation coefficient r value of the data gives a measure of how well the model fits the data. The closer the value of r is to 1, the better the fit. 102 Write mathematical models for direct variation p. 310 , direct variation as an nth power p. 311 , inverse variation p. 312 , and joint variation p. 313 . Direct variation: y kx for some nonzero constant k Direct variation as an nth power: y kx n for some constant k Inverse variation: y kx for some constant k Joint variation: z kxy for some constant k www.elsolucionario.net 103108 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 322 Chapter 3 Polynomial Functions 3 REVIEW EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. 3.1 In Exercises 1 and 2, graph each function. Compare the graph of each function with the graph of y x 2. 1. a b c d 2. a b c d f x 2x 2 gx 2x 2 hx x 2 2 kx x 22 f x x 2 4 gx 4 x 2 hx x 32 1 kx 2x 2 1 y 5 1 x 1 In Exercises 1520, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. y 2 y 16. 4, 1 2 2, 1 8 0, 3 2 4 6 17. 18. 19. 20. 2 Vertex: 1, 4; point: 2, 3 Vertex: 2, 3; point: 1, 6 Vertex: 32, 0; point: 92, 11 4 Vertex: 3, 3; point: 14, 45 2 3 4 5 6 7 8 a Write the area A of the rectangle as a function of x. b Determine the domain of the function in the context of the problem. c Create a table showing possible values of x and the corresponding area of the rectangle. d Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum area. e Write the area function in standard form to find analytically the dimensions that will produce the maximum area. 22. GEOMETRY The perimeter of a rectangle is 200 meters. a Draw a diagram that gives a visual representation of the problem. Label the length and width as x and y, respectively. b Write y as a function of x. Use the result to write the area as a function of x. c Of all possible rectangles with perimeters of 200 meters, find the dimensions of the one with the maximum area. 23. MAXIMUM REVENUE The total revenue R earned in dollars from producing a gift box of candles is given by R p 10p2 800p 6 x 4 x, y 2 gx x 2 2x f x 6x x 2 f x x 2 8x 10 hx 3 4x x 2 f t 2t 2 4t 1 f x x 2 8x 12 hx 4x 2 4x 13 f x x 2 6x 1 hx x 2 5x 4 f x 4x 2 4x 5 f x 13x 2 5x 4 f x 126x 2 24x 22 15. x + 2y 8 = 0 3 In Exercises 314, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and x intercept s . 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 21. NUMERICAL, GRAPHICAL, AND ANALYTICAL ANALYSIS A rectangle is inscribed in the region bounded by the x axis, the y axis, and the graph of x 2y 8 0, as shown in the figure. 2, 2 x 2 4 6 where p is the price per unit in dollars . a Find the revenues when the prices per box are 20, 25, and 30. b Find the unit price that will yield a maximum revenue. What is the maximum revenue Explain your results. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Review Exercises 24. MAXIMUM PROFIT A real estate office handles an apartment building that has 50 units. When the rent is 540 per month, all units are occupied. However, for each 30 increase in rent, one unit becomes vacant. Each occupied unit requires an average of 18 per month for service and repairs. What rent should be charged to obtain the maximum profit 25. MINIMUM COST A soft drink manufacturer has daily production costs of C 70,000 120x 0.055x 2 where C is the total cost in dollars and x is the number of units produced. How many units should be produced each day to yield a minimum cost 26. SOCIOLOGY The average age of the groom at a first marriage for a given age of the bride can be approximated by the model y 0.107x2 5.68x 48.5, 20 x 25 where y is the age of the groom and x is the age of the bride. Sketch a graph of the model. For what age of the bride is the average age of the groom 26 Source: U.S. Census Bureau 3.2 In Exercises 2732, sketch the graphs of y x n and the transformation. 27. 28. 29. 30. 31. 32. f x x 23 f x 4x 3 f x 6 x 4 f x 2x 84 f x x 55 f x 12x5 3 y x3, y x3, y x 4, y x 4, y x 5, y x 5, In Exercises 43 46, sketch the graph of the function by a applying the Leading Coefficient Test, b finding the zeros of the polynomial, c plotting sufficient solution points, and d drawing a continuous curve through the points. 43. 44. 45. 46. 47. 48. 49. 50. 51. 53. 54. f x 2x 2 5x 12 f x 12 x 3 2x 3 f x 3x 2 20x 32 f t t 3 3t f x x 3 8x 2 f x 18x 3 12x 2 gx x 4 x 3 12x 2 3x 3 x 2 3 0.25x 3 3.65x 6.12 x 4 5x 1 7x 4 3x 3 8x 2 2 56. 30x 2 3x 8 5x 3 4x 7 3x 2 5x 3 21x 2 25x 4 x 2 5x 1 4 3x x2 1 x 4 3x 3 4x 2 6x 3 x2 2 6x 4 10x 3 13x 2 5x 2 2x 2 1 In Exercises 57 60, use synthetic division to divide. gx 4x 4 3x 2 2 hx x7 8x 2 8x 6x 4 4x 3 27x 2 18x x2 0.1x 3 0.3x 2 0.5 58. x5 3 2x 25x 2 66x 48 59. x8 57. In Exercises 37 42, find all the real zeros of the polynomial function. Determine the multiplicity of each zero and the number of turning points of the graph of the function. Use a graphing utility to verify your answers. 37. 39. 40. 41. 42. f x f x f x f x 3.3 In Exercises 5156, use long division to divide. 55. 33. 34. 35. 36. f x x3 x2 2 gx 2x3 4x2 f x xx3 x2 5x 3 hx 3x2 x 4 In Exercises 4750, a use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. b Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. 52. In Exercises 3336, describe the right hand and left hand behavior of the graph of the polynomial function. 323 38. f x xx 32 60. 5x3 33x 2 50x 8 x4 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 324 Chapter 3 Polynomial Functions In Exercises 61 and 62, use synthetic division to determine whether the given values of x are zeros of the function. 61. f x 20x 4 9x 3 14x 2 3x a x 1 b x 34 c x 0 62. f x 3x 3 8x 2 20x 16 a x 4 b x 4 c x 23 d x 1 d x 1 In Exercises 63 and 64, use the Remainder Theorem and synthetic division to find each function value. In Exercises 65 68, a verify the given factor s of the function f, b find the remaining factors of f, c use your results to write the complete factorization of f, d list all real zeros of f, and e confirm your results by using a graphing utility to graph the function. 65. 66. 67. 68. Function f x 4x 2 25x 28 f x 2x 3 11x 2 21x 90 f x x 4 4x 3 7x 2 22x 24 f x x 4 11x 3 41x 2 61x 30 Factor s x 4 x 6 x 2x 3 x 2x 5 3.4 In Exercises 6974, find all the zeros of the function. 69. 70. 71. 72. 73. 74. f x 4xx 32 f x x 4x 92 f x x 2 11x 18 f x x 3 10x f x x 4x 6x 2ix 2i f x x 8x 52x 3 ix 3 i 75. f x 4x 3 8x 2 3x 15 76. f x 3x4 4x 3 5x 2 8 x3 3x 2 28x 60 4x 3 27x 2 11x 42 x 3 10x 2 17x 8 x 3 9x 2 24x 20 x 4 x 3 11x 2 x 12 25x 4 25x 3 154x 2 4x 24 Function Zero f x x4 h x x 3 2x 2 16x 32 g x 2x 4 3x 3 13x 2 37x 15 f x 4x 4 11x 3 14x2 6x x3 4x 2 i 4i 2i 1i In Exercises 8992, find all the zeros of the function and write the polynomial as a product of linear factors. 89. 90. 91. 92. f x x3 4x2 5x gx x3 7x2 36 gx x 4 4x3 3x2 40x 208 f x x 4 8x3 8x2 72x 153 In Exercises 9396, use a graphing utility to a graph the function, b determine the number of real zeros of the function, and c approximate the real zeros of the function to the nearest hundredth. 93. 94. 95. 96. f x x 4 2x 1 gx x 3 3x 2 3x 2 h x x 3 6x 2 12x 10 f x x 5 2x 3 3x 20 97. gx 5x 3 3x 2 6x 9 98. hx 2x 5 4x 3 2x 2 5 In Exercises 99 and 100, use synthetic division to verify the upper and lower bounds of the real zeros of f. In Exercises 7782, find all the rational zeros of the function. f x f x f x f x f x f x In Exercises 8588, use the given zero to find all the zeros of the function. In Exercises 97 and 98, use Descartess Rule of Signs to determine the possible numbers of positive and negative zeros of the function. In Exercises 75 and 76, use the Rational Zero Test to list all possible rational zeros of f. 77. 78. 79. 80. 81. 82. 83. 23, 4, 3i 84. 2, 3, 1 2i 85. 86. 87. 88. 63. f x x 4 10x 3 24x 2 20x 44 a f 3 b f 1 64. gt 2t 5 5t 4 8t 20 a g4 b g2 x3 In Exercises 83 and 84, find a polynomial function with real coefficients that has the given zeros. There are many correct answers. 99. f x 4x3 3x2 4x 3 a Upper: x 1 1 b Lower: x 4 100. f x 2x3 5x2 14x 8 a Upper: x 8 b Lower: x 4 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Review Exercises 3.5 101. COMPACT DISCS The values V in billions of dollars of shipments of compact discs in the United States from 2000 through 2007 are shown in the table. A linear model that approximates these data is V 0.742t 13.62 103. where t represents the year, with t 0 corresponding to 2000. Source: Recording Industry Association of America Year Value, V 104. 2000 2001 2002 2003 2004 2005 2006 2007 13.21 12.91 12.04 11.23 11.45 10.52 9.37 7.45 105. 106. a Plot the actual data and the model on the same set of coordinate axes. b How closely does the model represent the data 102. DATA ANALYSIS: TV USAGE The table shows the projected numbers of hours H of television usage in the United States from 2003 through 2011. Source: Communications Industry Forecast and Report Year Hours, H 2003 2004 2005 2006 2007 2008 2009 2010 2011 1615 1620 1659 1673 1686 1704 1714 1728 1742 107. 108. 325 c Use the model to estimate the projected number of hours of television usage in 2020. d Interpret the meaning of the slope of the linear model in the context of the problem. MEASUREMENT You notice a billboard indicating that it is 2.5 miles or 4 kilometers to the next restaurant of a national fast food chain. Use this information to find a mathematical model that relates miles to kilometers. Then use the model to find the numbers of kilometers in 2 miles and 10 miles. ENERGY The power P produced by a wind turbine is proportional to the cube of the wind speed S. A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find the output for a wind speed of 40 miles per hour. FRICTIONAL FORCE The frictional force F between the tires and the road required to keep a car on a curved section of a highway is directly proportional to the square of the speed s of the car. If the speed of the car is doubled, the force will change by what factor DEMAND A company has found that the daily demand x for its boxes of chocolates is inversely proportional to the price p. When the price is 5, the demand is 800 boxes. Approximate the demand when the price is increased to 6. TRAVEL TIME The travel time between two cities is inversely proportional to the average speed. A train travels between the cities in 3 hours at an average speed of 65 miles per hour. How long would it take to travel between the cities at an average speed of 80 miles per hour COST The cost of constructing a wooden box with a square base varies jointly as the height of the box and the square of the width of the box. A box of height 16 inches and of width 6 inches costs 28.80. How much would a box of height 14 inches and of width 8 inches cost EXPLORATION TRUE OR FALSE In Exercises 109 and 110, determine whether the statement is true or false. Justify your answer. a Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t 3 corresponding to 2003. b Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data. Then graph the model and the scatter plot you found in part a in the same viewing window. How closely does the model represent the data 109. A fourth degree polynomial with real coefficients can have 5, 8i, 4i, and 5 as its zeros. 110. If y is directly proportional to x, then x is directly proportional to y. 111. WRITING Explain how to determine the maximum or minimum value of a quadratic function. 112. WRITING Explain the connections between factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial equation. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 326 Chapter 3 Polynomial Functions 3 CHAPTER TEST See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Describe how the graph of g differs from the graph of f x x 2. 2 a gx 2 x 2 b gx x 32 2. Identify the vertex and intercepts of the graph of y x 2 4x 3. 3. Find an equation of the parabola shown in the figure at the left. 1 2 4. The path of a ball is given by y 20 x 3x 5, where y is the height in feet of the ball and x is the horizontal distance in feet from where the ball was thrown. a Find the maximum height of the ball. b Which number determines the height at which the ball was thrown Does changing this value change the coordinates of the maximum height of the ball Explain. 5. Determine the right hand and left hand behavior of the graph of the function h t 34t 5 2t 2. Then sketch its graph. 6. Divide using long division. 7. Divide using synthetic division. y 6 4 2 0, 3 x 4 2 2 4 6 8 4 6 3, 6 FIGURE FOR 3 3x 3 4x 1 x2 1 2x 4 5x 2 3 x2 8. Use synthetic division to show that x 3 is a zero of the function given by f x 2x 3 5x 2 6x 15. Use the result to factor the polynomial function completely and list all the real zeros of the function. In Exercises 9 and 10, find all the rational zeros of the function. 9. gt 2t 4 3t 3 16t 24 10. hx 3x 5 2x 4 3x 2 In Exercises 11 and 12, find a polynomial function with real coefficients that has the given zeros. There are many correct answers. 11. 0, 3, 2 i 12. 1 3 i, 2, 2 In Exercises 13 and 14, find all the zeros of the function. 13. f x 3x3 14x2 7x 10 14. f x x 4 9x2 22x 24 In Exercises 1517, find a mathematical model that represents the statement. In each case, determine the constant of proportionality. Year, t Salaries, S 4 5 6 7 8 1550 2150 2500 2750 3175 15. v varies directly as the square root of s. v 24 when s 16. 16. A varies jointly as x and y. A 500 when x 15 and y 8. 17. b varies inversely as a. b 32 when a 1.5. 18. The table at the left shows the median salaries S in thousands of dollars for baseball players on the Chicago Cubs from 2004 through 2008, where t 4 represents 2004. Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. How well does the model represent the data Source: USA Today www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS These two pages contain proofs of four important theorems about polynomial functions. The first two theorems are from Section 3.3, and the second two theorems are from Section 3.4. The Remainder Theorem p. 288 If a polynomial f x is divided by x k, the remainder is r f k. Proof From the Division Algorithm, you have f x x kqx r x and because either r x 0 or the degree of r x is less than the degree of x k, you know that r x must be a constant. That is, r x r. Now, by evaluating f x at x k, you have f k k kqk r 0qk r r. To be successful in algebra, it is important that you understand the connection among factors of a polynomial, zeros of a polynomial function, and solutions or roots of a polynomial equation. The Factor Theorem is the basis for this connection. The Factor Theorem p. 288 A polynomial f x has a factor x k if and only if f k 0. Proof Using the Division Algorithm with the factor x k, you have f x x kqx r x. By the Remainder Theorem, r x r f k, and you have f x x kqx f k where qx is a polynomial of lesser degree than f x. If f k 0, then f x x kqx and you see that x k is a factor of f x. Conversely, if x k is a factor of f x, division of f x by x k yields a remainder of 0. So, by the Remainder Theorem, you have f k 0. 327 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS Linear Factorization Theorem p. 293 If f x is a polynomial of degree n, where n 0, then f has precisely n linear factors f x anx c1x c2 . . . x cn The Fundamental Theorem of Algebra The Linear Factorization Theorem is closely related to the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra has a long and interesting history. In the early work with polynomial equations, The Fundamental Theorem of Algebra was thought to have been not true, because imaginary solutions were not considered. In fact, in the very early work by mathematicians such as Abu al Khwarizmi c. 800 A.D. , negative solutions were also not considered. Once imaginary numbers were accepted, several mathematicians attempted to give a general proof of the Fundamental Theorem of Algebra. These included Gottfried von Leibniz 1702 , Jean dAlembert 1746 , Leonhard Euler 1749 , JosephLouis Lagrange 1772 , and Pierre Simon Laplace 1795 . The mathematician usually credited with the first correct proof of the Fundamental Theorem of Algebra is Carl Friedrich Gauss, who published the proof in his doctoral thesis in 1799. where c1, c2, . . . , cn are complex numbers. Proof Using the Fundamental Theorem of Algebra, you know that f must have at least one zero, c1. Consequently, x c1 is a factor of f x, and you have f x x c1f1x. If the degree of f1x is greater than zero, you again apply the Fundamental Theorem to conclude that f1 must have a zero c2, which implies that f x x c1x c2f2x. It is clear that the degree of f1x is n 1, that the degree of f2x is n 2, and that you can repeatedly apply the Fundamental Theorem n times until you obtain f x anx c1x c2 . . . x cn where an is the leading coefficient of the polynomial f x. Factors of a Polynomial p. 297 Every polynomial of degree n 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. Proof To begin, you use the Linear Factorization Theorem to conclude that f x can be completely factored in the form f x d x c1x c2x c3 . . . x cn. If each ci is real, there is nothing more to prove. If any ci is complex ci a bi, b 0, then, because the coefficients of f x are real, you know that the conjugate cj a bi is also a zero. By multiplying the corresponding factors, you obtain x cix cj x a bix a bi x2 2ax a2 b2 where each coefficient is real. 328 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROBLEM SOLVING This collection of thought provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. a Find the zeros of each quadratic function gx. i gx x2 4x 12 ii gx x2 5x iii gx x2 3x 10 iv gx x2 4x 4 v gx x2 2x 6 vi gx x2 3x 4 b For each function in part a , use a graphing utility to graph f x x 2 gx. Verify that 2, 0 is an x intercept of the graph of f x. Describe any similarities or differences in the behavior of the six functions at this x intercept. c For each function in part b , use the graph of f x to approximate the other x intercepts of the graph. d Describe the connections that you find among the results of parts a , b , and c . 2. Quonset huts were developed during World War II. They were temporary housing structures that could be assembled quickly and easily. A Quonset hut is shaped like a half cylinder. A manufacturer has 600 square feet of material with which to build a Quonset hut. a The formula for the surface area of half a cylinder is S r2 rl, where r is the radius and l is the length of the hut. Solve this equation for l when S 600. 1 b The formula for the volume of the hut is V 2 r2l. Write the volume V of the Quonset hut as a polynomial function of r. c Use the function you wrote in part b to find the maximum volume of a Quonset hut with a surface area of 600 square feet. What are the dimensions of the hut 3. Show that if f x ax3 bx2 cx d then f k r, where r ak3 bk2 ck d using long division. In other words, verify the Remainder Theorem for a thirddegree polynomial function. 4. In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of y3 y2. To be able to use this table, the Babylonians sometimes had to manipulate the equation as shown below. ax3 bx2 c a3 x3 b3 a2 x2 b2 axb axb 3 2 a2 a2 c b3 c b3 Then they would find a2cb3 in the y3 y2 column of the table. Because they knew that the corresponding y value was equal to axb, they could conclude that x bya. a Calculate y3 y2 for y 1, 2, 3, . . . , 10. Record the values in a table. Use the table from part a and the method above to solve each equation. b x3 x2 252 c x3 2x2 288 d 3x3 x2 90 e 2x3 5x2 2500 f 7x3 6x2 1728 g 10x3 3x2 297 Using the methods from this chapter, verify your solution to each equation. 5. At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold 6. a Complete the table. Function Zeros Sum of zeros Product of zeros f1x x2 5x 6 f2x x3 7x 6 f3x x 4 2x3 x2 8x 12 f4x x5 3x4 9x3 25x2 6x b Use the table to make a conjecture relating sum of the zeros of a polynomial function to coefficients of the polynomial function. c Use the table to make a conjecture relating product of the zeros of a polynomial function to coefficients of the polynomial function. the the the the Original equation Multiply each side by a2 . b3 Rewrite. 329 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 7. Determine whether the statement is true or false. If false, provide one or more reasons why the statement is false and correct the statement. Let f x ax3 bx2 cx d, a 0 and let f 2 1. Then f x 2 qx x1 x1 where qx is a second degree polynomial. 8. The parabola shown in the figure has an equation of the form y ax2 bx c. Find the equation of this parabola by the following methods. a Find the equation analytically. b Use the regression feature of a graphing utility to find the equation. d Find the slope mh of the line joining 2, 4 and 2 h, f 2 h in terms of the nonzero number h. e Evaluate the slope formula from part d for h 1, 1, and 0.1. Compare these values with those in parts a c . f What can you conclude the slope mtan of the tangent line at 2, 4 to be Explain your answer. 10. A rancher plans to fence a rectangular pasture adjacent to a river see figure . The rancher has 100 meters of fencing, and no fencing is needed along the river. y y 2 4 2 4 6 2, 2 4, 0 1, 0 6 x x 8 0, 4 6, 10 9. One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point 2, 4 on the graph of the quadratic function f x x2, which is shown in the figure. y 5 4 y 2, 4 3 2 a Write the area A of the pasture as a function of x, the length of the side parallel to the river. What is the domain of Ax b Graph the function Ax and estimate the dimensions that yield the maximum area of the pasture. c Find the exact dimensions that yield the maximum area of the pasture by writing the quadratic function in standard form. 11. A wire 100 centimeters in length is cut into two pieces. One piece is bent to form a square and the other to form a circle. Let x equal the length of the wire used to form the square. 1 3 2 1 x 1 2 3 a Find the slope m1 of the line joining 2, 4 and 3, 9. Is the slope of the tangent line at 2, 4 greater than or less than the slope of the line through 2, 4 and 3, 9 b Find the slope m2 of the line joining 2, 4 and 1, 1. Is the slope of the tangent line at 2, 4 greater than or less than the slope of the line through 2, 4 and 1, 1 c Find the slope m3 of the line joining 2, 4 and 2.1, 4.41. Is the slope of the tangent line at 2, 4 greater than or less than the slope of the line through 2, 4 and 2.1, 4.41 a Write the function that represents the combined area of the two figures. b Determine the domain of the function. c Find the value s of x that yield a maximum area and a minimum area. d Explain your reasoning. 330 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Rational Functions and Conics 4.1 Rational Functions and Asymptotes 4.2 Graphs of Rational Functions 4.3 Conics 4.4 Translations of Conics 4 In Mathematics Functions defined by rational expressions are called rational functions. Conics are collections of points satisfying certain geometric properties. Rational functions and conics are used to model real life situations, such as the population growth of a deer herd, the concentration of a chemical in the bloodstream, or the path of a projectile. For instance, you can use a conic to model the path of a satellite as it escapes Earths gravity. See Exercise 42, page 368. Erik Simonsen Photographer s ChoiceGetty Images In Real Life IN CAREERS There are many careers that use rational functions and conics. Several are listed below. Game Commissioner Exercise 44, page 339 Aeronautical Engineer Exercise 95, page 360 Bridge Designer Exercise 45, page 359 Radio Navigator Exercise 96, page 361 331 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 332 Chapter 4 Rational Functions and Conics 4.1 RATIONAL FUNCTIONS AND ASYMPTOTES What you should learn Find the domains of rational functions. Find the vertical and horizontal asymptotes of graphs of rational functions. Use rational functions to model and solve real life problems. Why you should learn it ZQFotography,2009 Used under license from Shutterstock.com Rational functions can be used to model and solve real life problems relating to environmental scenarios. For instance, in Exercise 42 on page 338, a rational function shows how to determine the cost of supplying recycling bins in a pilot project. Introduction A rational function is a quotient of polynomial functions. It can be written in the form f x N x D x where Nx and Dx are polynomials and Dx is not the zero polynomial. In general, the domain of a rational function of x includes all real numbers except x values that make the denominator zero. Much of the discussion of rational functions will focus on their graphical behavior near the x values excluded from the domain. Example 1 Finding the Domain of a Rational Function Find the domain of f x 1 and discuss the behavior of f near any excluded x values. x Solution Because the denominator is zero when x 0, the domain of f is all real numbers except x 0. To determine the behavior of f near this excluded value, evaluate f x to the left and right of x 0, as indicated in the following tables. x 1 0.5 0.1 0.01 0.001 0 f x 1 2 10 100 1000 x 0 0.001 0.01 0.1 0.5 1 f x 1000 100 10 2 1 Note that as x approaches 0 from the left, f x decreases without bound. In contrast, as x approaches 0 from the right, f x increases without bound. The graph of f is shown in Figure 4.1. y f x = 1x 2 1 Note that the rational function given by f x x 1 2 1 1 x is also referred to as the reciprocal function discussed in Section 2.4. 1 FIGURE 4.1 Now try Exercise 5. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.1 333 Rational Functions and Asymptotes Vertical and Horizontal Asymptotes In Example 1, the behavior of f near x 0 is denoted as follows. y 2 f x f x = 1x 2 Vertical asymptote: x=0 1 as x f x decreases without bound as x approaches 0 from the left. x 1 1 as x 0 f x increases without bound as x approaches 0 from the right. The line x 0 is a vertical asymptote of the graph of f, as shown in Figure 4.2. From this figure, you can see that the graph of f also has a horizontal asymptotethe line y 0. This means that the values of f x 1x approach zero as x increases or decreases without bound. 2 Horizontal asymptote: y=0 1 f x FIGURE f x 0 f x 0 as x 0 as x 4.2 f x approaches 0 as x decreases without bound. f x approaches 0 as x increases without bound. Definitions of Vertical and Horizontal Asymptotes 1. The line x a is a vertical asymptote of the graph of f if f x as x or f x a, either from the right or from the left. 2. The line y b is a horizontal asymptote of the graph of f if f x b or x as x . Eventually as x , the distance between the horizontal or x asymptote and the points on the graph must approach zero. Figure 4.3 shows the vertical and horizontal asymptotes of the graphs of three rational functions. y f x = 2x + 1 x+1 3 Vertical asymptote: x = 1 2 a FIGURE y f x = 4 3 y 1 Horizontal asymptote: y=2 f x = 4 x2 + 1 4 Horizontal asymptote: y=0 3 2 2 1 1 x 2 1 1 x 1 b 2 Vertical asymptote: x=1 Horizontal asymptote: y=0 3 2 1 2 x 1 2 x 1 2 3 c 4.3 The graphs of f x 1x in Figure 4.2 and f x 2x 1x 1 in Figure 4.3 a are hyperbolas. You will study hyperbolas in Sections 4.3 and 4.4. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 334 Chapter 4 Rational Functions and Conics Vertical and Horizontal Asymptotes of a Rational Function Let f be the rational function given by f x an x n an1x n1 . . . a1x a 0 Nx Dx bm x m bm1x m1 . . . b1x b0 where Nx and Dx have no common factors. 1. The graph of f has vertical asymptotes at the zeros of Dx. 2. The graph of f has one or no horizontal asymptote determined by comparing the degrees of Nx and Dx. a. If n m, the graph of f has the line y 0 the x axis as a horizontal asymptote. b. If n m, the graph of f has the line y anbm ratio of the leading coefficients as a horizontal asymptote. c. If n m, the graph of f has no horizontal asymptote. Example 2 Finding Vertical and Horizontal Asymptotes Find all vertical and horizontal asymptotes of the graph of each rational function. y f x = 1 a. f x 2x 3x 2 + 1 2x 3x2 1 b. f x 2x2 x2 1 Solution x 1 1 Horizontal asymptote: y=0 1 FIGURE 4.4 2 f x = 2x x2 1 y 4 x2 1 0 3 2 Horizontal asymptote: y = 2 1 4 3 2 1 Vertical asymptote: x = 1 FIGURE 4.5 a. For this rational function, the degree of the numerator is less than the degree of the denominator, so the graph has the line y 0 as a horizontal asymptote. To find any vertical asymptotes, set the denominator equal to zero and solve the resulting equation for x. Because the equation 3x2 1 0 has no real solutions, you can conclude that the graph has no vertical asymptote. The graph of the function is shown in Figure 4.4. b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 1, so the graph has the line y 2 as a horizontal asymptote. To find any vertical asymptotes, set the denominator equal to zero and solve the resulting equation for x. x 1 2 3 4 Vertical asymptote: x=1 Set denominator equal to zero. x 1x 1 0 Factor. x10 x 1 Set 1st factor equal to 0. x10 x1 Set 2nd factor equal to 0. This equation has two real solutions, x 1 and x 1, so the graph has the lines x 1 and x 1 as vertical asymptotes. The graph of the function is shown in Figure 4.5. Now try Exercise 13. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.1 Example 3 Rational Functions and Asymptotes 335 Finding Vertical and Horizontal Asymptotes Find all vertical and horizontal asymptotes of the graph of f x x2 x 2 . x2 x 6 Solution For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of both the numerator and denominator is 1, so the graph has the line y 1 as a horizontal asymptote. To find any vertical asymptotes, first factor the numerator and denominator as follows. f x x2 x 2 x 1x 2 x 1 , x2 x 6 x 2x 3 x 3 x 2 By setting the denominator x 3 of the simplified function equal to zero, you can determine that the graph has the line x 3 as a vertical asymptote. Now try Exercise 29. Applications There are many examples of asymptotic behavior in real life. For instance, Example 4 shows how a vertical asymptote can be used to analyze the cost of removing pollutants from smokestack emissions. Example 4 Cost Benefit Model A utility company burns coal to generate electricity. The cost C in dollars of removing p of the smokestack pollutants is given by C 80,000p100 p for 0 p 100. Sketch the graph of this function. You are a member of a state legislature considering a law that would require utility companies to remove 90 of the pollutants from their smokestack emissions. The current law requires 85 removal. How much additional cost would the utility company incur as a result of the new law Solution Cost in thousands of dollars C The graph of this function is shown in Figure 4.6. Note that the graph has a vertical asymptote at p 100. Because the current law requires 85 removal, the current cost to the utility company is Smokestack Emissions 1000 800 C 90 600 80,000 p C= 100 p C 200 p 20 40 60 80 100 Percent of pollutants removed FIGURE 4.6 Evaluate C when p 85. If the new law increases the percent removal to 90, the cost will be 85 400 80,00085 453,333. 100 85 80,00090 720,000. 100 90 Evaluate C when p 90. So, the new law would require the utility company to spend an additional 720,000 453,333 266,667. Subtract 85 removal cost from 90 removal cost. Now try Exercise 41. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 336 Chapter 4 Exposure time in hours T Rational Functions and Conics Ultraviolet Radiation Example 5 8 Ultraviolet Radiation For a person with sensitive skin, the amount of time T in hours the person can be exposed to the sun with minimal burning can be modeled by 7 6 5 T= 4 0.37s + 23.8 s T 0.37s 23.8 , s 0 s 120 3 where s is the Sunsor Scale reading. The Sunsor Scale is based on the level of intensity of UVB rays. Source: Sunsor, Inc. T = 0.37 2 1 s 20 40 60 80 100 120 Sunsor Scale reading FIGURE 4.7 a. Find the amounts of time a person with sensitive skin can be exposed to the sun with minimal burning when s 10, s 25, and s 100. b. If the model were valid for all s 0, what would be the horizontal asymptote of this function, and what would it represent Solution a. When s 10, T 0.3710 23.8 10 2.75 hours. When s 25, T 0.3725 23.8 25 1.32 hours. When s 100, T 0.37100 23.8 100 0.61 hour. b. As shown in Figure 4.7, the horizontal asymptote is the line T 0.37. This line represents the shortest possible exposure time with minimal burning. Now try Exercise 43. CLASSROOM DISCUSSION Asymptotes of Graphs of Rational Functions Do you think it is possible for the graph of a rational function to cross its horizontal asymptote If so, how can you determine when the graph of a rational function will cross its horizontal asymptote Use the graphs of the following functions to investigate these questions. Write a summary of your conclusions. Explain your reasoning. x x2 1 x b. gx 2 x 3 x2 c. hx 3 2x x a. f x www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.1 4.1 EXERCISES 337 Rational Functions and Asymptotes See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. Functions of the form f x NxDx, where Nx and Dx are polynomials and Dx is not the zero polynomial, are called ________ ________. 2. If f x as x a from the left or the right, then x a is a ________ ________ of the graph of f. 3. If f x b as x , then y b is a ________ ________ of the graph of f. 4. The graph of f x 1x is called a ________. SKILLS AND APPLICATIONS In Exercises 58, a find the domain of the function, b complete each table, and c discuss the behavior of f near any excluded x values. x 0.5 0.9 0.99 In Exercises 1720, match the rational function with its graph. The graphs are labeled a , b , c , and d . y a 0.999 4 2 f x 2 x 2 x 1.5 1.1 1.01 8 6 6 4 2 1.5 1.1 1.01 1.001 y c x 2 4 4 y d 4 4 f x x 4 1.001 f x x y b 4 2 2 0.5 0.9 0.99 0.999 2 f x 1 x1 3x 2 7. f x 2 x 1 5. f x 1 x 12 4x 8. f x 2 x 1 6. f x In Exercises 916, find the domain of the function and identify any vertical and horizontal asymptotes. 4 9. f x 2 x 11. f x 13. f x 14. f x 15. f x 16. f x x 5x 5x x3 2 x 1 2x 2 x1 3x 2 1 2 x x9 3x 2 x 5 x2 1 1 10. f x x 23 3 7x 12. f x 3 2x 6 4 6 4 2 x 2 4 4 x5 x1 19. f x x4 17. f x 5 x2 x2 20. f x x4 18. f x In Exercises 2128, find the zeros if any of the rational function. 21. gx x2 1 x1 23. hx 2 5 x2 2 24. f x 1 3 x2 4 25. f x 1 3 x3 26. gx 4 2 x5 27. gx x3 8 x2 1 28. f x x3 1 x2 6 www.elsolucionario.net 22. f x x2 2 x3 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 338 Chapter 4 Rational Functions and Conics In Exercises 2936, find the domain of the function and identify any vertical and horizontal asymptotes. 29. f x x4 x2 16 31. f x 1 x2 2x 3 x2 x2 3x 4 33. f x 2 2x x 1 35. f x 30. f x x3 x2 9 32. f x 4 x2 3x 2 C x2 x2 x 2 34. f x 2 2x 5x 2 6x2 5x 6 3x2 8x 4 36. f x 6x2 11x 3 6x2 7x 3 ANALYTICAL AND NUMERICAL ANALYSIS In Exercises 37 40, a determine the domains of f and g, b simplify f and find any vertical asymptotes of f, c complete the table, and d explain how the two functions differ. x2 4 , 37. f x x2 x 4 2.5 2 1.5 1 f x gx 38. f x x x 2x 3 , x 2 3x 3 2 gx x 1 0 1 2 3 f x gx 39. f x x 2x 1 1 , gx 2x 2 x x 1 0.5 0 0.5 2 3 0 255p , 0 p 100. 100 p a Use a graphing utility to graph the cost function. b Find the costs of removing 10, 40, and 75 of the pollutants. c According to this model, would it be possible to remove 100 of the pollutants Explain. 42. RECYCLING In a pilot project, a rural township is given recycling bins for separating and storing recyclable products. The cost C in dollars of supplying bins to p of the population is given by C gx x 2 3 41. POLLUTION The cost C in millions of dollars of removing p of the industrial and municipal pollutants discharged into a river is given by 25,000p , 0 p 100. 100 p a Use a graphing utility to graph the cost function. b Find the costs of supplying bins to 15, 50, and 90 of the population. c According to this model, would it be possible to supply bins to 100 of the residents Explain. 43. DATA ANALYSIS: PHYSICS EXPERIMENT Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge see figure on next page . Known masses M ranging from 200 grams to 2000 grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time t in seconds of one oscillation for each mass is recorded in the table. 4 f x gx 40. f x x x2 0 2x 8 , 9x 20 1 2 3 gx 4 5 2 x5 6 f x gx www.elsolucionario.net Mass, M Time, t 200 400 600 800 1000 1200 1400 1600 1800 2000 0.450 0.597 0.721 0.831 0.906 1.003 1.008 1.168 1.218 1.338 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.1 339 Rational Functions and Asymptotes where P is the fraction of correct responses after n trials. a Complete the table for this model. What does it suggest 50 cm M n 1 2 3 4 5 6 7 8 9 10 P A model for the data that can be used to predict the time of one oscillation is t 38M 16,965 . 10M 5000 EXPLORATION a Use this model to create a table showing the predicted time for each of the masses shown in the table. b Compare the predicted times with the experimental times. What can you conclude c Use the model to approximate the mass of an object for which t 1.056 seconds. 44. POPULATION GROWTH The game commission introduces 100 deer into newly acquired state game lands. The population N of the herd is modeled by N 205 3t , t 0 1 0.04t 1.568x 0.001 , x 0 6.360x 1 where x is the quantity in milligrams of food supplied and y is the quantity in milligrams of food consumed. a Use a graphing utility to graph this model. b At what level of consumption will the moth become satiated 46. HUMAN MEMORY MODEL Psychologists have developed mathematical models to predict memory performance as a function of the number of trials n of a certain task. Consider the learning curve P TRUE OR FALSE In Exercises 47 and 48, determine whether the statement is true or false. Justify your answer. 47. A polynomial function can have infinitely many vertical asymptotes. 48. f x x 3 2x 2 5x 6 is a rational function. In Exercises 4952, a determine the value that the function f approaches as the magnitude of x increases. Is f x greater than or less than this functional value when b x is positive and large in magnitude and c x is negative and large in magnitude 1 x 2x 1 51. f x x3 49. f x 4 where t is the time in years. a Use a graphing utility to graph this model. b Find the populations when t 5, t 10, and t 25. c What is the limiting size of the herd as time increases 45. FOOD CONSUMPTION A biology class performs an experiment comparing the quantity of food consumed by a certain kind of moth with the quantity supplied. The model for the experimental data is given by y b According to this model, what is the limiting percent of correct responses as n increases 1 x3 2x 1 52. f x 2 x 1 50. f x 2 THINK ABOUT IT In Exercises 53 and 54, write a rational function f that has the specified characteristics. There are many correct answers. 53. Vertical asymptote: None Horizontal asymptote: y 2 54. Vertical asymptotes: x 2, x 1 Horizontal asymptote: None 55. THINK ABOUT IT Give an example of a rational function whose domain is the set of all real numbers. Give an example of a rational function whose domain is the set of all real numbers except x 15. Given a polynomial px, is it true that px the graph of the function given by f x 2 has x 4 a vertical asymptote at x 2 Why or why not 56. CAPSTONE 0.5 0.9n 1 , n 0 1 0.9n 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 340 Chapter 4 Rational Functions and Conics 4.2 GRAPHS OF RATIONAL FUNCTIONS What you should learn Analyze and sketch graphs of rational functions. Sketch graphs of rational functions that have slant asymptotes. Use graphs of rational functions to model and solve real life problems. Why you should learn it You can use rational functions to model average speed over a distance. For instance, see Exercise 85 on page 348. Analyzing Graphs of Rational Functions To sketch the graph of a rational function, use the following guidelines. Guidelines for Analyzing Graphs of Rational Functions Let f x NxDx, where Nx and Dx are polynomials. 1. Simplify f, if possible. 2. Find and plot the y intercept if any by evaluating f 0. 3. Find the zeros of the numerator if any by solving the equation Nx 0. Then plot the corresponding x intercepts. 4. Find the zeros of the denominator if any by solving the equation Dx 0. Then sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote if any by using the rule for finding the horizontal asymptote of a rational function. 6. Plot at least one point between and one point beyond each x intercept and vertical asymptote. Mike PowellGetty Images 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes. You may also want to test for symmetry when graphing rational functions, especially for simple rational functions. Recall from Section 2.4 that the graph of f x 1x is symmetric with respect to the origin. T E C H N O LO G Y Some graphing utilities have difficulty graphing rational functions that have vertical asymptotes. Often, the utility will connect parts of the graph that are not supposed to be connected. For instance, the screen on the left below shows the graph of f x 1x 2. Notice that the graph should consist of two unconnected portionsone to the left of x 2 and the other to the right of x 2. To eliminate this problem, you can try changing the mode of the graphing utility to dot mode. The problem with this is that the graph is then represented as a collection of dots as shown in the screen on the right rather than as a smooth curve. 5 5 5 5 5 www.elsolucionario.net 5 5 5 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.2 y g x = 3 x2 Horizontal 4 asymptote: y=0 Example 1 x 2 2 Vertical asymptote: x=2 4 FIGURE 6 4 341 Sketching the Graph of a Rational Function Sketch the graph of gx 2 Graphs of Rational Functions Solution 3 and state its domain. x2 y intercept: 0, 32 , because g0 32 x intercept: None, because 3 Vertical asymptote: x 2, zero of denominator 0 Horizontal asymptote: y 0, because degree of Nx degree of Dx 4.8 Additional points: x gx 4 1 2 3 5 0.5 3 Undefined 3 1 By plotting the intercepts, asymptotes, and a few additional points, you can obtain the graph shown in Figure 4.8. The domain of g is all real numbers except x 2. Now try Exercise 15. Note in the examples in this section that the vertical asymptotes are included in the table of additional points. This is done to emphasize numerically the behavior of the graph of the function. The graph of g in Example 1 is a vertical stretch and a right shift of the graph of f x 1x, because gx 3 x2 3 x 2 1 3f x 2. Example 2 Sketching the Graph of a Rational Function Sketch the graph of f x 2x 1 and state its domain. x Solution y 3 Horizontal asymptote: y=2 x 1 FIGURE 4.9 x intercept: 12, 0, because f 12 0 Vertical asymptote: x 0, zero of denominator Additional points: 1 Vertical asymptote: 2 x=0 None, because x 0 is not in the domain Horizontal asymptote: y 2, because degree of Nx degree of Dx 2 4 3 2 1 y intercept: 1 2 3 x 4 1 0 1 4 4 f x 2.25 3 Undefined 2 1.75 4 f x = 2x x 1 By plotting the intercepts, asymptotes, and a few additional points, you can obtain the graph shown in Figure 4.9. The domain of f is all real numbers except x 0. Now try Exercise 19. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 342 Chapter 4 Rational Functions and Conics Example 3 Sketching the Graph of a Rational Function Sketch the graph of f x Solution Vertical Vertical asymptote: asymptote: x = 1 y x=2 Factor the denominator to determine more easily the zeros of the denominator. f x 3 Horizontal asymptote: y=0 x x x 2 x 2 x 1x 2 2 y intercept: 0, 0, because f 0 0 1 x intercept: 0, 0, because f 0 0 Vertical asymptotes: x 1, x 2, zeros of denominator x 1 2 3 1 Horizontal asymptote: y 0, because degree of Nx degree of Dx 2 Additional points: f x = 3 1 0.5 1 2 3 0.3 Undefined 0.4 0.5 Undefined 0.75 x 3 FIGURE x . x2 x 2 x2 f x x x2 The graph is shown in Figure 4.10. 4.10 Now try Exercise 31. Example 4 Sketching the Graph of a Rational Function Sketch the graph of f x x2 x2 9 . 2x 3 Solution By factoring the numerator and denominator, you have f x y f x = Horizontal asymptote: y=1 4 3 x 3. y intercept: 0, 3, because f 0 3 x intercept: 3, 0, because f 3 0 Vertical asymptote: x 1, zero of simplified denominator Horizontal asymptote: y 1, because degree of Nx degree of Dx 3 2 1 1 2 3 4 5 FIGURE x2 9 x 2 2x 3 x2 9 x 3x 3 x 3 , x 2 2x 3 x 3x 1 x 1 Additional points: x 1 2 3 4 5 6 Vertical asymptote: x = 1 4.11 Hole at x 3 x 5 2 1 0.5 1 3 4 f x 0.5 1 Undefined 5 2 Undefined 1.4 The graph is shown in Figure 4.11. Notice that there is a hole in the graph at x 3 because the function is not defined when x 3. Now try Exercise 39. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.2 Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant or oblique asymptote. For example, the graph of Vertical asymptote: x = 1 8 6 4 2 2 4 x 2 4 6 f x 8 Slant asymptote: y=x2 x2 x x1 has a slant asymptote, as shown in Figure 4.12. To find the equation of a slant asymptote, use long division. For instance, by dividing x 1 into x 2 x, you obtain f x FIGURE 343 Slant Asymptotes 2 f x = x x x+1 y Graphs of Rational Functions x2 x 2 x2 . x1 x1 Slant asymptote y x 2 4.12 As x increases or decreases without bound, the remainder term 2x 1 approaches 0, so the graph of f approaches the line y x 2, as shown in Figure 4.12. Example 5 A Rational Function with a Slant Asymptote Sketch the graph of f x x2 x 2 . x1 Solution First write f x in two different ways. Factoring the numerator f x x 2 x 2 x 2x 1 x1 x1 allows you to recognize the x intercepts. Long division f x Slant asymptote: y=x y x2 x 2 2 x x1 x1 allows you to recognize that the line y x is a slant asymptote of the graph. y intercept: 0, 2, because f 0 2 4 x intercepts: 1, 0 and 2, 0, because f 1 0 and f 2 0 3 Vertical asymptote: x 1, zero of denominator 5 2 Slant asymptote: x 3 2 1 3 4 5 Vertical asymptote: x=1 FIGURE x f x 2 3 Additional points: yx 2 f x = x x 2 x1 2 0.5 1 1.5 3 1.33 4.5 Undefined 2.5 2 The graph is shown in Figure 4.13. Now try Exercise 61. 4.13 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 344 Chapter 4 Rational Functions and Conics Application Example 6 Finding a Minimum Area 1 12 A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are 112 inches wide. What should the dimensions of the page be so that the least amount of paper is used 1 in. x in. y 1 12 in. 1 in. FIGURE 4.14 Graphical Solution Numerical Solution Let A be the area to be minimized. From Figure 4.14, you can write Let A be the area to be minimized. From Figure 4.14, you can write A x 3 y 2. The printed area inside the margins is modeled by 48 xy or y 48x. To find the minimum area, rewrite the equation for A in terms of just one variable by substituting 48x for y. A x 3 x 48 2 A x 3 y 2. The printed area inside the margins is modeled by 48 xy or y 48x. To find the minimum area, rewrite the equation for A in terms of just one variable by substituting 48x for y. A x 3 48x 2 x 3x48 2x, x 0 Use the table feature of a graphing utility to create a table of values for the function x 348 2x , x 0 x y1 The graph of this rational function is shown in Figure 4.15. Because x represents the width of the printed area, you need consider only the portion of the graph for which x is positive. Using a graphing utility, you can approximate the minimum value of A to occur when x 8.5 inches. The corresponding value of y is 488.5 5.6 inches. So, the dimensions should be x 3 11.5 inches by y 2 7.6 inches. x 348 2x x beginning at x 1. From the table, you can see that the minimum value of y1 occurs when x is somewhere between 8 and 9, as shown in Figure 4.16. To approximate the minimum value of y1 to one decimal place, change the table so that it starts at x 8 and increases by 0.1. The minimum value of y1 occurs when x 8.5, as shown in Figure 4.17. The corresponding value of y is 488.5 5.6 inches. So, the dimensions should be x 3 11.5 inches by y 2 7.6 inches. 200 A= x + 3 48 + 2x ,x0 x 0 24 FIGURE 4.16 FIGURE 4.17 0 FIGURE 4.15 Now try Exercise 79. If you go on to take a course in calculus, you will learn an analytic technique for finding the exact value of x that produces a minimum area. In this case, that value is x 62 8.485. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.2 4.2 EXERCISES Graphs of Rational Functions 345 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. For the rational function given by f x NxDx, if the degree of Nx is exactly one more than the degree of Dx, then the graph of f has a ________ or oblique ________. 2. The graph of g x 3x 2 has a ________ asymptote at x 2. SKILLS AND APPLICATIONS In Exercises 3 6, use the graph of f x 2x to sketch the graph of g. y f x = 4 2 x 2 2 3. gx 4 x 2 5. gx x 4 2 4. gx x5 1 6. gx x2 In Exercises 710, use the graph of f x 3x 2 to sketch the graph of g. 17. 18. 19. 21. 25. 2 x 2 3 1 x2 3 9. gx x 12 7. gx 27. 4 8. gx 10. gx 3 x2 1 x2 In Exercises 1114, use the graph of f x 4x3 to sketch the graph of g. y 4 f x = 43 x 2 x 4 2 4 1 x3 1 gx 6x 1 3x Px 1x 3 f x 2 2 x 1 2t f t t x gx 2 x 9 1 f x x 22 2 hx 2 x x 2 3x f x 2 x 2x 3 16. f x 23. f x = 32 x 1 x2 1 hx x4 7 2x Cx 2x 1 gx 2 x2 x2 f x 2 x 9 x2 hx 2 x 9 4s gs 2 s 4 4x 1 gx xx 4 2x f x 2 x 3x 4 15. f x y 2 4 2 x3 2 14. gx 3 x 12. gx In Exercises 1544, a state the domain of the function, b identify all intercepts, c find any vertical and horizontal asymptotes, and d plot additional solution points as needed to sketch the graph of the rational function. x 2 4 x 23 4 13. gx 3 x 11. gx 29. 31. 33. hx x2 5x 4 x2 4 35. f x 6x x 2 5x 14 36. f x 37. f x 38. f x www.elsolucionario.net 20. 22. 24. 26. 28. 30. 32. 34. gx x2 2x 8 x2 9 3x 2 1 2x 15 x2 2x 2 5x 3 x 3 2x 2 x 2 x3 x2 x 2 2x 2 5x 6 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 346 Chapter 4 39. f x 41. f x Rational Functions and Conics x2 3x x6 40. f x 5x 4 x2 x 12 2x2 5x 2 2x2 x 6 42. f x 3x2 8x 4 2x2 3x 2 x2 t2 1 43. f t t1 x2 36 44. f x x6 a Determine the domains of f and g. b Simplify f and find any vertical asymptotes of the graph of f. c Compare the functions by completing the table. d Use a graphing utility to graph f and g in the same viewing window. e Explain why the graphing utility may not show the difference in the domains of f and g. 45. f x x 1, x1 3 gx x 1 2 1.5 1 0.5 0 x 2x 2 , x 2 2x 1 0 gx x 1 1.5 2 2.5 f x gx x2 1 x 54. h x x2 x1 1 60. 3 61. x2 , 1 gx x 2 2x x 0.5 0 0.5 1 1.5 2 gx x 53. g x 59. f x 48. f x 1 x2 x 58. gx x 52. f x 57. f x 47. f x 2x 2 1 x 56. gx x 51. f x x2 0 2x 6 , gx 2 7x 12 x4 1 2 3 4 5 t2 1 t5 x2 f x 3x 1 x3 f x 2 x 4 x3 gx 2 2x 8 x3 1 f x 2 x x x4 x f x x3 x2 x 1 f x x1 55. f t f x 46. f x x2 9 x 2 x 5 50. gx x 49. hx ANALYTICAL, NUMERICAL, AND GRAPHICAL ANALYSIS In Exercises 4548, do the following. x2 In Exercises 4964, a state the domain of the function, b identify all intercepts, c identify any vertical and slant asymptotes, and d plot additional solution points as needed to sketch the graph of the rational function. 6 62. f x 2x 2 5x 5 x2 63. f x 2x3 x2 2x 1 x2 3x 2 64. f x 2x3 x2 8x 4 x2 3x 2 3 In Exercises 6568, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. x 2 5x 8 x3 1 3x 2 x 3 67. gx x2 65. f x www.elsolucionario.net 2x 2 x x1 12 2x x 2 68. hx 24 x 66. f x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.2 GRAPHICAL REASONING In Exercises 6972, a use the graph to determine any x intercepts of the graph of the rational function and b set y 0 and solve the resulting equation to confirm your result in part a . x1 x3 69. y 70. y y y 6 6 4 4 2 2 x 2 4 6 8 4 71. y 2x x3 2 x 2 4 6 8 4 1 x x 72. y x 3 y 2 x Graphs of Rational Functions d As the tank is filled, what happens to the rate at which the concentration of brine is increasing What percent does the concentration of brine appear to approach 78. GEOMETRY A rectangular region of length x and width y has an area of 500 square meters. a Write the width y as a function of x. b Determine the domain of the function based on the physical constraints of the problem. c Sketch a graph of the function and determine the width of the rectangle when x 30 meters. 79. PAGE DESIGN A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom margins are 1 inch deep and the margins on each side are 2 inches wide see figure . 1 in. y 4 8 2 4 x 4 2 4 8 4 2 in. 4 8 GRAPHICAL REASONING In Exercises 7376, a use a graphing utility to graph the rational function and determine any x intercepts of the graph and b set y 0 and solve the resulting equation to confirm your result in part a . 75. y x 6 x1 9 76. y x x 77. CONCENTRATION OF A MIXTURE A 1000 liter tank contains 50 liters of a 25 brine solution. You add x liters of a 75 brine solution to the tank. a Show that the concentration C, the proportion of brine to total solution, in the final mixture is C 2 in. y x 4 1 in. x 4 1 4 73. y x5 x 2 3 74. y 20 x1 x 347 3x 50 . 4x 50 a Show that the total area A on the page is A b Determine the domain of the function based on the physical constraints of the problem. c Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. 80. PAGE DESIGN A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are 112 inches wide. What should the dimensions of the page be so that the least amount of paper is used In Exercises 81 and 82, use a graphing utility to graph the function and locate any relative maximum or minimum points on the graph. 81. f x b Determine the domain of the function based on the physical constraints of the problem. c Sketch a graph of the concentration function. 2xx 11 . x4 3x 1 x2 x 1 82. Cx x www.elsolucionario.net 32 x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 348 Chapter 4 Rational Functions and Conics 83. MINIMUM COST The ordering and transportation cost C in thousands of dollars for the components used in manufacturing a product is given by C 100 x 200 2 x , x 1 x 30 where x is the order size in hundreds . Use a graphing utility to graph the cost function. From the graph, estimate the order size that minimizes cost. 84. MINIMUM COST The cost C of producing x units of a product is given by and the average cost per unit is given by C 0.2x 2 10x 5 , C x 0. x x Sketch the graph of the average cost function and estimate the number of units that should be produced to minimize the average cost per unit. 85. AVERAGE SPEED A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 100 miles away. The average speeds for going and returning were x and y miles per hour, respectively. 25x . a Show that y x 25 b Determine the vertical and horizontal asymptotes of the graph of the function. c Use a graphing utility to graph the function. d Complete the table. 30 35 40 45 50 55 60 y e Are the results in the table what you expected Explain. f Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip Explain. 86. MEDICINE The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given by C 3t t , t 0. t 3 50 2 EXPLORATION TRUE OR FALSE In Exercises 87 90, determine whether the statement is true or false. Justify your answer. C 0.2x 2 10x 5 x a Determine the horizontal asymptote of the graph of the function and interpret its meaning in the context of the problem. b Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest. c Use a graphing utility to determine when the concentration is less than 0.345. 87. If the graph of a rational function f has a vertical asymptote at x 5, it is possible to sketch the graph without lifting your pencil from the paper. 88. The graph of a rational function can never cross one of its asymptotes. 2x3 89. The graph of f x has a slant asymptote. x1 90. Every rational function has a horizontal asymptote. THINK ABOUT IT In Exercises 91 and 92, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. 6 2x 3x x2 x 2 92. gx x1 91. hx 93. WRITING Given a rational function f, how can you determine whether f has a slant asymptote If f has a slant asymptote, explain the process for finding it. 94. CAPSTONE Write a rational function satisfying the following criteria. Then sketch a graph of your function. Vertical asymptote: x 2 Slant asymptote: y x 1 Zero of the function: x 2 PROJECT: DEPARTMENT OF DEFENSE To work an extended application analyzing the total numbers of Department of Defense personnel from 1980 through 2007, visit this texts website at academic.cengage.com. Data Source: U.S. Department of Defense www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 Conics 349 4.3 CONICS What you should learn Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas vertex at origin . Recognize, graph, and write equations of ellipses center at origin . Recognize, graph, and write equations of hyperbolas center at origin . Introduction Conic sections were discovered during the classical Greek period, 600 to 300 B.C. This early Greek study was largely concerned with the geometric properties of conics. It was not until the early 17th century that the broad applicability of conics became apparent and played a prominent role in the early development of calculus. A conic section or simply conic is the intersection of a plane and a doublenapped cone. Notice in Figure 4.18 that in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a degenerate conic, as shown in Figure 4.19. Why you should learn it Conics have been used for hundreds of years to model and solve engineering problems. For instance, in Exercise 45 on page 359, a parabola can be used to model the cables of the Golden Gate Bridge. Circle Ellipse 4.18 Basic Conics Parabola Hyperbola Cosmo CondinaGetty Images FIGURE Point Line FIGURE Two Intersecting Lines 4.19 Degenerate Conics There are several ways to approach the study of conics. You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general second degree equation Ax 2 Bxy Cy 2 Dx Ey F 0. However, you will study a third approach, in which each of the conics is defined as a locus collection of points satisfying a certain geometric property. For example, in Section 1.1 you saw how the definition of a circle as the collection of all points x, y that are equidistant from a fixed point h, k led easily to the standard form of the equation of a circle x h2 y k 2 r 2. Equation of a circle Recall from Section 1.1 that the center of a circle is at h, k and that the radius of the circle is r. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 350 Chapter 4 Rational Functions and Conics Parabolas In Section 3.1, you learned that the graph of the quadratic function f x ax 2 bx c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. y Definition of a Parabola A parabola is the set of all points x, y in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line. See Figure 4.20. The vertex is the midpoint between the focus and the directrix. The axis of the parabola is the line passing through the focus and the vertex. d2 Focus d1 Vertex d1 d2 Directrix x FIGURE 4.20 Parabola Standard Equation of a Parabola Vertex at Origin The standard form of the equation of a parabola with vertex at 0, 0 and directrix y p is x 2 4py, p 0. Vertical axis For directrix x p, the equation is y 2 4px, p 0. Horizontal axis The focus is on the axis p units directed distance from the vertex. For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 376. Notice that a parabola can have a vertical or a horizontal axis and that a parabola is symmetric with respect to its axis. Examples of each are shown in Figure 4.21. x 2 = 4py, p = 0 y y Vertex 0, 0 y 2 = 4px, p = 0 x , y Focus p, 0 Focus 0, p p x , y Vertex 0, 0 x x p p Directrix: y = p a Parabola with vertical axis FIGURE p Directrix: x = p b Parabola with horizontal axis 4.21 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 Example 1 Conics 351 Finding the Focus of a Parabola Find the focus of the parabola whose equation is y 2x 2. Solution Because the squared term in the equation involves x, you know that the axis is vertical, and the equation is of the form y Focus 0, 18 x 1 1 You can write the original equation in this form as follows. y = 2x 2 1 x2 y 2 1 8 y x2 4 2 FIGURE x 2 4py. 1 Write in standard form. So, p 18. Because p is negative, the parabola opens downward see Figure 4.22 , and the focus of the parabola is 4.22 0, p 0, y 2 1 . 8 Focus Now try Exercise 21. y 2 = 8x 1 Vertex 1 1 Example 2 Focus 2, 0 2 3 4 0, 0 Find the standard form of the equation of the parabola with vertex at the origin and focus at 2, 0. Solution 2 FIGURE A Parabola with a Horizontal Axis x The axis of the parabola is horizontal, passing through 0, 0 and 2, 0, as shown in Figure 4.23. So, the standard form is 4.23 y 2 4px. Because the focus is p 2 units from the vertex, the equation is Light source at focus y 2 42x y 2 8x. Focus Axis Parabolic reflector: Light is reflected in parallel rays. FIGURE 4.24 Now try Exercise 27. Parabolas occur in a wide variety of applications. For instance, a parabolic reflector can be formed by revolving a parabola about its axis. The resulting surface has the property that all incoming rays parallel to the axis are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversely, the light rays emanating from the focus of a parabolic reflector used in a flashlight are all parallel to one another, as shown in Figure 4.24. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 352 Chapter 4 Rational Functions and Conics Ellipses x , y Vertex d1 Focus d2 Major axis Definition of an Ellipse An ellipse is the set of all points x, y in a plane the sum of whose distances from two distinct fixed points foci is constant. See Figure 4.25. Focus Center Minor axis Vertex d1 + d 2 is constant. FIGURE 4.25 The line through the foci intersects the ellipse at two points vertices . The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the major axis at the center is the minor axis. See Figure 4.25. You can visualize the definition of an ellipse by imagining two thumbtacks placed at the foci, as shown in Figure 4.26. If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse. The standard form of the equation of an ellipse takes one of two forms, depending on whether the major axis is horizontal or vertical. Standard Equation of an Ellipse Center at Origin FIGURE 4.26 The standard form of the equation of an ellipse centered at the origin with major and minor axes of lengths 2a and 2b where 0 b a is x2 y2 21 2 a b x2 y2 2 1. 2 b a or The vertices and foci lie on the major axis, a and c units, respectively, from the center, as shown in Figure 4.27. Moreover, a, b, and c are related by the equation c 2 a 2 b2. x2 y2 + =1 a2 b2 y x2 y2 + =1 b2 a2 0, b y 0, a 0, c 0, 0 c, 0 x c, 0 a, 0 0, 0 b, 0 a, 0 0, b x 0, a a Major axis is horizontal; minor axis is vertical. FIGURE b, 0 0, c b Major axis is vertical; minor axis is horizontal. 4.27 In Figure 4.27 a , note that because the sum of the distances from a point on the ellipse to the two foci is constant, it follows that Sum of distances from 0, b to foci sum of distances from a, 0 to foci 2b 2 c2 a c a c b2 c2 a c2 a 2 b 2. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 y Example 3 Conics 353 Finding the Standard Equation of an Ellipse 3 Find the standard form of the equation of the ellipse shown in Figure 4.28. Solution 1 2, 0 2 2, 0 x 1 1 2 1 x2 y2 1. a2 b2 3 FIGURE From Figure 4.28, the foci occur at 2, 0 and 2, 0. So, the center of the ellipse is 0, 0, the major axis is horizontal, and the ellipse has an equation of the form Standard form Also from Figure 4.28, the length of the major axis is 2a 6. This implies that a 3. Moreover, the distance from the center to either focus is c 2. Finally, 4.28 b2 a 2 c 2 32 22 9 4 5. Substituting a 2 32 and b2 5 yields the following equation in standard form. 2 T E C H N O LO G Y Conics can be graphed using a graphing utility by first solving for y. You may have to graph the conic using two separate equations. For example, you can graph the ellipse from Example 4 by graphing both y2 x2 1 32 5 2 This equation simplifies to x2 y2 1. 9 5 Now try Exercise 63. Example 4 y1 36 4x2 Sketch the ellipse given by 4x 2 y 2 36, and identify the vertices. and y2 36 4x2 Solution in the same viewing window. 4x 2 y 2 36 4x 2 36 y 0, 6 x2 y2 + =1 32 62 4 2 3, 0 6 3, 0 x 2 4 2 2 4 0, 6 FIGURE Sketching an Ellipse 4.29 4 6 y2 36 36 36 Write original equation. Divide each side by 36. x2 y2 1 9 36 Simplify. x2 y2 21 32 6 Write in standard form. Because the denominator of the y 2 term is larger than the denominator of the x 2 term, you can conclude that the major axis is vertical. Moreover, because a2 62, the endpoints of the major axis lie six units up and down from the center 0, 0. So, the vertices of the ellipse are 0, 6 and 0, 6. Similarly, because the denominator of the x2 term is b2 32, the endpoints of the minor axis or co vertices lie three units to the right and left of the center at 3, 0 and 3, 0. The ellipse is shown in Figure 4.29. Now try Exercise 53. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 354 Chapter 4 d1 Rational Functions and Conics Hyperbolas Focus x, y The definition of a hyperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is constant, whereas for a hyperbola the difference of the distances between the foci and a point on the hyperbola is constant. d2 Focus d 2 d 1 is a positive constant. Definition of a Hyperbola A hyperbola is the set of all points x, y in a plane the difference of whose distances from two distinct fixed points foci is a positive constant. See Figure 4.30 a . a Branch Vertex c a Center Transverse axis The graph of a hyperbola has two disconnected parts branches . The line through the two foci intersects the hyperbola at two points vertices . The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola. See Figure 4.30 b . Vertex Branch Standard Equation of a Hyperbola Center at Origin b FIGURE The standard form of the equation of a hyperbola with center at the origin where a 0 and b 0 is 4.30 x2 y2 21 a2 b Transverse axis is horizontal. y 2 x2 1. a 2 b2 Transverse axis is vertical. or The vertices and foci are, respectively, a and c units from the center. Moreover, a, b, and c are related by the equation b2 c 2 a 2. See Figure 4.31. Transverse axis WARNING CAUTION Be careful when finding the foci of ellipses and hyperbolas. Notice that the relationships between a, b, and c differ slightly. y x2 y2 =1 a 2 b2 y 0, c 0, b a, 0 a, 0 c, 0 c, 0 Transverse axis x 0, a b, 0 b, 0 a2 x 0, a Finding the foci of an ellipse: c2 y2 x2 =1 a 2 b2 0, b b2 0, c Finding the foci of a hyperbola: c2 a2 b2 a FIGURE b 4.31 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 y Example 5 355 Conics Finding the Standard Equation of a Hyperbola 3 Find the standard form of the equation of the hyperbola with foci at 3, 0 and 3, 0 and vertices at 2, 0 and 2, 0, as shown in Figure 4.32. 3 1 2 3 4.32 x 1 1 FIGURE 2, 0 3, 0 3 Solution From the graph, you can determine that c 3, because the foci are three units from the center. Moreover, a 2 because the vertices are two units from the center. So, it follows that b2 c 2 a2 32 22 94 5. Because the transverse axis is horizontal, the standard form of the equation is x 2 y2 1. a 2 b2 Finally, substitute a2 22 and b2 5 to obtain 2 x2 y2 1 22 5 2 Write in standard form. x 2 y2 1. 4 5 Simplify. Now try Exercise 85. An important aid in sketching the graph of a hyperbola is the determination of its asymptotes, as shown in Figure 4.33. Each hyperbola has two asymptotes that intersect at the center of the hyperbola. Furthermore, the asymptotes pass through the corners of a rectangle of dimensions 2a by 2b. The line segment of length 2b joining 0, b and 0, b or b, 0 and b, 0 is the conjugate axis of the hyperbola. x2 y2 2=1 a2 b y y2 x2 2=1 a2 b y Asymptote: y = ab x 0, a 0, b a, 0 a, 0 0, b Transverse axis b, 0 Asymptote: y = ax b b, 0 Conjugate axis x 0, a Asymptote: y = ab x a Transverse axis is horizontal; conjugate axis is vertical. FIGURE x Transverse axis 1 3, 0 2, 0 Conjugate axis 2 Asymptote: y= ax b b Transverse axis is vertical; conjugate axis is horizontal. 4.33 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 356 Chapter 4 Rational Functions and Conics Asymptotes of a Hyperbola Center at Origin The asymptotes of a hyperbola with center at 0, 0 are b y x a and b y x a Transverse axis is horizontal. a y x b and a y x. b Transverse axis is vertical. or Example 6 Sketching a Hyperbola Sketch the hyperbola whose equation is 4x2 y2 16. Graphical Solution Algebraic Solution 16 Write original equation. 4x 2 y2 16 16 16 16 Divide each side by 16. 4x 2 y2 Solve the equation of the hyperbola for y as follows. 4x 2 y 2 16 4x 2 16 y2 4x2 16 y x2 y2 1 4 16 Simplify. x2 y2 21 22 4 Write in standard form. Then use a graphing utility to graph y1 4x2 16 Because the x 2 term is positive, you can conclude that the transverse axis is horizontal and the vertices occur at 2, 0 and 2, 0. Moreover, the endpoints of the conjugate axis occur at 0, 4 and 0, 4, and you can sketch the rectangle shown in Figure 4.34. Finally, by drawing the asymptotes through the corners of this rectangle, you can complete the sketch shown in Figure 4.35. Note that the asymptotes are y 2x and y 2x. y2 4x2 16 in the same viewing window. Be sure to use a square setting. From the graph in Figure 4.36, you can see that the transverse axis is horizontal. You can use the zoom and trace features to approximate the vertices to be 2, 0 and 2, 0. y y 6 8 8 6 and 0, 4 y1 = 6 9 2, 0 6 4 9 x2 y2 =1 22 42 2, 0 4 4x 2 16 x 6 6 6 x 4 4 6 FIGURE y2 = 4x 2 16 4.36 0, 4 6 6 FIGURE 4.34 FIGURE 4.35 Now try Exercise 81. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 Example 7 4 4 Finding the Standard Equation of a Hyperbola Solution 2 0, 3 Because the transverse axis is vertical, the asymptotes are of the forms a y x b x 2 2 2 4 0, 3 and a y x. b Using the fact that y 2x and y 2x, you can determine that y = 2x a 2. b 4 FIGURE 357 Find the standard form of the equation of the hyperbola that has vertices at 0, 3 and 0, 3 and asymptotes y 2x and y 2x, as shown in Figure 4.37. y y = 2x Conics Because a 3, you can determine that b 32. Finally, you can conclude that the hyperbola has the following equation. 4.37 y2 x2 1 32 3 2 2 y2 x2 1 9 9 4 Write in standard form. Simplify. Now try Exercise 87. 4.3 EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. A ________ is the intersection of a plane and a double napped cone. 2. The equation x h2 y k2 r 2 is the standard form of the equation of a ________ with center ________ and radius ________. 3. A ________ is the set of all points x, y in a plane that are equidistant from a fixed line, called the ________, and a fixed point, called the ________, not on the line. 4. The ________ of a parabola is the midpoint between the focus and the directrix. 5. The line that passes through the focus and the vertex of a parabola is called the ________ of the parabola. 6. An ________ is the set of all points x, y in a plane, the sum of whose distances from two distinct fixed points, called________, is constant. 7. The chord joining the vertices of an ellipse is called the ________ ________, and its midpoint is the ________ of the ellipse. 8. The chord perpendicular to the major axis at the center of an ellipse is called the ________ ________ of the ellipse. 9. A ________ is the set of all points x, y in a plane, the difference of whose distances from two distinct fixed points, called ________, is a positive constant. 10. The line segment connecting the vertices of a hyperbola is called the ________ ________, and the midpoint of the line segment is the ________ of the hyperbola. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 358 Chapter 4 Rational Functions and Conics SKILLS AND APPLICATIONS In Exercises 1120, match the equation with its graph. If the graph of an equation is not shown, write not shown. The graphs are labeled a , b , c , d , e , f , g , and h . y a y b 4 2 6 4 4 x 2 x 8 4 4 8 4 y c y d 6 4 2 x 4 4 2 2 4 2 4 6 y 27. 29. 31. 33. 35. 36. 37. 38. Focus: 2, 0 28. Focus: 0, 2 Focus: 0, 12 30. Focus: 32, 0 Directrix: y 1 32. Directrix: y 2 Directrix: x 1 34. Directrix: x 4 Passes through the point 4, 6; horizontal axis Passes through the point 2, 2; vertical axis Passes through the point 2, 14 ; vertical axis Passes through the point 12, 4; horizontal axis In Exercises 39 42, find the standard form of the equation of the parabola and determine the coordinates of the focus. y 39. 4 6 6 e x In Exercises 2738, find the standard form of the equation of the parabola with the given characteristic s and vertex at the origin. 8 3, 6 6 y f 4 2, 6 4 4 8 2 2 x 2 2 4 4 6 4 x 2 2 4 6 2 x 2 4 6 6 11. 13. 15. 17. 19. 2 2 2 x 2 2y y 2 2x 9x 2 y 2 9 9x 2 y 2 9 x 2 y 2 25 x 2 4 6 21. y 23. y 2 6x 25. x 2 12y 0 4 8 x 4 6 8 10 x 8 4 8, 4 5, 3 12 43. FLASHLIGHT The light bulb in a flashlight is at the focus of the parabolic reflector, 1.5 centimeters from the vertex of the reflector see figure . Write an equation for a cross section of the flashlights reflector with its focus on the positive x axis and its vertex at the origin. x 2 2y y 2 2x x 2 9y 2 9 y 2 9x 2 9 x 2 y 2 16 22. y 4x 2 24. y 2 3x 26. x y 2 0 y 42. y y 1.5 cm Receiver x In Exercises 2126, find the vertex and focus of the parabola and sketch its graph. 1 2 2x 8 4 4 4 12. 14. 16. 18. 20. 2 6 4 2 y h 4 x 2 y 4 y x 4 4 41. g y 40. 3.5 ft x FIGURE FOR 43 FIGURE FOR 44 44. SATELLITE ANTENNA Write an equation for a cross section of the parabolic satellite dish antenna shown in the figure. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 45. SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended in the shape of a parabola between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway at the midpoint between the towers. a Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. b Write an equation that models the cables. c Complete the table by finding the height y of the suspension cables over the roadway at a distance of x meters from the center of the bridge. In Exercises 5766, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. y 57. 0 200 400 500 2 2 4 Not drawn to scale a Find an equation of the parabola. Assume that the origin is at the center of the beam. b How far from the center of the beam is the 1 deflection 2 inch 4 47. 49. 51. 53. 1 2 1 x y2 1 55. 16 81 48. 50. 52. 54. 56. x2 y2 1 121 144 x2 y2 1 4 14 x2 y2 1 28 64 4x 2 9y 2 36 1 2 1 x y2 1 100 49 5, 0 0, 6 y 8 0, 32 0, 72 7, 0 x x 8 4 0, 32 4 0, 72 8 7, 0 8 Vertices: 5, 0; foci: 2, 0 Vertices: 0, 8; foci: 0, 4 Foci: 5, 0; major axis of length 14 Foci: 2, 0; major axis of length 10 Vertices: 0, 5; passes through the point 4, 2 Vertical major axis; passes through the points 0, 4 and 2, 0 67. ARCHITECTURE A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of 2 feet at the center and a width of 6 feet along the base see figure . The contractor draws the outline of the ellipse on the wall by the method shown in Figure 4.26. Give the required positions of the tacks and the length of the string. y 3 In Exercises 4756, find the center and vertices of the ellipse and sketch its graph. x2 y2 1 25 16 x2 y2 1 259 169 x2 y2 1 36 7 4x 2 y 2 1 2 4 6 60. 4 61. 62. 63. 64. 65. 66. x 2 2, 0 4 64 ft 6 y Height, y 1 in. x 4 0, 2 59. 0, 6 4 2 4 600 46. BEAM DEFLECTION A simply supported beam see figure is 64 feet long and has a load at the center. The deflection of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. 5, 0 0, 2 1, 0 1, 0 4 y 58. 4 2, 0 Distance, x 359 Conics 1 3 2 1 x 1 2 3 68. ARCHITECTURE A semielliptical arch over a tunnel for a one way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. a Sketch the arch of the tunnel on a rectangular coordinate system with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. b Find an equation of the semielliptical arch over the tunnel. c You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 360 Chapter 4 Rational Functions and Conics 69. ARCHITECTURE Repeat Exercise 68 for a semielliptical arch with a major axis of 40 feet and a height at the center of 15 feet. The dimensions of the truck are 10 feet wide by 14 feet high. 70. GEOMETRY A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve see figure . Show that the length of each latus rectum is 2b 2a. 93. ART A sculpture has a hyperbolic cross section see figure . y 16 2, 13 2, 13 8 1, 0 1, 0 4 x 3 2 2 4 3 4 8 y 2, 13 16 Latera recta F1 F2 x In Exercises 7174, sketch the graph of the ellipse, using the latera recta see Exercise 70 . x2 y 2 1 4 1 73. 9x 2 4y 2 36 71. x2 y2 1 9 16 74. 5x 2 3y 2 15 72. 2, 13 a Write an equation that models the curved sides of the sculpture. b Each unit on the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 5 feet. 94. OPTICS A hyperbolic mirror used in some telescopes has the property that a light ray directed at the focus will be reflected to the other focus. The focus of a hyperbolic mirror see figure has coordinates 24, 0. Find the vertex of the mirror if its mount at the top edge of the mirror has coordinates 24, 24. y 24, 24 In Exercises 7584, find the center and vertices of the hyperbola and sketch its graph, using asymptotes as sketching aids. 77. y 2 x2 1 1 4 y2 x2 1 79. 49 196 x2 y2 1 76. 9 16 y 2 x2 1 78. 9 1 x2 y2 1 80. 36 4 81. 4y 2 x 2 1 82. 4y 2 9x 2 36 75. x 2 y 2 1 83. 1 2 1 2 y x 1 36 100 84. 1 2 1 2 x y 1 144 169 In Exercises 8592, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 85. 86. 87. 88. 89. 90. 91. 92. x 24, 0 Vertices: 0, 2; foci: 0, 6 Vertices: 4, 0; foci: 5, 0 Vertices: 1, 0; asymptotes: y 3x Vertices: 0, 3; asymptotes: y 3x Foci: 0, 8; asymptotes: y 4x 3 Foci: 10, 0; asymptotes: y 4x Vertices: 0, 3; passes through the point 2, 5 Vertices: 2, 0; passes through the point 3, 3 24, 0 95. AERONAUTICS When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, the sound waves intersect the ground in a hyperbola with the airplane directly above its center see figure . A sonic boom is heard along the hyperbola. You hear a sonic boom that is audible along a hyperbola with the equation x2 y2 1 100 4 where x and y are measured in miles. What is the shortest horizontal distance you could be from the airplane www.elsolucionario.net Shock wave Ground Not drawn to scale http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.3 96. NAVIGATION Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light 186,000 miles per second . The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations 300 miles apart are positioned on a rectangular coordinate system at points with coordinates 150, 0 and 150, 0 and that a ship is traveling on a path with coordinates x, 75, as shown in the figure. Find the x coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 micro seconds 0.001 second . y 150 75 150 x 75 75 150 EXPLORATION TRUE OR FALSE In Exercises 97100, determine whether the statement is true or false. Justify your answer. 97. The equation x 2 y 2 144 represents a circle. 98. The major axis of the ellipse y 2 16x 2 64 is vertical. 99. It is possible for a parabola to intersect its directrix. 100. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical. 101. Consider the ellipse x2 a2 y2 b2 a The area of the ellipse is given by A ab. Write the area of the ellipse as a function of a. b Find the equation of an ellipse with an area of 264 square centimeters. c Complete the table using your equation from part a , and make a conjecture about the shape of the ellipse with maximum area. 8 361 d Use a graphing utility to graph the area function and use the graph to support your conjecture in part c . 102. CAPSTONE Identify the conic. Explain your reasoning. a 4x2 4y2 16 0 b 4y2 5x2 20 0 c 3y2 6x 0 d 2x2 4y2 12 0 e 4x2 y2 16 0 f 2x2 12y 0 103. THINK ABOUT IT How can you tell if an ellipse is a circle from the equation 104. THINK ABOUT IT Is the graph of x 2 4y4 4 an ellipse Explain. 105. THINK ABOUT IT The graph of x 2 y 2 0 is a degenerate conic. Sketch this graph and identify the degenerate conic. 106. THINK ABOUT IT Which part of the graph of the ellipse 4x2 9y2 36 is represented by each equation Do not graph. a x 324 y2 b y 239 x2 107. WRITING At the beginning of this section, you learned that each type of conic section can be formed by the intersection of a plane and a double napped cone. Write a short paragraph describing examples of physical situations in which hyperbolas are formed. 108. WRITING Write a paragraph discussing the changes in the shape and orientation of the graph of the ellipse x2 y2 1 a2 42 as a increases from 1 to 8. 1, a b 20. a Conics 9 10 11 12 13 109. Use the definition of an ellipse to derive the standard form of the equation of an ellipse. 110. Use the definition of a hyperbola to derive the standard form of the equation of a hyperbola. 111. An ellipse can be drawn using two thumbtacks placed at the foci of the ellipse, a string of fixed length greater than the distance between the tacks , and a pencil, as shown in Figure 4.26. Try doing this. Vary the length of the string and the distance between the thumbtacks. Explain how to obtain ellipses that are almost circular. Explain how to obtain ellipses that are long and narrow. A www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 362 Chapter 4 Rational Functions and Conics 4.4 TRANSLATIONS OF CONICS What you should learn Recognize equations of conics that have been shifted vertically or horizontally in the plane. Write and graph equations of conics that have been shifted vertically or horizontally in the plane. Why you should learn it In some real life applications, it is not convenient to use conics whose centers or vertices are at the origin. For instance, in Exercise 41 on page 368, a parabola can be used to model the maximum sales for Texas Instruments, Inc. Vertical and Horizontal Shifts of Conics In Section 4.3 you looked at conic sections whose graphs were in standard position. In this section you will study the equations of conic sections that have been shifted vertically or horizontally in the plane. Standard Forms of Equations of Conics Circle: Center h, k; radius r x h2 y k 2 r 2 Ellipse: Center h, k Major axis length 2a; minor axis length 2b y h , k x h2 y k2 1. a2 b2 2a h , k 2b x x Hyperbola: Center h, k Transverse axis length 2a; conjugate axis length 2b y x h 2 y k 2 =1 a2 b2 If you let a b, then the equation can be rewritten as h , k x h2 y k2 a2 which is the standard form of the equation of a circle with radius r a see Section 1.1 . Geometrically, when a b for an ellipse, the major and minor axes are of equal length, and so the graph is a circle see Example 1 a . x h 2 y k 2 + =1 b2 a2 2b 2a Consider the equation of the ellipse y x h 2 y k 2 + =1 a2 b2 y h , k 2b 2a y k 2 x h 2 =1 a2 b2 2a 2b x x Parabola: Vertex h, k Directed distance from vertex to focus p y y 2 x h = 4 p y k p0 2 y k = 4 p x h Focus: h, k + p Vertex: h , k p0 Vertex: h, k x www.elsolucionario.net Focus: h + p , k x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.4 Example 1 363 Translations of Conics Equations of Conic Sections Identify each conic. Then describe the translation of the graph of the conic. a. x 12 y 22 32 c. b. x 32 y 22 1 12 32 x 22 y 12 1 32 22 d. x 22 41 y 3 Solution y x 1 2 + y + 2 2 = 32 2 x 2 6 1, 2 x 2 2 y 12 1 32 22 3 6 FIGURE 4.38 Circle y 6 4 x 2 2 y 1 2 + =1 32 22 2 x 2 6 2 FIGURE 4.39 Ellipse is an ellipse whose center is the point 2, 1. The major axis of the ellipse is horizontal and of length 23 6, and the minor axis of the ellipse is vertical and of length 22 4, as shown in Figure 4.39. The graph of the ellipse has been shifted two units to the right and one unit upward from standard position. c. The graph of x 3 2 y 22 1 12 32 3 2, 1 a. The graph of x 12 y 22 32 is a circle whose center is the point 1, 2 and whose radius is 3, as shown in Figure 4.38. The graph of the circle has been shifted one unit to the right and two units downward from standard position. b. The graph of is a hyperbola whose center is the point 3, 2. The transverse axis is horizontal and of length 21 2, and the conjugate axis is vertical and of length 23 6, as shown in Figure 4.40. The graph of the hyperbola has been shifted three units to the right and two units upward from standard position. d. The graph of x 22 41 y 3 is a parabola whose vertex is the point 2, 3. The axis of the parabola is vertical. The focus is one unit above or below the vertex. Moreover, because p 1, it follows that the focus lies below the vertex, as shown in Figure 4.41. The graph of the parabola has been reflected in the x axis, shifted two units to the right and three units upward from standard position. y y x 3 2 y 2 2 =1 12 32 6 4 4 2 2 3, 2 3 FIGURE 2, 3 p = 1 2, 2 x 6 2 x 2 2 = 4 1 y 3 6 x 2 8 2 1 4.40 Hyperbola 4 FIGURE 4.41 Parabola Now try Exercise 11. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 364 Chapter 4 Rational Functions and Conics Equations of Conics in Standard Form y Example 2 Finding the Standard Equation of a Parabola 2 1 1, 1 Find the vertex and focus of the parabola x 2 2x 4y 3 0. x 2 1 1 2 3 Solution 4 Complete the square to write the equation in standard form. 1, 0 2 3 x 2 2x 4y 3 0 2 x 1 = 4 1 y 1 x 2 2x 4y 3 4 FIGURE Write original equation. x 2 2x 1 4y 3 1 4.42 Add 1 to each side. x 12 4y 4 Write in completed square form. x 1 41 y 1 Write in standard form, x h2 4p y k. 2 Note in Example 2 that p is the directed distance from the vertex to the focus. Because the axis of the parabola is vertical and p 1, the focus is one unit below the vertex, and the parabola opens downward. Group terms. From this standard form, it follows that h 1, k 1, and p 1. Because the axis is vertical and p is negative, the parabola opens downward. The vertex is h, k 1, 1 and the focus is h, k p 1, 0. See Figure 4.42. Now try Exercise 31. Example 3 Sketching an Ellipse Sketch the ellipse x 2 4y 2 6x 8y 9 0. Solution Complete the square to write the equation in standard form. x 2 4y 2 6x 8y 9 0 x2 6x 4y 2 8y 9 x 6x 4 y 2y 9 2 2 Write original equation. Group terms. Factor 4 out of y terms. x 6x 9 4 y 2y 1 9 9 41 2 y 4 x + 3 2 y 1 2 + =1 22 12 5, 1 3 3, 2 1 3, 0 5 4 3 2 x 1 1 FIGURE 4.43 Add 9 and 41 4 to each side. x 32 4 y 12 4 Write in completed square form. x 32 4 y 12 1 4 4 Divide each side by 4. x 32 y 12 1 22 12 1, 1 2 3, 1 2 x h2 y k2 1 a2 b2 From this standard form, it follows that the center is h, k 3, 1. Because the denominator of the x term is a 2 22, the endpoints of the major axis lie two units to the right and left of the center. Similarly, because the denominator of the y term is b2 12, the endpoints of the minor axis lie one unit up and down from the center. The ellipse is shown in Figure 4.43. Now try Exercise 47. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.4 Example 4 Translations of Conics 365 Sketching a Hyperbola Sketch the hyperbola y 2 4x 2 4y 24x 41 0. Solution Complete the square to write the equation in standard form. y 2 4x 2 4y 24x 41 0 y 4y 4x 24x 41 2 2 y2 4y 4 x2 6x 41 Write original equation. Group terms. Factor 4 out of x terms. y 4y 4 4x 6x 9 41 4 49 2 2 y 22 4x 32 9 Write in completed square form. y 22 4x 32 1 9 9 Divide each side by 9. y 22 x 32 1 9 9 4 y + 2 y 3 2 2 y 22 x 32 1 32 3 2 2 2 x 3 =1 32 2 3, 1 2 x 4 2 4 6 Change 4 to 1 1 4 . y k2 x h2 1 a2 b2 From this standard form, it follows that the transverse axis is vertical and the center lies at h, k 3, 2. Because the denominator of the y term is a2 32, you know that the vertices occur three units above and below the center. 2 2 Add 4 and subtract 49 36. 6 3, 2 3, 5 3, 1 and 3, 5 Vertices To sketch the hyperbola, draw a rectangle whose top and bottom pass through the 2 vertices. Because the denominator of the x term is b2 32 , locate the sides of the rectangle 32 units to the right and left of the center, as shown in Figure 4.44. Finally, sketch the asymptotes by drawing lines through the opposite corners of the rectangle. Using these asymptotes, you can complete the graph of the hyperbola, as shown in Figure 4.44. Now try Exercise 67. FIGURE 4.44 To find the foci in Example 4, first find c. c2 a2 b2 9 9 45 4 4 c 35 2 Because the transverse axis is vertical, the foci lie c units above and below the center. 3, 2 325 and 3, 2 325 www.elsolucionario.net Foci http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 366 Chapter 4 Rational Functions and Conics y Example 5 Writing the Equation of an Ellipse 4 2, 4 Write the standard form of the equation of the ellipse whose vertices are 2, 2 and 2, 4. The length of the minor axis of the ellipse is 4, as shown in Figure 4.45. 3 2 Solution 4 1 The center of the ellipse lies at the midpoint of its vertices. So, the center is x 1 1 1 3 4 5 h, k 2, 1. Center Because the vertices lie on a vertical line and are six units apart, it follows that the major axis is vertical and has a length of 2a 6. So, a 3. Moreover, because the minor axis has a length of 4, it follows that 2b 4, which implies that b 2. So, the standard form of the ellipse is as follows. 2, 2 2 FIGURE 2 4.45 x h2 y k2 1 b2 a2 Major axis is vertical. x 22 y 12 1 22 32 Write in standard form. Now try Exercise 51. Hyperbolic orbit Vertex Elliptical orbit Sun p Parabolic orbit FIGURE 4.46 An interesting application of conic sections involves the orbits of comets in our solar system. Of the 610 comets identified prior to 1970, 245 have elliptical orbits, 295 have parabolic orbits, and 70 have hyperbolic orbits. For example, Halleys comet has an elliptical orbit, and reappearance of this comet can be predicted every 76 years. The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in Figure 4.46. If p is the distance between the vertex and the focus in meters , and v is the speed of the comet at the vertex in meters per second , then the type of orbit is determined as follows. 2GM p 2GM 2. Parabola: v p 2GM 3. Hyperbola: v p 1. Ellipse: v In each of these relations, M 1.989 1030 kilograms the mass of the sun and G 6.67 1011 cubic meter per kilogram second squared the universal gravitational constant . CLASSROOM DISCUSSION Identifying Equations of Conics Use the Internet to research information about the orbits of comets in our solar system. What can you find about the orbits of comets that have been identified since 1970 Write a summary of your results. Identify your source. Does it seem reliable www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.4 4.4 EXERCISES Translations of Conics 367 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Match the description of the conic with its standard equation. The equations are labeled a , b , c , d , e , and f . a x h2 y k2 1 a2 b2 b x h2 y k2 1 a2 b2 d x h2 y k2 1 b2 a2 e x h2 4p y k 1. Hyperbola with horizontal transverse axis 4. Hyperbola with vertical transverse axis c y k2 x h2 1 a2 b2 f y k2 4px h 2. Ellipse with vertical major axis 5. Ellipse with horizontal major axis 3. Parabola with vertical axis 6. Parabola with horizontal axis SKILLS AND APPLICATIONS In Exercises 712, describe the translation of the graph of the conic. 7. x 22 y 12 4 8. y 12 42x 2 y y 4 3 2 1 2 x 4 9. 6 1 2 2 4 6 4 y 32 x 12 1 4 10. x 22 y 12 1 9 4 y y 4 4 2 x 4 2 2 4 6 2 4 6 11. x 2 1 x 2 4 6 4 x 42 y 22 1 9 16 12. x 22 y 32 1 4 9 y y 2 4 2 6 8 x 2 8 6 4 2 x 4 2 4 4 In Exercises 1318, identify the center and radius of the circle. 13. x 2 y 2 49 14. x 2 y 2 1 15. x 42 y 52 36 16. x 82 y 12 144 17. x 12 y 2 10 18. x 2 y 122 24 In Exercises 1924, write the equation of the circle in standard form, and then identify its center and radius. 19. 20. 21. 22. 23. 24. x 2 y 2 2x 6y 9 0 x 2 y 2 10x 6y 25 0 x2 y 2 8x 0 2x 2 2y 2 2x 2y 7 0 4x 2 4y 2 12x 24y 41 0 9x 2 9y 2 54x 36y 17 0 In Exercises 2532, find the vertex, focus, and directrix of the parabola, and sketch its graph. 25. 26. 27. 29. 31. 32. x 12 8 y 2 0 x 2 y 42 0 y 12 2 2x 5 28. x 12 2 4 y 3 30. 4x y 2 2y 33 0 y 14x 2 2x 5 2 y 6y 8x 25 0 y 2 4y 4x 0 In Exercises 3338, find the standard form of the equation of the parabola with the given characteristics. 33. 34. 35. 36. 37. 38. 39. Vertex: 3, 2; focus: 1, 2 Vertex: 1, 2; focus: 1, 0 Vertex: 0, 4; directrix: y 2 Vertex: 2, 1; directrix: x 1 Focus: 4, 4; directrix: x 4 Focus: 0, 0; directrix: y 4 PROJECTILE MOTION A cargo plane is flying at an altitude of 30,000 feet and a speed of 540 miles per hour 792 feet per second . How many feet will a supply crate dropped from the plane travel horizontally before it hits the ground if the path of the crate is modeled by x2 39,204 y 30,000 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 368 Chapter 4 Rational Functions and Conics 40. PATH OF A PROJECTILE The path of a softball is modeled by 12.5 y 7.125 x 6.252. The coordinates x and y are measured in feet, with x 0 corresponding to the position from which the ball was thrown. a Use a graphing utility to graph the trajectory of the softball. b Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory. 41. SALES The sales S in billions of dollars for Texas Instruments, Inc. for the years 2002 through 2008 are shown in the table. Source: Texas Instruments, Inc. Year Sales, S 2002 2003 2004 2005 2006 2007 2008 8.4 9.8 12.6 13.4 14.3 13.8 12.5 Circular orbit y Parabolic path 4100 miles x Not drawn to scale FIGURE FOR 42 a Find the escape velocity of the satellite. b Find an equation of its path assume that the radius of Earth is 4000 miles . In Exercises 4350, find the center, foci, and vertices of the ellipse, and sketch its graph. x 12 y 52 1 9 25 x 62 y 72 44. 1 4 16 y 42 45. x 22 1 14 x 32 46. y 82 1 259 47. 9x 2 25y 2 36x 50y 52 0 48. 16x 2 25y 2 32x 50y 16 0 49. 9x 2 4y 2 36x 24y 36 0 50. 9x 2 4y 2 36x 8y 31 0 43. a Use a graphing utility to find an equation of the parabola y at2 bt c that models the data. Write the equation in standard form. Let t represent the year, with t 2 corresponding to 2002. b Find the coordinates of the vertex and interpret its meaning in the context of the problem. c Use a graphing utility to graph the function. d Use the trace feature of the graphing utility to approximate graphically the year in which sales were maximum. e Use the table feature of the graphing utility to approximate numerically the year in which sales were maximum. f Compare the results of parts b , d , and e . What did you learn by using all three approaches 42. SATELLITE ORBIT A satellite in a 100 mile high circular orbit around Earth has a velocity of approximately 17,500 miles per hour see figure . If this velocity is multiplied by 2, the satellite will have the minimum velocity necessary to escape Earths gravity and it will follow a parabolic path with the center of Earth as the focus. In Exercises 5158, find the standard form of the equation of the ellipse with the given characteristics. Vertices: 3, 3, 3, 3; minor axis of length 2 Vertices: 2, 3, 6, 3; minor axis of length 6 Foci: 0, 0, 4, 0; major axis of length 8 Foci: 0, 0, 0, 8; major axis of length 16 Center: 0, 4; a 2c; vertices: 4, 4, 4, 4 Center: 3, 2; a 3c; foci: 1, 2, 5, 2 Vertices: 0, 2, 4, 2; endpoints of the minor axis: 2, 3, 2, 1 58. Vertices: 5, 0, 5, 12; endpoints of the minor axis: 0, 6, 10, 6 51. 52. 53. 54. 55. 56. 57. In Exercises 59 and 60, e is called the eccentricity of an ellipse, and is defined by e ca. It measures the flatness of the ellipse. 59. Find the standard form of the equation of the ellipse with vertices 5, 0 and eccentricity e 35. 60. Find the standard form of the equation of the ellipse with vertices 0, 8 and eccentricity e 12. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 4.4 61. PLANETARY MOTION The dwarf planet Pluto moves in an elliptical orbit with the sun at one of the foci, as shown in the figure. The length of half of the major axis, a, is 3.67 109 miles, and the eccentricity is 0.249. Find the smallest distance perihelion and the greatest distance aphelion of Pluto from the center of the sun. y Pluto x Sun a Not drawn to scale 62. AUSTRALIAN FOOTBALL In Australia, football by Australian Rules or rugby is played on elliptical fields. The field can be a maximum of 155 meters wide and a maximum of 185 meters long. Let the center of a field of maximum size be represented by the point 0, 77.5. Write the standard form of the equation of the ellipse that represents this field. Source: Australian Football League In Exercises 6370, find the center, foci, and vertices of the hyperbola, and sketch its graph, using the asymptotes as an aid. 63. 64. 65. 66. 67. 68. 69. 70. x 22 y 12 1 16 9 x 12 y 42 1 144 25 y 62 x 22 1 y 12 x 32 1 14 19 x 2 9y 2 2x 54y 85 0 16y 2 x 2 2x 64y 62 0 9x 2 y 2 36x 6y 18 0 x 2 9y 2 36y 72 0 Vertices: Vertices: Vertices: Vertices: Vertices: Vertices: Vertices: Vertices: 369 In Exercises 7988, identify the conic by writing its equation in standard form. Then sketch its graph. 79. 80. 81. 83. 84. 85. 86. 87. 88. x 2 y 2 6x 4y 9 0 x 2 4y 2 6x 16y 21 0 y 2 x 2 4y 0 82. y 2 4y 4x 0 2 16y 128x 8y 7 0 4x 2 y 2 4x 3 0 9x2 16y 2 36x 128y 148 0 25x 2 10x 200y 119 0 16x2 16y 2 16x 24y 3 0 4x 2 3y 2 8x 24y 51 0 EXPLORATION TRUE OR FALSE In Exercises 89 and 90, determine whether the statement is true or false. Justify your answer. 89. The conic represented by the equation 3x 2 2y 2 18x 16y 58 0 is an ellipse. 90. The graphs of x 2 10y 10x 5 0 and x 2 16y 2 10x 32y 23 0 do not intersect. x2 y2 2 2 1. a b a Show that the equation of the ellipse can be written as 91. Consider the ellipse x h2 y k2 2 1 2 a a 1 e2 where e is the eccentricity see Exercises 59 and 60 . b Use a graphing utility to graph the ellipse x 22 y 32 1 4 41 e2 for e 0.95, 0.75, 0.5, 0.25, and 0. Make a conjecture about the change in the shape of the ellipse as e approaches 0. In Exercises 7178, find the standard form of the equation of the hyperbola with the given characteristics. 71. 72. 73. 74. 75. 76. 77. 78. Translations of Conics 0, 2, 0, 0; foci: 0, 3, 0, 1 1, 2, 5, 2; foci: 0, 2, 6, 2 2, 0, 6, 0; foci: 0, 0, 8, 0 2, 3, 2, 3; foci: 2, 5, 2, 5 2, 3, 2, 3; passes through the point 0, 5 2, 1, 2, 1; passes through the point 4, 3 0, 2, 6, 2; asymptotes: y 23x, y 4 23x 3, 0, 3, 4; asymptotes: y 23x, y 4 23x 92. CAPSTONE equations. Compare the graphs of the following a x 12 y 22 1 16 4 b x 12 y 22 1 4 16 c x 12 y 22 1 16 4 d x 12 y 22 1 4 16 e x 12 y 22 1 16 16 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 370 Chapter 4 Rational Functions and Conics What Did You Learn ExplanationExamples Find the domains of rational functions p. 332 . A rational function is a quotient of polynomial functions. It can be written in the form f x NxDx, where Nx and Dx are polynomials and Dx is not the zero polynomial. In general, the domain of a rational function of x includes all real numbers except x values that make the denominator zero. 14 Find the vertical and horizontal asymptotes of graphs of rational functions p. 333 . The line x a is a vertical asymptote of the graph of f if f x or f x as x a, either from the right or from the left. The line y b is a horizontal asymptote of the graph of f if f x b as x or x . 510 Use rational functions to model and solve real life problems p. 335 . A rational function can be used to model the cost of removing a given percent of smokestack pollutants at a utility company that burns coal. See Example 4. 11, 12 Analyze and sketch graphs of rational functions p. 340 . Let f x NxDx, where Nx and Dx are polynomials. 1. Simplify f, if possible. 2. Find and plot the y intercept if any by evaluating f 0. 3. Find the zeros of the numerator if any by solving the equation Nx 0. Then plot the corresponding x intercepts. 4. Find the zeros of the denominator if any by solving Dx 0. Then sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote if any . 6. Plot at least one point between and one point beyond each x intercept and vertical asymptote. 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes. 1324 Sketch graphs of rational functions that have slant asymptotes p. 343 . Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant asymptote. 2530 Use graphs of rational functions to model and solve real life problems p. 344 . The graph of a rational function can be used to model the printed area of a rectangular page that is to be minimized, and to find the page dimensions so that the least amount of paper is used. See Example 6. 3134 Recognize the four basic conics: circle, ellipse, parabola, and hyperbola p. 349 . A conic section or simply conic is the intersection of a plane and a double napped cone. 3542 Review Exercises Section 4.3 Section 4.2 Section 4.1 4 CHAPTER SUMMARY Circle Ellipse Parabola www.elsolucionario.net Hyperbola http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Summary What Did You Learn ExplanationExamples Recognize, graph, and write equations of parabolas vertex at origin p. 350 . The standard form of the equation of a parabola with vertex at 0, 0 and directrix y p is x2 4py, p 0. Vertical axis For directrix x p, the equation is y2 4px, p 0. Horizontal axis The focus is on the axis p units from the vertex. 4350 Recognize, graph, and write equations of ellipses center at origin p. 352 . The standard form of the equation of an ellipse centered at the origin with major and minor axes of lengths 2a and 2b where 0 b a is x2 y2 Major axis is horizontal. 2 2 1 a b x2 y2 or Major axis is vertical. 2 2 1. b a The vertices and foci lie on the major axis, a and c units, respectively, from the center. Moreover, a, b, and c are related by the equation c2 a2 b2. 5158 Recognize, graph, and write equations of hyperbolas center at origin p. 354 . The standard form of the equation of a hyperbola with center at the origin where a 0 and b 0 is x2 y2 21 Transverse axis is horizontal. 2 a b 2 2 y x or 1. Transverse axis is vertical. a2 b2 The vertices and foci are, respectively, a and c units from the center. Moreover, a, b, and c are related by the equation b2 c2 a2. 5962 Recognize equations of conics that have been shifted vertically or horizontally in the plane p. 362 and write and graph equations of conics that have been shifted vertically or horizontally in the plane p. 364 . Circle: The graph of x 22 y 12 52 is a circle whose center is the point 2, 1 and whose radius is 5. The graph has been shifted two units to the right and one unit downward from standard position. x 12 y 22 Ellipse: The graph of 1 is an ellipse 42 32 whose center is the point 1, 2. The major axis is horizontal and of length 8, and the minor axis is vertical and of length 6. The graph has been shifted one unit to the right and two units upward from standard position. x 52 y 42 Hyperbola: The graph of 1 is a 12 22 hyperbola whose center is the point 5, 4. The transverse axis is horizontal and of length 2, and the conjugate axis is vertical and of length 4. The graph has been shifted five units to the right and four units upward from standard position. Parabola: The graph of x 12 41 y 6 is a parabola whose vertex is the point 1, 6. The axis of the parabola is vertical. Because p 1, the focus lies below the vertex. The graph has been reflected in the x axis, shifted one unit to the right and six units upward from standard position. 6387 Section 4.3 Section 4.4 371 www.elsolucionario.net Review Exercises http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 372 Chapter 4 Rational Functions and Conics 4 REVIEW EXERCISES 4.1 In Exercises 14, find the domain of the rational function. 1. f x 3x x 10 2. f x 4x3 2 5x 3. f x 8 x 2 10x 24 4. f x x2 x 2 x2 4 In Exercises 510, identify any vertical and horizontal asymptotes. 5. f x 4 x3 x2 7. gx 2 x 4 5x 20 9. hx 2 x 2x 24 6. f x 2x 2 5x 3 x2 2 1 8. gx x 32 x3 4x2 10. hx 2 x 3x 2 11. AVERAGE COST A business has a production cost of C 0.5x 500 for producing x units of a product. The average cost per unit, C, is given by C C 0.5x 500 , x x 528p , 0 p 100. 100 p a Use a graphing utility to graph the cost function. b Find the costs of seizing 25, 50, and 75 of the drug. c According to this model, would it be possible to seize 100 of the drug 4.2 In Exercises 1324, a state the domain of the function, b identify all intercepts, c find any vertical and horizontal asymptotes, and d plot additional solution points as needed to sketch the graph of the rational function. 3 2x 2 2x 15. gx 1x 5x2 17. px 2 4x 1 13. f x 4 x x4 16. hx x7 2x 18. f x 2 x 4 14. f x x x2 1 6x 2 21. f x 2 x 1 19. f x 23. f x 6x2 11x 3 3x2 x 9 x 32 2x 2 22. y 2 x 4 6x2 7x 2 24. f x 4x2 1 20. hx In Exercises 2530, a state the domain of the function, b identify all intercepts, c identify any vertical and slant asymptotes, and d plot additional solution points as needed to sketch the graph of the rational function. 25. f x 2x3 x2 1 26. f x x2 1 x1 27. f x x 2 3x 10 x2 28. f x x3 x 25 29. f x 3x3 2x2 3x 2 3x2 x 4 30. f x 3x3 4x2 12x 16 3x2 5x 2 2 31. AVERAGE COST The cost of producing x units of a product is C, and the average cost per unit C is given by x 0. Determine the average cost per unit as x increases without bound. Find the horizontal asymptote. 12. SEIZURE OF ILLEGAL DRUGS The cost C in millions of dollars for the federal government to seize p of an illegal drug as it enters the country is given by C See www.CalcChat.com for worked out solutions to odd numbered exercises. C C 100,000 0.9x , x 0. x x a Graph the average cost function. b Find the average costs of producing x 1000, 10,000, and 100,000 units. c By increasing the level of production, what is the smallest average cost per unit you can obtain Explain your reasoning. 32. PAGE DESIGN A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom margins are 2 inches deep and the margins on each side are 2 inches wide. a Draw a diagram that gives a visual representation of the problem. b Show that the total area A of the page is A 2x2x 7 . x4 c Determine the domain of the function based on the physical constraints of the problem. d Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 373 Review Exercises 33. PHOTOSYNTHESIS The amount y of CO2 uptake in milligrams per square decimeter per hour at optimal temperatures and with the natural supply of CO2 is approximated by the model 50. SUSPENSION BRIDGE Each cable of a suspension bridge is suspended in the shape of a parabola between two towers see figure . y 18.47x 2.96 y , x 0 0.23x 1 30 20 where x is the light intensity in watts per square meter . Use a graphing utility to graph the function and determine the limiting amount of CO2 uptake. 34. MEDICINE The concentration C of a medication in the bloodstream t hours after injection into muscle tissue is given by Ct 2t 1t2 4, t 0. a Determine the horizontal asymptote of the graph of the function and interpret its meaning in the context of the problem. b Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest. 4.3 In Exercises 3542, identify the conic. 35. y 2 16x 37. 39. x2 64 x2 y2 4 36. 16x 2 y 2 16 1 20y 0 y2 x2 1 41. 49 144 38. x2 y2 1 1 36 40. x2 y2 400 x2 y2 1 42. 49 144 In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic s and vertex at the origin. 10 60 40 20 x 20 40 60 10 a Find the coordinates of the focus. b Write an equation that models the cables. In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 51. 52. 53. 54. 55. 56. Vertices: 9, 0; minor axis of length 6 Vertices: 0, 10; minor axis of length 2 Vertices: 0, 6; passes through the point 2, 2 Vertices: 7, 0; foci: 6, 0 Foci: 14, 0; minor axis of length 10 Foci: 3, 0; major axis of length 12 57. ARCHITECTURE A semielliptical archway is to be formed over the entrance to an estate see figure . The arch is to be set on pillars that are 10 feet apart and is to have a height atop the pillars of 4 feet. Where should the foci be placed in order to sketch the arch 43. Passes through the point 3, 6; horizontal axis 44. Passes through the point 4, 2; vertical axis 45. Focus: 6, 0 46. Focus: 0, 7 47. Directrix: y 3 60, 20 60, 20 4 ft 10 ft 48. Directrix: x 3 49. SATELLITE ANTENNA A cross section of a large parabolic antenna see figure is modeled by y x2200, 100 x 100. The receiving and transmitting equipment is positioned at the focus. Find the coordinates of the focus. y 100 50 y2 x2 1. 324 196 Find the longest distance across the pool, the shortest distance, and the distance between the foci. 150 2 y = x 100 200 58. WADING POOL You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as Focus x 50 100 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 374 Chapter 4 Rational Functions and Conics In Exercises 5962, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 59. Vertices: 0, 1; foci: 0, 5 60. Vertices: 4, 0; foci: 6, 0 61. Vertices: 1, 0; asymptotes: y 2x 62. Vertices: 0, 2; asymptotes: y 85. ARCHITECTURE A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters see figure . How wide is the archway at ground level y 4, 10 2 x 5 y 8 ft 0, 12 4, 10 d x x 8 ft 4.4 In Exercises 6366, find the standard form of the equation of the parabola with the given characteristics. 63. 64. 65. 66. Vertex: 8, 8; directrix: y 1 Focus: 0, 5; directrix: x 6 Vertex: 4, 2; focus: 4, 0 Vertex: 2, 0; focus: 0, 0 FIGURE FOR In Exercises 6770, find the standard form of the equation of the ellipse with the given characteristics. 67. 68. 69. 70. Vertices: 0, 3, 12, 3; passes through the point 6, 0 Center: 0, 4; vertices: 0, 0, 0, 8 Vertices: 3, 0, 7, 0; foci: 0, 0, 4, 0 Vertices: 2, 0, 2, 4; foci: 2, 1, 2, 3 In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 71. Vertices: 6, 7; 1 1 asymptotes: y 2 x 7, y 2 x 7 72. Vertices: 0, 0, 0, 4; passes through the point 2, 25 1 73. Vertices: 10, 3, 6, 3; foci: 12, 3, 8, 3 74. Vertices: 2, 2, 2, 2; foci: 4, 2, 4, 2 75. Foci: 0, 0, 8, 0; asymptotes: y 2x 4 76. Foci: 3, 2; asymptotes: y 2x 3 In Exercises 77 84, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. 77. 78. 79. 80. 81. 82. 83. 84. x 2 6x 2y 9 0 y 2 12y 8x 20 0 x 2 y 2 2x 4y 5 0 16x 2 16y 2 16x 24y 3 0 x 2 9y 2 10x 18y 25 0 4x 2 y 2 16x 15 0 9x 2 y 2 72x 8y 119 0 x 2 9y 2 10x 18y 7 0 4 ft 85 FIGURE FOR 86 86. ARCHITECTURE A church window see figure is bounded above by a parabola and below by the arc of a circle. a Find equations for the parabola and the circle. b Complete the table by filling in the vertical distance d between the circle and the parabola for each given value of x. x 0 1 2 3 4 d 87. RUNNING PATH Let 0, 0 represent a water fountain located in a city park. Each day you run through the park along a path given by x2 y2 200x 52,500 0 where x and y are measured in meters. a What type of conic is your path Explain your reasoning. b Write the equation of the path in standard form. Sketch a graph of the equation. c After you run, you walk to the water fountain. If you stop running at 100, 150, how far must you walk for a drink of water EXPLORATION TRUE OR FALSE In Exercises 88 and 89, determine whether the statement is true or false. Justify your answer. 88. The domain of a rational function can never be the set of all real numbers. 89. The graph of the equation Ax 2 Bxy Cy 2 Dx Ey F 0 can be a single point. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Chapter Test 4 CHAPTER TEST 375 See www.CalcChat.com for worked out solutions to odd numbered exercises. Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 13, find the domain of the function and identify any asymptotes. 1. y 3x x1 2. f x 3 x2 3 x2 3. gx x 2 7x 12 x3 In Exercises 49, identify any intercepts and asymptotes of the graph of the function. Then sketch a graph of the function. 0, y 2, 1 1 FIGURE FOR x x, 0 2 0, 16 6, 14 x 8 8 FIGURE FOR x1 x2 x 12 7. f x 2x2 5x 12 x2 16 8. f x 2x2 9 5x2 9 9. gx 2x3 7x2 4x 4 x2 x 2 12. y 2 4x 0 14. x2 10x 2y 19 0 y2 15. x 2 1 4 17. x2 3y2 2x 36y 100 0 16 20 767,640 km Earth Perigee FIGURE FOR 6. f x In Exercises 1217, graph the conic and identify the center, vertices, and foci, if applicable. 8 8 x2 2 x1 2 . x2 b Write the area A of the triangle as a function of x. Determine the domain of the function in the context of the problem. c Graph the area function. Estimate the minimum area of the triangle from the graph. y 24 5. gx a Verify that y 1 11 6, 14 4 1 x2 10. A rectangular page is designed to contain 36 square inches of print. The margins at the top and bottom of the page are 2 inches deep. The margins on each side are 1 inch wide. What should the dimensions of the page be so that the least amount of paper is used 11. A triangle is formed by the coordinate axes and a line through the point 2, 1, as shown in the figure. y 1 4. hx 21 768,800 km Apogee Moon 13. x2 y2 10x 4y 4 0 16. y2 x2 1 4 18. Find an equation of the ellipse with vertices 0, 2 and 8, 2 and minor axis of length 4. 19. Find an equation of the hyperbola with vertices 0, 3 and asymptotes y 32x. 20. A parabolic archway is 16 meters high at the vertex. At a height of 14 meters, the width of the archway is 12 meters, as shown in the figure. How wide is the archway at ground level 21. The moon orbits Earth in an elliptical path with the center of Earth at one focus, as shown in the figure. The major and minor axes of the orbit have lengths of 768,800 kilometers and 767,640 kilometers, respectively. Find the smallest distance perigee and the greatest distance apogee from the center of the moon to the center of Earth. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROOFS IN MATHEMATICS You can use the definition of a parabola to derive the standard form of the equation of a parabola whose directrix is parallel to the x axis or to the y axis. Standard Equation of a Parabola Vertex at Origin p. 350 Parabolic Patterns The standard form of the equation of a parabola with vertex at 0, 0 and directrix y p is There are many natural occurrences of parabolas in real life. For instance, the famous astronomer Galileo discovered in the 17th century that an object that is projected upward and obliquely to the pull of gravity travels in a parabolic path. Examples of this are the center of gravity of a jumping dolphin and the path of water molecules of a drinking water fountain. x 2 4py, p 0. Vertical axis For directrix x p, the equation is y 2 4px, p 0. Horizontal axis The focus is on the axis p units directed distance from the vertex. Proof For the first case, suppose the directrix y p is parallel to the x axis. In the figure, you assume that p 0, and because p is the directed distance from the vertex to the focus, the focus must lie above the vertex. Because the point x, y is equidistant from 0, p and y p, you can apply the Distance Formula to obtain x 02 y p2 y p y Focus: 0, p 2 2 2 Square each side. 2 x 4py. x, y x p Directrix: y = p Parabola with vertical axis Simplify. x p2 y 02 x p Distance Formula x p y x p 2 x2 x, y Focus: p, 0 Expand. A proof of the second case is similar to the proof of the first case. Suppose the directrix x p is parallel to the y axis. In the figure, you assume that p 0, and because p is the directed distance from the vertex to the focus, the focus must lie to the right of the vertex. Because the point x, y is equidistant from p, 0 and x p, you can apply the Distance Formula as follows. y p 2 2 p Vertex: 0, 0 2 x y 2py p y 2py p 2 Vertex: 0, 0 Distance Formula x y p y p 2 2px p2 2 2 y2 y2 4px x2 2px Square each side. p2 Expand. Simplify. x p Directrix: x = p Parabola with horizontal axis 376 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com PROBLEM SOLVING This collection of thought provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Match the graph of the rational function given by Object blurry ax b f x cx d Object clear Near point with the given conditions. a b y Far point y FIGURE FOR x c 3 x d y y x i a 0 ii a 0 iii b 0 b 0 c 0 c 0 d 0 d 0 2. Consider the function given by f x Object blurry x a b c d 0 0 0 0 iv a b c d 0 0 0 a Determine the effect on the graph of f if b 0 and a is varied. Consider cases in which a is positive and a is negative. b Determine the effect on the graph of f if a 0 and b is varied. 3. The endpoints of the interval over which distinct vision is possible is called the near point and far point of the eye see figure . With increasing age, these points normally change. The table shows the approximate near points y in inches for various ages x in years . Near point, y 16 32 44 50 60 3.0 4.7 9.8 19.7 39.4 a Use the regression feature of a graphing utility to find a quadratic model for the data. Use a graphing utility to plot the data and graph the model in the same viewing window. b Find a rational model for the data. Take the reciprocals of the near points to generate the points x, 1y. Use the regression feature of a graphing utility to find a linear model for the data. The resulting line has the form 1 ax b. y 0 ax . x b2 Age, x Solve for y. Use a graphing utility to plot the data and graph the model in the same viewing window. c Use the table feature of a graphing utility to create a table showing the predicted near point based on each model for each of the ages in the original table. How well do the models fit the original data d Use both models to estimate the near point for a person who is 25 years old. Which model is a better fit e Do you think either model can be used to predict the near point for a person who is 70 years old Explain. 377 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 4. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long. a Find an equation that models the shape of the room. b How far apart are the two foci c What is the area of the floor of the room The area of an ellipse is A ab. 5. Use the figure to show that d2 d1 2a. y d2 y y1 x1 x x1. 2p a What is the slope of the tangent line b For each parabola in Exercise 8, find the equations of the tangent lines at the endpoints of the chord. Use a graphing utility to graph the parabola and tangent lines. 10. A tour boat travels between two islands that are 12 miles apart see figure . For each trip between the islands, there is enough fuel for a 20 mile trip. x , y d1 x c, 0 c For each parabola in part a , find the length of the chord passing through the focus and parallel to the directrix. How can the length of this chord be determined directly from the standard form of the equation of the parabola d Explain how the result of part c can be used as a sketching aid when graphing parabolas. 9. Let x1, y1 be the coordinates of a point on the parabola x2 4py. The equation of the line that just touches the parabola at the point x1, y1, called a tangent line, is given by c, 0 a, 0 a, 0 6. Find an equation of a hyperbola such that for any point on the hyperbola, the difference between its distances from the points 2, 2 and 10, 2 is 6. 7. The filament of a light bulb is a thin wire that glows when electricity passes through it. The filament of a car headlight is at the focus of a parabolic reflector, which sends light out in a straight beam. Given that the filament is 1.5 inches from the vertex, find an equation for the cross section of the reflector. A reflector is 7 inches wide. How deep is it 7 in. 1.5 in. 8. Consider the parabola x 2 4py. a Use a graphing utility to graph the parabola for p 1, p 2, p 3, and p 4. Describe the effect on the graph when p increases. b Locate the focus for each parabola in part a . Island 1 Island 2 12 mi Not drawn to scale a Explain why the region in which the boat can travel is bounded by an ellipse. b Let 0, 0 represent the center of the ellipse. Find the coordinates of the center of each island. c The boat travels from one island, straight past the other island to one vertex of the ellipse, and back to the second island. How many miles does the boat travel Use your answer to find the coordinates of the vertex. d Use the results of parts b and c to write an equation of the ellipse that bounds the region in which the boat can travel. 11. Prove that the graph of the equation Ax 2 Cy 2 Dx Ey F 0 is one of the following except in degenerate cases . a b c d Conic Circle Parabola Ellipse Hyperbola Condition AC A 0 or C 0 but not both AC 0 AC 0 378 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 5 Exponential and Logarithmic Functions 5.1 Exponential Functions and Their Graphs 5.2 Logarithmic Functions and Their Graphs 5.3 Properties of Logarithms 5.4 Exponential and Logarithmic Equations 5.5 Exponential and Logarithmic Models In Mathematics Exponential functions involve a constant base and a variable exponent. The inverse of an exponential function is a logarithmic function. Exponential and logarithmic functions are widely used in describing economic and physical phenomena such as compound interest, population growth, memory retention, and decay of radioactive material. For instance, a logarithmic function can be used to relate an animals weight and its lowest galloping speed. See Exercise 95, page 406. Juniors Bildarchiv Alamy In Real Life IN CAREERS There are many careers that use exponential and logarithmic functions. Several are listed below. Astronomer Example 7, page 404 Archeologist Example 3, page 422 Psychologist Exercise 136, page 417 Forensic Scientist Exercise 75, page 430 379 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 380 Chapter 5 Exponential and Logarithmic Functions 5.1 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Monkey Business Images LtdStockbrokerPhotoLibrary Exponential functions can be used to model and solve real life problems. For instance, in Exercise 76 on page 390, an exponential function is used to model the concentration of a drug in the bloodstream. So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functionsexponential functions and logarithmic functions. These functions are examples of transcendental functions. Definition of Exponential Function The exponential function f with base a is denoted by f x a x 1, and x is any real number. where a 0, a The base a 1 is excluded because it yields f x 1x 1. This is a constant function, not an exponential function. You have evaluated a x for integer and rational values of x. For example, you know that 43 64 and 412 2. However, to evaluate 4x for any real number x, you need to interpret forms with irrational exponents. For the purposes of this text, it is sufficient to think of a2 where 2 1.41421356 as the number that has the successively closer approximations a1.4, a1.41, a1.414, a1.4142, a1.41421, . . . . Example 1 Evaluating Exponential Functions Use a calculator to evaluate each function at the indicated value of x. Function a. f x 2 x b. f x 2x c. f x 0.6x Value x 3.1 x 3 x2 Solution Function Value a. f 3.1 23.1 b. f 2 3 c. f 2 0.632 Graphing Calculator Keystrokes 3.1 ENTER 2 ENTER 2 Why you should learn it Exponential Functions Recognize and evaluate exponential functions with base a. Graph exponential functions and use the One to One Property. Recognize, evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real life problems. .6 What you should learn 3 2 ENTER Display 0.1166291 0.1133147 0.4647580 Now try Exercise 7. When evaluating exponential functions with a calculator, remember to enclose fractional exponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.1 Exponential Functions and Their Graphs 381 Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics, as shown in Examples 2, 3, and 5. Example 2 Graphs of y a x In the same coordinate plane, sketch the graph of each function. a. f x 2x You can review the techniques for sketching the graph of an equation in Section 1.1. y b. gx 4x Solution The table below lists some values for each function, and Figure 5.1 shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of gx 4x is increasing more rapidly than the graph of f x 2x. g x = 4x 3 2 1 0 1 2 2 x 1 8 1 4 1 2 1 2 4 4x 1 64 1 16 1 4 1 4 16 x 16 14 12 10 8 6 Now try Exercise 17. 4 f x = 2x 2 x 4 3 2 1 2 FIGURE 1 2 3 4 The table in Example 2 was evaluated by hand. You could, of course, use a graphing utility to construct tables with even more values. Example 3 5.1 G x = 4 x Graphs of y ax In the same coordinate plane, sketch the graph of each function. y a. Fx 2x 16 14 b. Gx 4x Solution 12 The table below lists some values for each function, and Figure 5.2 shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of Gx 4x is decreasing more rapidly than the graph of Fx 2x. 10 8 6 4 F x = 4 3 2 1 2 FIGURE 2 1 0 1 2 3 2x 4 2 1 1 2 1 4 1 8 4x 16 4 1 1 4 1 16 1 64 x 2 x x 1 2 3 4 5.2 Now try Exercise 19. In Example 3, note that by using one of the properties of exponents, the functions F x 2x and Gx 4x can be rewritten with positive exponents. F x 2x 1 1 2x 2 x and Gx 4x www.elsolucionario.net 1 1 4x 4 x http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 382 Chapter 5 Exponential and Logarithmic Functions Comparing the functions in Examples 2 and 3, observe that Fx 2x f x and Gx 4x gx. Consequently, the graph of F is a reflection in the y axis of the graph of f. The graphs of G and g have the same relationship. The graphs in Figures 5.1 and 5.2 are typical of the exponential functions y a x and y ax. They have one y intercept and one horizontal asymptote the x axis , and they are continuous. The basic characteristics of these exponential functions are summarized in Figures 5.3 and 5.4. y Notice that the range of an exponential function is 0, , which means that a x 0 for all values of x. y = ax 0, 1 x FIGURE 5.3 y y = a x 0, 1 x FIGURE Graph of y a x, a 1 Domain: , Range: 0, y intercept: 0, 1 Increasing x axis is a horizontal asymptote ax 0 as x . Continuous Graph of y ax, a 1 Domain: , Range: 0, y intercept: 0, 1 Decreasing x axis is a horizontal asymptote ax 0 as x . Continuous 5.4 From Figures 5.3 and 5.4, you can see that the graph of an exponential function is always increasing or always decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one to one functions. You can use the following One to One Property to solve simple exponential equations. For a 0 and a Example 4 a. 9 32 2 1 b. 1 x 2 1, ax ay if and only if x y. One to One Property Using the One to One Property 3x1 3x1 x1 x Original equation 9 32 One to One Property Solve for x. 8 2x 23 x 3 Now try Exercise 51. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.1 383 Exponential Functions and Their Graphs In the following example, notice how the graph of y a x can be used to sketch the graphs of functions of the form f x b a xc. Example 5 You can review the techniques for transforming the graph of a function in Section 2.5. Transformations of Graphs of Exponential Functions Each of the following graphs is a transformation of the graph of f x 3x. a. Because gx 3x1 f x 1, the graph of g can be obtained by shifting the graph of f one unit to the left, as shown in Figure 5.5. b. Because hx 3x 2 f x 2, the graph of h can be obtained by shifting the graph of f downward two units, as shown in Figure 5.6. c. Because kx 3x f x, the graph of k can be obtained by reflecting the graph of f in the x axis, as shown in Figure 5.7. d. Because j x 3x f x, the graph of j can be obtained by reflecting the graph of f in the y axis, as shown in Figure 5.8. y y 2 3 f x = 3 x g x = 3 x + 1 1 2 x 2 1 2 FIGURE 1 f x = 3 x h x = 3 x 2 2 1 5.5 Horizontal shift FIGURE 5.6 Vertical shift y y 2 1 4 3 f x = 3 x x 2 1 1 2 k x = 3 x 2 FIGURE 2 1 x 1 1 5.7 Reflection in x axis 2 j x = 3 x f x = 3 x 1 x 2 1 FIGURE 1 2 5.8 Reflection in y axis Now try Exercise 23. Notice that the transformations in Figures 5.5, 5.7, and 5.8 keep the x axis as a horizontal asymptote, but the transformation in Figure 5.6 yields a new horizontal asymptote of y 2. Also, be sure to note how the y intercept is affected by each transformation. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 384 Chapter 5 Exponential and Logarithmic Functions The Natural Base e y In many applications, the most convenient choice for a base is the irrational number e 2.718281828 . . . . 3 1, e This number is called the natural base. The function given by f x e x is called the natural exponential function. Its graph is shown in Figure 5.9. Be sure you see that for the exponential function f x e x, e is the constant 2.718281828 . . . , whereas x is the variable. 2 f x = e x 1, e 1 0, 1 Example 6 2, e 2 2 FIGURE x 1 1 Use a calculator to evaluate the function given by f x e x at each indicated value of x. a. b. c. d. 5.9 Evaluating the Natural Exponential Function x 2 x 1 x 0.25 x 0.3 Solution y a. b. c. d. 8 f x = 2e 0.24x 7 6 5 Function Value f 2 e2 f 1 e1 f 0.25 e0.25 f 0.3 e0.3 Graphing Calculator Keystrokes ex 2 ENTER ex 1 ENTER ex 0.25 ENTER ex 0.3 ENTER Display 0.1353353 0.3678794 1.2840254 0.7408182 Now try Exercise 33. 4 3 Example 7 Graphing Natural Exponential Functions 1 x 4 3 2 1 FIGURE 1 2 3 4 Sketch the graph of each natural exponential function. a. f x 2e0.24x b. gx 12e0.58x 5.10 Solution y To sketch these two graphs, you can use a graphing utility to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figures 5.10 and 5.11. Note that the graph in Figure 5.10 is increasing, whereas the graph in Figure 5.11 is decreasing. 8 7 6 5 4 2 g x = 12 e 0.58x 1 4 3 2 1 FIGURE 5.11 3 2 1 0 1 2 3 f x 0.974 1.238 1.573 2.000 2.542 3.232 4.109 gx 2.849 1.595 0.893 0.500 0.280 0.157 0.088 x 3 x 1 2 3 4 Now try Exercise 41. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.1 Exponential Functions and Their Graphs 385 Applications One of the most familiar examples of exponential growth is an investment earning continuously compounded interest. On page 135 in Section 1.6, you were introduced to the formula for the balance in an account that is compounded n times per year. Using exponential functions, you can now develop that formula and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once per year. If the interest is added to the principal at the end of the year, the new balance P1 is P1 P Pr P1 r. This pattern of multiplying the previous principal by 1 r is then repeated each successive year, as shown below. Year 0 1 2 3 .. . t Balance After Each Compounding PP P1 P1 r P2 P11 r P1 r1 r P1 r2 P3 P21 r P1 r21 r P1 r3 .. . Pt P1 rt To accommodate more frequent quarterly, monthly, or daily compounding of interest, let n be the number of compoundings per year and let t be the number of years. Then the rate per compounding is rn and the account balance after t years is AP 1 m 1 1 m m r n . nt Amount balance with n compoundings per year If you let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per year, let m nr. This produces r n P 1 r mr 1 m AP 1 1 2 10 2.59374246 100 2.704813829 1,000 2.716923932 P 1 10,000 2.718145927 100,000 2.718268237 1,000,000 2.718280469 10,000,000 2.718281693 e P nt Amount with n compoundings per year mrt Substitute mr for n. mrt Simplify. 1 m . 1 m rt Property of exponents As m increases without bound, the table at the left shows that 1 1mm e as m . From this, you can conclude that the formula for continuous compounding is A Pert. Substitute e for 1 1mm. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 386 Chapter 5 Exponential and Logarithmic Functions WARNING CAUTION Be sure you see that the annual interest rate must be written in decimal form. For instance, 6 should be written as 0.06. Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r in decimal form is given by the following formulas. 1. For n compoundings per year: A P 1 r n nt 2. For continuous compounding: A Pe rt Example 8 Compound Interest A total of 12,000 is invested at an annual interest rate of 9. Find the balance after 5 years if it is compounded a. quarterly. b. monthly. c. continuously. Solution a. For quarterly compounding, you have n 4. So, in 5 years at 9, the balance is AP 1 r n nt Formula for compound interest 12,000 1 0.09 4 4 5 Substitute for P, r, n, and t. 18,726.11. Use a calculator. b. For monthly compounding, you have n 12. So, in 5 years at 9, the balance is AP 1 r n nt 12,000 1 Formula for compound interest 0.09 12 12 5 18,788.17. Substitute for P, r, n, and t. Use a calculator. c. For continuous compounding, the balance is A Pe rt 12,000e0.09 5 18,819.75. Formula for continuous compounding Substitute for P, r, and t. Use a calculator. Now try Exercise 59. In Example 8, note that continuous compounding yields more than quarterly or monthly compounding. This is typical of the two types of compounding. That is, for a given principal, interest rate, and time, continuous compounding will always yield a larger balance than compounding n times per year. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.1 Example 9 387 Exponential Functions and Their Graphs Radioactive Decay The half life of radioactive radium 226Ra is about 1599 years. That is, for a given amount of radium, half of the original amount will remain after 1599 years. After another 1599 years, one quarter of the original amount will remain, and so on. Let y represent the mass, in grams, of a quantity of radium. The quantity present after t t1599 years, then, is y 2512 . a. What is the initial mass when t 0 b. How much of the initial mass is present after 2500 years Graphical Solution Algebraic Solution 12 1 25 2 a. y 25 Use a graphing utility to graph y 2512 t1599 t1599 Write original equation. a. Use the value feature or the zoom and trace features of the graphing utility to determine that when x 0, the value of y is 25, as shown in Figure 5.12. So, the initial mass is 25 grams. b. Use the value feature or the zoom and trace features of the graphing utility to determine that when x 2500, the value of y is about 8.46, as shown in Figure 5.13. So, about 8.46 grams is present after 2500 years. 01599 Substitute 0 for t. 25 Simplify. So, the initial mass is 25 grams. 12 1 25 2 t1599 b. y 25 25 . 12 8.46 Write original equation. 30 30 25001599 Substitute 2500 for t. 1.563 Simplify. Use a calculator. 0 So, about 8.46 grams is present after 2500 years. 5000 0 FIGURE 0 5000 0 5.12 FIGURE 5.13 Now try Exercise 73. CLASSROOM DISCUSSION Identifying Exponential Functions Which of the following functions generated the two tables below Discuss how you were able to decide. What do these functions have in common Are any of them the same If so, explain why. 1 b. f2x 8 2 1 c. f3x 2x3 e. f5x 7 2x f. f6x 82x x a. f1x 2x3 d. f4x 2 7 1 x x 1 0 1 2 3 x 2 1 0 1 2 gx 7.5 8 9 11 15 hx 32 16 8 4 2 Create two different exponential functions of the forms y ab x and y c x d with y intercepts of 0, 3. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 388 Chapter 5 5.1 Exponential and Logarithmic Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. Polynomial and rational functions are examples of ________ functions. Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions. You can use the ________ Property to solve simple exponential equations. The exponential function given by f x e x is called the ________ ________ function, and the base e is called the ________ base. 5. To find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula ________. 6. To find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula ________. SKILLS AND APPLICATIONS In Exercises 712, evaluate the function at the indicated value of x. Round your result to three decimal places. 7. 8. 9. 10. 11. 12. Function f x 0.9x Value x 1.4 f x 2.3x f x 5x 5x f x 23 g x 50002x f x 2001.212x x x x x x 17. f x 12 19. f x 6x 21. f x 2 x1 x 3 2 3 10 1.5 24 y 6 6 4 4 2 x 2 2 4 2 y c 4 2 x 2 6 4 4 13. f x 2x 15. f x 2x 2 4 6 0, 1 4 2 2 30. y 3x 32. y 4x1 2 In Exercises 3338, evaluate the function at the indicated value of x. Round your result to three decimal places. 2 4 x In Exercises 2932, use a graphing utility to graph the exponential function. 29. y 2x 31. y 3x2 1 y 6 x 3 x, gx 3 x 1 4 x, gx 4 x3 2 x, gx 3 2 x 10 x, gx 10 x3 2 d 2 23. f x 24. f x 25. f x 26. f x x 2 0, 2 x 7 7 27. f x 2 , gx 2 28. f x 0.3 x, gx 0.3 x 5 0, 14 0, 1 4 y b 18. f x 12 20. f x 6 x 22. f x 4 x3 3 In Exercises 2328, use the graph of f to describe the transformation that yields the graph of g. In Exercises 1316, match the exponential function with its graph. The graphs are labeled a , b , c , and d . a In Exercises 1722, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 2 14. f x 2x 1 16. f x 2x2 x 4 33. 34. 35. 36. 37. 38. Function hx ex f x e x f x 2e5x f x 1.5e x2 f x 5000e0.06x f x 250e0.05x www.elsolucionario.net x x x x x Value 34 3.2 10 240 6 x 20 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.1 In Exercises 3944, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 39. f x e x 41. f x 3e x4 43. f x 2e x2 4 40. f x e x 42. f x 2e0.5x 44. f x 2 e x5 In Exercises 4550, use a graphing utility to graph the exponential function. 45. y 1.085x 47. st 2e0.12t 49. gx 1 ex 46. y 1.085x 48. st 3e0.2t 50. hx e x2 In Exercises 5158, use the One to One Property to solve the equation for x. 51. 3x1 27 52. 2x3 16 53. 2 32 55. e3x2 e3 2 57. ex 3 e2x 1 x 1 54. 5x2 125 56. e2x1 e4 2 58. ex 6 e5x COMPOUND INTEREST In Exercises 5962, complete the table to determine the balance A for P dollars invested at rate r for t years and compounded n times per year. n 1 2 4 12 365 Continuous A 59. 60. 61. 62. P 1500, r 2, t 10 years P 2500, r 3.5, t 10 years P 2500, r 4, t 20 years P 1000, r 6, t 40 years COMPOUND INTEREST In Exercises 6366, complete the table to determine the balance A for 12,000 invested at rate r for t years, compounded continuously. t 10 20 30 40 50 A 63. r 4 65. r 6.5 64. r 6 66. r 3.5 67. TRUST FUND On the day of a childs birth, a deposit of 30,000 is made in a trust fund that pays 5 interest, compounded continuously. Determine the balance in this account on the childs 25th birthday. Exponential Functions and Their Graphs 389 68. TRUST FUND A deposit of 5000 is made in a trust fund that pays 7.5 interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive 69. INFLATION If the annual rate of inflation averages 4 over the next 10 years, the approximate costs C of goods or services during any year in that decade will be modeled by Ct P1.04 t, where t is the time in years and P is the present cost. The price of an oil change for your car is presently 23.95. Estimate the price 10 years from now. 70. COMPUTER VIRUS The number V of computers infected by a computer virus increases according to the model Vt 100e4.6052t, where t is the time in hours. Find the number of computers infected after a 1 hour, b 1.5 hours, and c 2 hours. 71. POPULATION GROWTH The projected populations of California for the years 2015 through 2030 can be modeled by P 34.696e0.0098t, where P is the population in millions and t is the time in years , with t 15 corresponding to 2015. Source: U.S. Census Bureau a Use a graphing utility to graph the function for the years 2015 through 2030. b Use the table feature of a graphing utility to create a table of values for the same time period as in part a . c According to the model, when will the population of California exceed 50 million 72. POPULATION The populations P in millions of Italy from 1990 through 2008 can be approximated by the model P 56.8e0.0015t, where t represents the year, with t 0 corresponding to 1990. Source: U.S. Census Bureau, International Data Base a According to the model, is the population of Italy increasing or decreasing Explain. b Find the populations of Italy in 2000 and 2008. c Use the model to predict the populations of Italy in 2015 and 2020. 73. RADIOACTIVE DECAY Let Q represent a mass of radioactive plutonium 239Pu in grams , whose halflife is 24,100 years. The quantity of plutonium present 1 t24,100 . after t years is Q 162 a Determine the initial quantity when t 0 . b Determine the quantity present after 75,000 years. c Use a graphing utility to graph the function over the interval t 0 to t 150,000. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 390 Chapter 5 Exponential and Logarithmic Functions 74. RADIOACTIVE DECAY Let Q represent a mass of carbon 14 14C in grams , whose half life is 5715 years. The quantity of carbon 14 present after t years is t5715 Q 1012 . a Determine the initial quantity when t 0 . b Determine the quantity present after 2000 years. c Sketch the graph of this function over the interval t 0 to t 10,000. 75. DEPRECIATION After t years, the value of a wheelchair conversion van that originally cost 30,500 depreciates so that each year it is worth 78 of its value for the previous year. a Find a model for Vt, the value of the van after t years. b Determine the value of the van 4 years after it was purchased. 76. DRUG CONCENTRATION Immediately following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After t hours, the concentration is 75 of the level of the previous hour. a Find a model for Ct, the concentration of the drug after t hours. b Determine the concentration of the drug after 8 hours. 84. Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. a f x x 2ex b gx x23x 85. GRAPHICAL ANALYSIS Use a graphing utility to graph y1 1 1xx and y2 e in the same viewing window. Using the trace feature, explain what happens to the graph of y1 as x increases. 86. GRAPHICAL ANALYSIS Use a graphing utility to graph f x 1 0.5 x x gx e0.5 and in the same viewing window. What is the relationship between f and g as x increases and decreases without bound 87. GRAPHICAL ANALYSIS Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. a y1 2x, y2 x2 b y1 3x, y2 x3 88. THINK ABOUT IT Which functions are exponential a 3x b 3x 2 c 3x d 2x 89. COMPOUND INTEREST Use the formula r n nt EXPLORATION AP 1 TRUE OR FALSE In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer. to calculate the balance of an account when P 3000, r 6, and t 10 years, and compounding is done a by the day, b by the hour, c by the minute, and d by the second. Does increasing the number of compoundings per year result in unlimited growth of the balance of the account Explain. 77. The line y 2 is an asymptote for the graph of f x 10 x 2. 271,801 78. e 99,990 THINK ABOUT IT In Exercises 79 82, use properties of exponents to determine which functions if any are the same. 79. f x 3x2 gx 3x 9 hx 193x 81. f x 164x x2 gx 14 hx 16 22x 90. CAPSTONE The figure shows the graphs of y 2x, y ex, y 10x, y 2x, y ex, and y 10x. Match each function with its graph. The graphs are labeled a through f . Explain your reasoning. 80. f x 4x 12 gx 22x6 hx 644x 82. f x ex 3 gx e3x hx y c 10 b d 8 e 6 a e x3 83. Graph the functions given by y 3x and y 4x and use the graphs to solve each inequality. a 4x 3x b 4x 3x 2 1 f x 1 2 PROJECT: POPULATION PER SQUARE MILE To work an extended application analyzing the population per square mile of the United States, visit this texts website at academic.cengage.com. Data Source: U.S. Census Bureau www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.2 Logarithmic Functions and Their Graphs 391 5.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS What you should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real life problems. Logarithmic Functions In Section 2.7, you studied the concept of an inverse function. There, you learned that if a function is one to onethat is, if the function has the property that no horizontal line intersects the graph of the function more than oncethe function must have an inverse function. By looking back at the graphs of the exponential functions introduced in Section 5.1, you will see that every function of the form f x a x passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. Why you should learn it Logarithmic functions are often used to model scientific observations. For instance, in Exercise 97 on page 400, a logarithmic function is used to model human memory. Definition of Logarithmic Function with Base a For x 0, a 0, and a 1, y loga x if and only if x a y. The function given by f x loga x Read as log base a of x. Ariel SkelleyCorbis is called the logarithmic function with base a. The equations y loga x and x ay are equivalent. The first equation is in logarithmic form and the second is in exponential form. For example, the logarithmic equation 2 log3 9 can be rewritten in exponential form as 9 32. The exponential equation 53 125 can be rewritten in logarithmic form as log5 125 3. When evaluating logarithms, remember that a logarithm is an exponent. This means that loga x is the exponent to which a must be raised to obtain x. For instance, log2 8 3 because 2 must be raised to the third power to get 8. Example 1 Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x. a. f x log2 x, x 32 c. f x log4 x, x 2 Solution a. f 32 log2 32 5 b. f 1 log3 1 0 c. f 2 log4 2 12 1 1 d. f 100 2 log10 100 b. f x log3 x, x 1 1 d. f x log10 x, x 100 because 25 32. because 30 1. because 412 4 2. 1 because 102 101 2 100 . Now try Exercise 23. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 392 Chapter 5 Exponential and Logarithmic Functions The logarithmic function with base 10 is called the common logarithmic function. It is denoted by log10 or simply by log. On most calculators, this function is denoted by LOG . Example 2 shows how to use a calculator to evaluate common logarithmic functions. You will learn how to use a calculator to calculate logarithms to any base in the next section. Example 2 Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function given by f x log x at each value of x. b. x 13 a. x 10 c. x 2.5 d. x 2 Solution a. b. c. d. Function Value f 10 log 10 f 13 log 13 f 2.5 log 2.5 f 2 log2 Graphing Calculator Keystrokes LOG 10 ENTER 1 3 LOG ENTER LOG 2.5 ENTER LOG 2 ENTER Display 1 0.4771213 0.3979400 ERROR Note that the calculator displays an error message or a complex number when you try to evaluate log2. The reason for this is that there is no real number power to which 10 can be raised to obtain 2. Now try Exercise 29. The following properties follow directly from the definition of the logarithmic function with base a. Properties of Logarithms 1. loga 1 0 because a0 1. 2. loga a 1 because a1 a. 3. loga a x x and a log a x x Inverse Properties 4. If loga x loga y, then x y. One to One Property Example 3 Using Properties of Logarithms a. Simplify: log 4 1 b. Simplify: log7 7 c. Simplify: 6 log 6 20 Solution a. Using Property 1, it follows that log4 1 0. b. Using Property 2, you can conclude that log7 7 1. c. Using the Inverse Property Property 3 , it follows that 6 log 6 20 20. Now try Exercise 33. You can use the One to One Property Property 4 to solve simple logarithmic equations, as shown in Example 4. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.2 Example 4 Logarithmic Functions and Their Graphs 393 Using the One to One Property a. log3 x log3 12 Original equation x 12 One to One Property b. log2x 1 log 3x 2x 1 3x 1 x c. log4x2 6 log4 10 x2 6 10 x2 16 x 4 Now try Exercise 85. Graphs of Logarithmic Functions To sketch the graph of y loga x, you can use the fact that the graphs of inverse functions are reflections of each other in the line y x. Example 5 Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. y f x = a. f x 2x 2x b. gx log2 x 10 Solution a. For f x 2x, construct a table of values. By plotting these points and connecting y=x 8 them with a smooth curve, you obtain the graph shown in Figure 5.14. 6 g x = log 2 x 4 x 2 1 0 1 2 3 1 4 1 2 1 2 4 8 f x 2x 2 2 4 6 8 10 x b. Because gx log2 x is the inverse function of f x 2x, the graph of g is obtained by plotting the points f x, x and connecting them with a smooth curve. The graph of g is a reflection of the graph of f in the line y x, as shown in Figure 5.14. 2 FIGURE 2 5.14 Now try Exercise 37. y 5 4 Example 6 Vertical asymptote: x = 0 3 Sketch the graph of the common logarithmic function f x log x. Identify the vertical asymptote. f x = log x 2 1 Solution x 1 1 2 3 4 5 6 7 8 9 10 2 FIGURE Sketching the Graph of a Logarithmic Function 5.15 Begin by constructing a table of values. Note that some of the values can be obtained without a calculator by using the Inverse Property of Logarithms. Others require a calculator. Next, plot the points and connect them with a smooth curve, as shown in Figure 5.15. The vertical asymptote is x 0 y axis . Without calculator With calculator x 1 100 1 10 1 10 2 5 8 f x log x 2 1 0 1 0.301 0.699 0.903 Now try Exercise 43. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 394 Chapter 5 Exponential and Logarithmic Functions The nature of the graph in Figure 5.15 is typical of functions of the form f x loga x, a 1. They have one x intercept and one vertical asymptote. Notice how slowly the graph rises for x 1. The basic characteristics of logarithmic graphs are summarized in Figure 5.16. y 1 y = loga x 1, 0 x 1 2 1 FIGURE Graph of y loga x, a 1 Domain: 0, Range: , x intercept: 1, 0 Increasing One to one, therefore has an inverse function y axis is a vertical asymptote loga x as x 0 . Continuous Reflection of graph of y a x about the line y x 5.16 The basic characteristics of the graph of f x a x are shown below to illustrate the inverse relation between f x a x and gx loga x. Domain: , y intercept: 0,1 Range: 0, x axis is a horizontal asymptote a x 0 as x . In the next example, the graph of y loga x is used to sketch the graphs of functions of the form f x b logax c. Notice how a horizontal shift of the graph results in a horizontal shift of the vertical asymptote. Example 7 You can use your understanding of transformations to identify vertical asymptotes of logarithmic functions. For instance, in Example 7 a , the graph of gx f x 1 shifts the graph of f x one unit to the right. So, the vertical asymptote of gx is x 1, one unit to the right of the vertical asymptote of the graph of f x. Shifting Graphs of Logarithmic Functions The graph of each of the functions is similar to the graph of f x log x. a. Because gx logx 1 f x 1, the graph of g can be obtained by shifting the graph of f one unit to the right, as shown in Figure 5.17. b. Because hx 2 log x 2 f x, the graph of h can be obtained by shifting the graph of f two units upward, as shown in Figure 5.18. y y 1 2 f x = log x 1, 0 1 1 You can review the techniques for shifting, reflecting, and stretching graphs in Section 2.5. FIGURE x 1, 2 h x = 2 + log x 1 f x = log x 2, 0 x g x = log x 1 5.17 1, 0 FIGURE 2 5.18 Now try Exercise 45. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.2 Logarithmic Functions and Their Graphs 395 The Natural Logarithmic Function By looking back at the graph of the natural exponential function introduced on page 384 in Section 5.1, you will see that f x e x is one to one and so has an inverse function. This inverse function is called the natural logarithmic function and is denoted by the special symbol ln x, read as the natural log of x or el en of x. Note that the natural logarithm is written without a base. The base is understood to be e. y The Natural Logarithmic Function f x = e x 3 The function defined by y=x 2 1, 1e f x loge x ln x, 1, e is called the natural logarithmic function. e, 1 0, 1 x 2 x 0 1 1, 0 2 1 , 1 e 3 1 2 g x = f 1 x = ln x Reflection of graph of f x e x about the line y x FIGURE 5.19 The definition above implies that the natural logarithmic function and the natural exponential function are inverse functions of each other. So, every logarithmic equation can be written in an equivalent exponential form, and every exponential equation can be written in logarithmic form. That is, y ln x and x e y are equivalent equations. Because the functions given by f x e x and gx ln x are inverse functions of each other, their graphs are reflections of each other in the line y x. This reflective property is illustrated in Figure 5.19. On most calculators, the natural logarithm is denoted by LN , as illustrated in Example 8. Example 8 Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given by f x ln x for each value of x. a. b. c. d. x2 x 0.3 x 1 x 1 2 Solution Function Value WARNING CAUTION Notice that as with every other logarithmic function, the domain of the natural logarithmic function is the set of positive real numbersbe sure you see that ln x is not defined for zero or for negative numbers. a. b. c. d. Graphing Calculator Keystrokes f 2 ln 2 f 0.3 ln 0.3 LN f 1 ln1 f 1 2 ln1 2 LN LN LN 2 .3 ENTER ENTER 1 ENTER 1 2 ENTER Display 0.6931472 1.2039728 ERROR 0.8813736 Now try Exercise 67. In Example 8, be sure you see that ln1 gives an error message on most calculators. Some calculators may display a complex number. This occurs because the domain of ln x is the set of positive real numbers see Figure 5.19 . So, ln1 is undefined. The four properties of logarithms listed on page 392 are also valid for natural logarithms. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 396 Chapter 5 Exponential and Logarithmic Functions Properties of Natural Logarithms 1. ln 1 0 because e0 1. 2. ln e 1 because e1 e. 3. ln e x x and e ln x x Inverse Properties 4. If ln x ln y, then x y. One to One Property Example 9 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplify each expression. a. ln 1 e b. e ln 5 c. ln 1 3 d. 2 ln e Solution 1 ln e1 1 e ln 1 0 c. 0 3 3 a. ln Inverse Property b. e ln 5 5 Inverse Property Property 1 d. 2 ln e 21 2 Property 2 Now try Exercise 71. Example 10 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f x lnx 2 b. gx ln2 x c. hx ln x 2 Solution a. Because lnx 2 is defined only if x 2 0, it follows that the domain of f is 2, . The graph of f is shown in Figure 5.20. b. Because ln2 x is defined only if 2 x 0, it follows that the domain of g is , 2. The graph of g is shown in Figure 5.21. c. Because ln x 2 is defined only if x 2 0, it follows that the domain of h is all real numbers except x 0. The graph of h is shown in Figure 5.22. y y f x = ln x 2 2 g x =1ln 2 x x 1 2 2 3 4 2 x 1 5.20 FIGURE 5.21 x 2 2 4 2 1 4 h x = ln x 2 5 1 3 FIGURE 4 2 1 1 y 4 FIGURE 5.22 Now try Exercise 75. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.2 Logarithmic Functions and Their Graphs 397 Application Example 11 Human Memory Model Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model f t 75 6 lnt 1, 0 t 12, where t is the time in months. a. What was the average score on the original t 0 exam b. What was the average score at the end of t 2 months c. What was the average score at the end of t 6 months Algebraic Solution Graphical Solution a. The original average score was Use a graphing utility to graph the model y 75 6 lnx 1. Then use the value or trace feature to approximate the following. f 0 75 6 ln0 1 Substitute 0 for t. 75 6 ln 1 Simplify. 75 60 Property of natural logarithms 75. Solution b. After 2 months, the average score was f 2 75 6 ln2 1 Substitute 2 for t. 75 6 ln 3 Simplify. 75 61.0986 Use a calculator. 68.4. Solution 100 100 0 c. After 6 months, the average score was f 6 75 6 ln6 1 a. When x 0, y 75 see Figure 5.23 . So, the original average score was 75. b. When x 2, y 68.4 see Figure 5.24 . So, the average score after 2 months was about 68.4. c. When x 6, y 63.3 see Figure 5.25 . So, the average score after 6 months was about 63.3. 12 0 Substitute 6 for t. 75 6 ln 7 Simplify. 75 61.9459 Use a calculator. 63.3. Solution FIGURE 0 12 0 5.23 FIGURE 5.24 100 0 12 0 FIGURE 5.25 Now try Exercise 97. CLASSROOM DISCUSSION Analyzing a Human Memory Model Use a graphing utility to determine the time in months when the average score in Example 11 was 60. Explain your method of solving the problem. Describe another way that you can use a graphing utility to determine the answer. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 398 Chapter 5 5.2 Exponential and Logarithmic Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. 5. 6. The inverse function of the exponential function given by f x ax is called the ________ function with base a. The common logarithmic function has base ________ . The logarithmic function given by f x ln x is called the ________ logarithmic function and has base ________. The Inverse Properties of logarithms and exponentials state that log a ax x and ________. The One to One Property of natural logarithms states that if ln x ln y, then ________. The domain of the natural logarithmic function is the set of ________ ________ ________ . SKILLS AND APPLICATIONS In Exercises 714, write the logarithmic equation in exponential form. For example, the exponential form of log5 25 2 is 52 25. 7. 9. 11. 13. log4 16 2 1 log9 81 2 2 log32 4 5 log64 8 12 8. 10. 12. 14. log7 343 3 1 log 1000 3 3 log16 8 4 log8 4 23 In Exercises 1522, write the exponential equation in logarithmic form. For example, the logarithmic form of 23 8 is log2 8 3. 15. 17. 19. 21. 5 125 8114 3 1 62 36 240 1 3 16. 18. 20. 22. 13 169 9 32 27 1 43 64 103 0.001 2 35. log 36. 9log915 In Exercises 3744, find the domain, x intercept, and vertical asymptote of the logarithmic function and sketch its graph. 37. f x log4 x 39. y log3 x 2 41. f x log6x 2 x 43. y log 7 23. 24. 25. 26. 27. 28. g x logb x y a 29. x 31. x 12.5 3 3 2 2 1 Value x 64 x5 x1 x 10 x a2 3 x 1 1 4 3 2 1 1 2 y c x b3 1 2 y d 4 3 3 2 2 1 x 1 1 500 30. x 32. x 96.75 In Exercises 3336, use the properties of logarithms to simplify the expression. 34. log3.2 1 2 1 1 x 1 1 1 2 3 4 y e 1 2 3 3 4 2 y f 3 3 2 2 1 33. log11 117 y b x In Exercises 2932, use a calculator to evaluate f x log x at the indicated value of x. Round your result to three decimal places. 7 8 44. y logx In Exercises 4550, use the graph of gx log3 x to match the given function with its graph. Then describe the relationship between the graphs of f and g. The graphs are labeled a , b , c , d , e , and f . In Exercises 2328, evaluate the function at the indicated value of x without using a calculator. Function f x log2 x f x log25 x f x log8 x f x log x g x loga x 38. gx log6 x 40. hx log4x 3 42. y log5x 1 4 1 x 1 1 1 2 2 www.elsolucionario.net 3 4 x 1 1 1 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.2 45. f x log3 x 2 47. f x log3x 2 49. f x log31 x 46. f x log3 x 48. f x log3x 1 50. f x log3x In Exercises 5158, write the logarithmic equation in exponential form. 51. 53. 55. 57. ln 12 0.693 . . . ln 7 1.945 . . . ln 250 5.521 . . . ln 1 0 52. 54. 56. 58. ln 25 0.916 . . . ln 10 2.302 . . . ln 1084 6.988 . . . ln e 1 In Exercises 59 66, write the exponential equation in logarithmic form. 59. 61. 63. 65. e4 54.598 . . . e12 1.6487 . . . e0.9 0.406 . . . ex 4 60. 62. 64. 66. e2 7.3890 . . . e13 1.3956 . . . e4.1 0.0165 . . . e2x 3 In Exercises 6770, use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places. 67. 68. 69. 70. Function f x ln x f x 3 ln x g x 8 ln x g x ln x Value x 18.42 x 0.74 x 0.05 x 12 87. log2x 1 log 15 89. lnx 4 ln 12 91. lnx2 2 ln 23 72. x e4 74. x e52 In Exercises 7578, find the domain, x intercept, and vertical asymptote of the logarithmic function and sketch its graph. 75. f x lnx 4 77. gx lnx 76. hx lnx 5 78. f x ln3 x In Exercises 7984, use a graphing utility to graph the function. Be sure to use an appropriate viewing window. 79. f x logx 9 81. f x lnx 1 83. f x ln x 8 t 16.625 ln 85. log5x 1 log5 6 86. log2x 3 log2 9 The model x 750, x x 750 approximates the length of a home mortgage of 150,000 at 6 in terms of the monthly payment. In the model, t is the length of the mortgage in years and x is the monthly payment in dollars. a Use the model to approximate the lengths of a 150,000 mortgage at 6 when the monthly payment is 897.72 and when the monthly payment is 1659.24. b Approximate the total amounts paid over the term of the mortgage with a monthly payment of 897.72 and with a monthly payment of 1659.24. c Approximate the total interest charges for a monthly payment of 897.72 and for a monthly payment of 1659.24. d What is the vertical asymptote for the model Interpret its meaning in the context of the problem. 94. COMPOUND INTEREST A principal P, invested at 5 12 and compounded continuously, increases to an amount K times the original principal after t years, where t is given by t ln K0.055. a Complete the table and interpret your results. 1 K 2 4 6 8 10 12 t b Sketch a graph of the function. 95. CABLE TELEVISION The numbers of cable television systems C in thousands in the United States from 2001 through 2006 can be approximated by the model C 10.355 0.298t ln t, 1 t 6 where t represents the year, with t 1 corresponding to 2001. Source: Warren Communication News a Complete the table. 80. f x logx 6 82. f x lnx 2 84. f x 3 ln x 1 In Exercises 8592, use the One to One Property to solve the equation for x. 88. log5x 3 log 12 90. lnx 7 ln 7 92. lnx2 x ln 6 93. MONTHLY PAYMENT In Exercises 7174, evaluate gx ln x at the indicated value of x without using a calculator. 71. x e5 73. x e56 399 Logarithmic Functions and Their Graphs t 1 2 3 4 5 6 C b Use a graphing utility to graph the function. c Can the model be used to predict the numbers of cable television systems beyond 2006 Explain. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 400 Chapter 5 Exponential and Logarithmic Functions 96. POPULATION The time t in years for the world population to double if it is increasing at a continuous rate of r is given by t ln 2r. a Complete the table and interpret your results. r 0.005 0.010 0.015 0.020 0.025 0.030 105. THINK ABOUT IT Complete the table for f x 10 x. c What was the average score after 4 months d What was the average score after 10 months 98. SOUND INTENSITY The relationship between the number of decibels and the intensity of a sound I in watts per square meter is 10 log 10 . 1 2 1 100 1 10 1 10 100 f x Compare the two tables. What is the relationship between f x 10 x and f x log x 106. GRAPHICAL ANALYSIS Use a graphing utility to graph f and g in the same viewing window and determine which is increasing at the greater rate as x approaches . What can you conclude about the rate of growth of the natural logarithmic function a f x ln x, gx x 4 b f x ln x, gx x 107. a Complete the table for the function given by f x ln xx. 1 x 5 10 102 104 106 f x I 12 a Determine the number of decibels of a sound with an intensity of 1 watt per square meter. b Determine the number of decibels of a sound with an intensity of 102 watt per square meter. c The intensity of the sound in part a is 100 times as great as that in part b . Is the number of decibels 100 times as great Explain. EXPLORATION TRUE OR FALSE In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer. 99. You can determine the graph of f x log6 x by graphing gx 6 x and reflecting it about the x axis. 100. The graph of f x log3 x contains the point 27, 3. In Exercises 101104, sketch the graphs of f and g and describe the relationship between the graphs of f and g. What is the relationship between the functions f and g f x 3x, f x 5x, f x e x, f x 8 x, 0 Complete the table for f x log x. x b Use a graphing utility to graph the function. 97. HUMAN MEMORY MODEL Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given by the human memory model f t 80 17 logt 1, 0 t 12, where t is the time in months. a Use a graphing utility to graph the model over the specified domain. b What was the average score on the original exam t 0 1 f x t 101. 102. 103. 104. 2 x gx log3 x gx log5 x gx ln x gx log8 x b Use the table in part a to determine what value f x approaches as x increases without bound. c Use a graphing utility to confirm the result of part b . 108. CAPSTONE The table of values was obtained by evaluating a function. Determine which of the statements may be true and which must be false. x y 1 0 2 1 8 3 a y is an exponential function of x. b y is a logarithmic function of x. c x is an exponential function of y. d y is a linear function of x. 109. WRITING Explain why loga x is defined only for 0 a 1 and a 1. In Exercises 110 and 111, a use a graphing utility to graph the function, b use the graph to determine the intervals in which the function is increasing and decreasing, and c approximate any relative maximum or minimum values of the function. 110. f x ln x www.elsolucionario.net 111. hx lnx 2 1 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.3 Properties of Logarithms 401 5.3 PROPERTIES OF LOGARITHMS What you should learn Use the change of base formula to rewrite and evaluate logarithmic expressions. Use properties of logarithms to evaluate or rewrite logarithmic expressions. Use properties of logarithms to expand or condense logarithmic expressions. Use logarithmic functions to model and solve real life problems. Why you should learn it Logarithmic functions can be used to model and solve real life problems. For instance, in Exercises 8790 on page 406, a logarithmic function is used to model the relationship between the number of decibels and the intensity of a sound. Change of Base Most calculators have only two types of log keys, one for common logarithms base 10 and one for natural logarithms base e . Although common logarithms and natural logarithms are the most frequently used, you may occasionally need to evaluate logarithms with other bases. To do this, you can use the following change of base formula. Change of Base Formula Let a, b, and x be positive real numbers such that a can be converted to a different base as follows. Base b logb x loga x logb a 1. Then loga x Base e ln x loga x ln a One way to look at the change of base formula is that logarithms with base a are simply constant multiples of logarithms with base b. The constant multiplier is 1logb a. Example 1 a. log4 25 Changing Bases Using Common Logarithms log 25 log 4 log a x 1.39794 0.60206 Use a calculator. 2.3219 Dynamic Graphics Jupiter Images Base 10 log x loga x log a 1 and b b. log2 12 log x log a Simplify. log 12 1.07918 3.5850 log 2 0.30103 Now try Exercise 7 a . Example 2 a. log4 25 Changing Bases Using Natural Logarithms ln 25 ln 4 loga x 3.21888 1.38629 Use a calculator. 2.3219 b. log2 12 ln x ln a Simplify. ln 12 2.48491 3.5850 ln 2 0.69315 Now try Exercise 7 b . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 402 Chapter 5 Exponential and Logarithmic Functions Properties of Logarithms You know from the preceding section that the logarithmic function with base a is the inverse function of the exponential function with base a. So, it makes sense that the properties of exponents should have corresponding properties involving logarithms. For instance, the exponential property a0 1 has the corresponding logarithmic property loga 1 0. WARNING CAUTION There is no general property that can be used to rewrite logau v. Specifically, logau v is not equal to loga u loga v. Properties of Logarithms Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. Logarithm with Base a 1. Product Property: logauv loga u loga v 2. Quotient Property: loga 3. Power Property: u loga u loga v v loga u n n loga u Natural Logarithm lnuv ln u ln v ln u ln u ln v v ln u n n ln u For proofs of the properties listed above, see Proofs in Mathematics on page 440. Example 3 Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. a. ln 6 HISTORICAL NOTE b. ln Solution The Granger Collection a. ln 6 ln2 John Napier, a Scottish mathematician, developed logarithms as a way to simplify some of the tedious calculations of his day. Beginning in 1594, Napier worked about 20 years on the invention of logarithms. Napier was only partially successful in his quest to simplify tedious calculations. Nonetheless, the development of logarithms was a step forward and received immediate recognition. 2 27 b. ln 3 Rewrite 6 as 2 3. ln 2 ln 3 Product Property 2 ln 2 ln 27 27 Quotient Property ln 2 ln 33 Rewrite 27 as 33. ln 2 3 ln 3 Power Property Now try Exercise 27. Example 4 Using Properties of Logarithms Find the exact value of each expression without using a calculator. 3 a. log5 5 b. ln e6 ln e2 Solution 3 a. log5 5 log5 513 13 log5 5 13 1 13 b. ln e6 ln e2 ln e6 ln e4 4 ln e 41 4 e2 Now try Exercise 29. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.3 Properties of Logarithms 403 Rewriting Logarithmic Expressions The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because these properties convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively. Example 5 Expanding Logarithmic Expressions Expand each logarithmic expression. a. log4 5x3y b. ln 3x 5 7 Solution a. log4 5x3y log4 5 log4 x 3 log4 y Product Property log4 5 3 log4 x log4 y b. ln 3x 5 7 ln Power Property 3x 5 7 12 Rewrite using rational exponent. ln3x 512 ln 7 Quotient Property 1 ln3x 5 ln 7 2 Power Property Now try Exercise 53. In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions. Example 6 Condensing Logarithmic Expressions Condense each logarithmic expression. a. 12 log x 3 logx 1 c. 13 log2 x log2x 1 b. 2 lnx 2 ln x Solution a. 1 2 log x 3 logx 1 log x12 logx 13 logx x 1 3 b. 2 lnx 2 ln x lnx 2 ln x 2 ln You can review rewriting radicals and rational exponents in Section P.2. x 2 x Power Property Product Property Power Property 2 Quotient Property c. 13 log2 x log2x 1 13 log2xx 1 log2 xx 113 log2 xx 1 3 Product Property Power Property Rewrite with a radical. Now try Exercise 75. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 404 Chapter 5 Exponential and Logarithmic Functions Application One method of determining how the x and y values for a set of nonlinear data are related is to take the natural logarithm of each of the x and y values. If the points are graphed and fall on a line, then you can determine that the x and y values are related by the equation ln y m ln x where m is the slope of the line. Example 7 Finding a Mathematical Model The table shows the mean distance from the sun x and the period y the time it takes a planet to orbit the sun for each of the six planets that are closest to the sun. In the table, the mean distance is given in terms of astronomical units where Earths mean distance is defined as 1.0 , and the period is given in years. Find an equation that relates y and x. Planets Near the Sun y Period in years 25 20 Mercury Venus 15 10 Jupiter Earth 5 Mars x 2 4 6 8 Mean distance, x Period, y Mercury Venus Earth Mars Jupiter Saturn 0.387 0.723 1.000 1.524 5.203 9.537 0.241 0.615 1.000 1.881 11.860 29.460 10 Mean distance in astronomical units FIGURE 5.26 Solution The points in the table above are plotted in Figure 5.26. From this figure it is not clear how to find an equation that relates y and x. To solve this problem, take the natural logarithm of each of the x and y values in the table. This produces the following results. ln y ln y = 1 Venus Mercury 5.27 Mercury Venus Earth Mars Jupiter Saturn ln x 0.949 0.324 0.000 0.421 1.649 2.255 ln y 1.423 0.486 0.000 0.632 2.473 3.383 Now, by plotting the points in the second table, you can see that all six of the points appear to lie in a line see Figure 5.27 . Choose any two points to determine the slope of the line. Using the two points 0.421, 0.632 and 0, 0, you can determine that the slope of the line is Jupiter 2 Earth Planet Saturn 3 FIGURE Planet Saturn 30 3 2 ln x Mars ln x 1 2 3 m 0.632 0 3 1.5 . 0.421 0 2 By the point slope form, the equation of the line is Y 32 X, where Y ln y and X ln x. You can therefore conclude that ln y 32 ln x. Now try Exercise 91. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.3 5.3 EXERCISES Properties of Logarithms 405 See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY In Exercises 13, fill in the blanks. 1. To evaluate a logarithm to any base, you can use the ________ formula. 2. The change of base formula for base e is given by loga x ________. 3. You can consider loga x to be a constant multiple of logb x; the constant multiplier is ________. In Exercises 46, match the property of logarithms with its name. 4. logauv loga u loga v 5. ln u n n ln u u 6. loga loga u loga v v a Power Property b Quotient Property c Product Property SKILLS AND APPLICATIONS In Exercises 714, rewrite the logarithm as a ratio of a common logarithms and b natural logarithms. 7. 9. 11. 13. log5 16 log15 x 3 logx 10 log2.6 x 8. 10. 12. 14. log3 47 log13 x logx 34 log 7.1 x In Exercises 1522, evaluate the logarithm using the change of base formula. Round your result to three decimal places. 15. 17. 19. 21. log3 7 log12 4 log9 0.1 log15 1250 16. 18. 20. 22. log7 4 log14 5 log20 0.25 log3 0.015 In Exercises 2328, use the properties of logarithms to rewrite and simplify the logarithmic expression. 23. log4 8 1 25. log5 250 27. ln5e6 24. log242 9 26. log 300 6 28. ln 2 e 34 log3 9 4 log2 8 log4 162 log22 30. 32. 34. 36. 45. ln 4x 47. log8 x 4 5 x 51. ln z 53. ln xyz2 49. log5 55. ln zz 12, z 1 a 1 9 x y , a 1 y 61. ln x z 59. ln 1 log5 125 3 log6 6 log3 813 log327 38. 3 ln e4 4 e3 40. ln 42. 2 ln e 6 ln e 5 44. log4 2 log4 32 In Exercises 4566, use the properties of logarithms to expand the expression as a sum, difference, andor constant multiple of logarithms. Assume all variables are positive. 57. log2 In Exercises 2944, find the exact value of the logarithmic expression without using a calculator. If this is not possible, state the reason. 29. 31. 33. 35. 37. ln e4.5 1 39. ln e 41. ln e 2 ln e5 43. log5 75 log5 3 3 2 x2 y 2z 3 4 65. ln x3x2 3 63. log5 www.elsolucionario.net 46. log3 10z y 48. log10 2 1 50. log6 3 z 3 52. ln t 54. log 4x2 y x2 1 56. ln ,x 1 x3 58. ln 6 1 x2 y3 x 2 y 62. log x z 60. ln 2 4 3 xy4 z5 2 66. ln x x 2 64. log10 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 406 Chapter 5 Exponential and Logarithmic Functions In Exercises 6784, condense the expression to the logarithm of a single quantity. 67. 69. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. ln 2 ln x 68. ln y ln t log4 z log4 y 70. log5 8 log5 t 2 log2 x 4 log2 y 2 3 log7z 2 1 4 log3 5x 4 log6 2x log x 2 logx 1 2 ln 8 5 lnz 4 log x 2 log y 3 log z 3 log3 x 4 log3 y 4 log3 z ln x lnx 1 lnx 1 4ln z lnz 5 2 lnz 5 1 2 3 2 lnx 3 ln x lnx 1 23 ln x lnx 1 ln x 1 1 3 log8 y 2 log8 y 4 log8 y 1 1 2 log4x 1 2 log4x 1 6 log4 x CURVE FITTING In Exercises 9194, find a logarithmic equation that relates y and x. Explain the steps used to find the equation. 91. 92. 93. 94. In Exercises 85 and 86, compare the logarithmic quantities. If two are equal, explain why. log2 32 32 , log2 , log2 32 log2 4 log2 4 4 1 86. log770, log7 35, 2 log7 10 85. x 1 2 3 4 5 6 y 1 1.189 1.316 1.414 1.495 1.565 x 1 2 3 4 5 6 y 1 1.587 2.080 2.520 2.924 3.302 x 1 2 3 4 5 6 y 2.5 2.102 1.9 1.768 1.672 1.597 x 1 2 3 4 5 6 y 0.5 2.828 7.794 16 27.951 44.091 95. GALLOPING SPEEDS OF ANIMALS Four legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table to find a logarithmic equation that relates an animals weight x in pounds and its lowest galloping speed y in strides per minute . SOUND INTENSITY In Exercises 8790, use the following information. The relationship between the number of decibels and the intensity of a sound l in watts per square meter is given by 10 log Weight, x Galloping speed, y 25 35 50 75 500 1000 191.5 182.7 173.8 164.2 125.9 114.2 10 . I 12 87. Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensity of 106 watt per square meter. 88. Find the difference in loudness between an average office with an intensity of 1.26 107 watt per square meter and a broadcast studio with an intensity of 3.16 1010 watt per square meter. 89. Find the difference in loudness between a vacuum cleaner with an intensity of 104 watt per square meter and rustling leaves with an intensity of 1011 watt per square meter. 90. You and your roommate are playing your stereos at the same time and at the same intensity. How much louder is the music when both stereos are playing compared with just one stereo playing 96. NAIL LENGTH The approximate lengths and diameters in inches of common nails are shown in the table. Find a logarithmic equation that relates the diameter y of a common nail to its length x. Length, x Diameter, y Length, x Diameter, y 1 0.072 4 0.203 2 0.120 5 0.238 3 0.148 6 0.284 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.3 97. COMPARING MODELS A cup of water at an initial temperature of 78C is placed in a room at a constant temperature of 21C. The temperature of the water is measured every 5 minutes during a half hour period. The results are recorded as ordered pairs of the form t, T , where t is the time in minutes and T is the temperature in degrees Celsius . t, T 1 21. Use a graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form 1 at b. T 21 Solve for T, and use a graphing utility to graph the rational function and the original data points. e Why did taking the logarithms of the temperatures lead to a linear scatter plot Why did taking the reciprocals of the temperatures lead to a linear scatter plot EXPLORATION 98. PROOF 99. PROOF u logb u logb v. v Prove that logb un n logb u. Prove that logb 407 100. CAPSTONE A classmate claims that the following are true. a lnu v ln u ln v lnuv b lnu v ln u ln v ln u v c ln un nln u ln un Discuss how you would demonstrate that these claims are not true. 0, 78.0, 5, 66.0, 10, 57.5, 15, 51.2, 20, 46.3, 25, 42.4, 30, 39.6 a The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points t, T and t, T 21. b An exponential model for the data t, T 21 is given by T 21 54.40.964t. Solve for T and graph the model. Compare the result with the plot of the original data. c Take the natural logarithms of the revised temperatures. Use a graphing utility to plot the points t, lnT 21 and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form lnT 21 at b. Solve for T, and verify that the result is equivalent to the model in part b . d Fit a rational model to the data. Take the reciprocals of the y coordinates of the revised data points to generate the points Properties of Logarithms TRUE OR FALSE In Exercises 101106, determine whether the statement is true or false given that f x ln x. Justify your answer. 101. 102. 103. 104. 105. 106. f 0 0 f ax f a f x, a 0, x 0 f x 2 f x f 2, x 2 1 f x 2 f x If f u 2 f v, then v u2. If f x 0, then 0 x 1. In Exercises 107112, use the change of base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. 107. 108. 109. 110. 111. 112. f x f x f x f x f x f x log2 x log4 x log12 x log14 x log11.8 x log12.4 x 113. THINK ABOUT IT x f x ln , 2 Consider the functions below. gx ln x , ln 2 hx ln x ln 2 Which two functions should have identical graphs Verify your answer by sketching the graphs of all three functions on the same set of coordinate axes. 114. GRAPHICAL ANALYSIS Use a graphing utility to graph the functions given by y1 ln x lnx 3 x and y2 ln in the same viewing window. Does x3 the graphing utility show the functions with the same domain If so, should it Explain your reasoning. 115. THINK ABOUT IT For how many integers between 1 and 20 can the natural logarithms be approximated given the values ln 2 0.6931, ln 3 1.0986, and ln 5 1.6094 Approximate these logarithms do not use a calculator . www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 408 Chapter 5 Exponential and Logarithmic Functions 5.4 EXPONENTIAL AND LOGARITHMIC EQUATIONS What you should learn Solve simple exponential and logarithmic equations. Solve more complicated exponential equations. Solve more complicated logarithmic equations. Use exponential and logarithmic equations to model and solve real life problems. Why you should learn it Exponential and logarithmic equations are used to model and solve life science applications. For instance, in Exercise 132 on page 417, an exponential function is used to model the number of trees per acre given the average diameter of the trees. Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential and logarithmic functions. In this section, you will study procedures for solving equations involving these exponential and logarithmic functions. There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One to One Properties and was used to solve simple exponential and logarithmic equations in Sections 5.1 and 5.2. The second is based on the Inverse Properties. For a 0 and a 1, the following properties are true for all x and y for which log a x and loga y are defined. One to One Properties a x a y if and only if x y. loga x loga y if and only if x y. Inverse Properties a log a x x loga a x x James MarshallCorbis Example 1 Solving Simple Equations Original Equation a. 2 x 32 b. ln x ln 3 0 x c. 13 9 d. e x 7 e. ln x 3 f. log x 1 g. log3 x 4 Rewritten Equation Solution Property 2 x 25 ln x ln 3 3x 32 ln e x ln 7 e ln x e3 10 log x 101 3log3 x 34 x5 x3 x 2 x ln 7 x e3 1 x 101 10 x 81 One to One One to One One to One Inverse Inverse Inverse Inverse Now try Exercise 17. The strategies used in Example 1 are summarized as follows. Strategies for Solving Exponential and Logarithmic Equations 1. Rewrite the original equation in a form that allows the use of the One to One Properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.4 Exponential and Logarithmic Equations 409 Solving Exponential Equations Example 2 Solving Exponential Equations Solve each equation and approximate the result to three decimal places, if necessary. a. ex e3x4 b. 32 x 42 2 Solution ex e3x4 Write original equation. x2 One to One Property 2 a. x2 3x 4 3x 4 0 x 1x 4 0 Write in general form. Factor. x 1 0 x 1 Set 1st factor equal to 0. x 4 0 x 4 Set 2nd factor equal to 0. The solutions are x 1 and x 4. Check these in the original equation. b. Another way to solve Example 2 b is by taking the natural log of each side and then applying the Power Property, as follows. 32x 42 2x 14 ln 2x ln 14 32 x 42 log2 14 Divide each side by 3. 2x log2 14 Take log base 2 of each side. x log2 14 x As you can see, you obtain the same result as in Example 2 b . ln 14 3.807 ln 2 Inverse Property Change of base formula The solution is x log2 14 3.807. Check this in the original equation. x ln 2 ln 14 ln 14 x 3.807 ln 2 Write original equation. 2x Now try Exercise 29. In Example 2 b , the exact solution is x log2 14 and the approximate solution is x 3.807. An exact answer is preferred when the solution is an intermediate step in a larger problem. For a final answer, an approximate solution is easier to comprehend. Example 3 Solving an Exponential Equation Solve e x 5 60 and approximate the result to three decimal places. Solution Remember that the natural logarithmic function has a base of e. e x 5 60 e x 55 ln e x ln 55 x ln 55 4.007 Write original equation. Subtract 5 from each side. Take natural log of each side. Inverse Property The solution is x ln 55 4.007. Check this in the original equation. Now try Exercise 55. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 410 Chapter 5 Exponential and Logarithmic Functions Example 4 Solving an Exponential Equation Solve 232t5 4 11 and approximate the result to three decimal places. Solution 232t5 4 11 Write original equation. 232t5 15 32t5 Remember that to evaluate a logarithm such as log3 7.5, you need to use the change of base formula. log3 7.5 Add 4 to each side. 15 2 Divide each side by 2. log3 32t5 log3 15 2 Take log base 3 of each side. 2t 5 log3 15 2 Inverse Property 2t 5 log3 7.5 t ln 7.5 1.834 ln 3 5 1 log3 7.5 2 2 t 3.417 5 2 Add 5 to each side. Divide each side by 2. Use a calculator. 1 2 The solution is t log3 7.5 3.417. Check this in the original equation. Now try Exercise 57. When an equation involves two or more exponential expressions, you can still use a procedure similar to that demonstrated in Examples 2, 3, and 4. However, the algebra is a bit more complicated. Example 5 Solving an Exponential Equation of Quadratic Type Solve e 2x 3e x 2 0. Algebraic Solution Graphical Solution e 2x 3e x 2 0 Write original equation. e x2 3e x 2 0 Write in quadratic form. e x 2e x 1 0 ex 20 x ln 2 ex 1 0 x0 Factor. Set 1st factor equal to 0. Use a graphing utility to graph y e2x 3ex 2. Use the zero or root feature or the zoom and trace features of the graphing utility to approximate the values of x for which y 0. In Figure 5.28, you can see that the zeros occur at x 0 and at x 0.693. So, the solutions are x 0 and x 0.693. y = e 2x 3e x + 2 Solution 3 Set 2nd factor equal to 0. Solution The solutions are x ln 2 0.693 and x 0. Check these in the original equation. 3 3 1 FIGURE 5.28 Now try Exercise 59. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.4 Exponential and Logarithmic Equations 411 Solving Logarithmic Equations To solve a logarithmic equation, you can write it in exponential form. ln x 3 Logarithmic form e ln x e 3 Exponentiate each side. x e3 Exponential form This procedure is called exponentiating each side of an equation. Example 6 Solving Logarithmic Equations a. ln x 2 WARNING CAUTION Remember to check your solutions in the original equation when solving equations to verify that the answer is correct and to make sure that the answer lies in the domain of the original equation. Original equation e ln x e 2 x Exponentiate each side. e2 Inverse Property b. log35x 1 log3x 7 Original equation 5x 1 x 7 One to One Property 4x 8 Add x and 1 to each side. x2 Divide each side by 4. c. log63x 14 log6 5 log6 2x log6 3x 5 14 log 6 Original equation 2x Quotient Property of Logarithms 3x 14 2x 5 One to One Property 3x 14 10x Cross multiply. 7x 14 Isolate x. x2 Divide each side by 7. Now try Exercise 83. Example 7 Solving a Logarithmic Equation Solve 5 2 ln x 4 and approximate the result to three decimal places. Graphical Solution Algebraic Solution 5 2 ln x 4 Write original equation. 2 ln x 1 ln x 1 2 eln x e12 Subtract 5 from each side. Divide each side by 2. Use a graphing utility to graph y1 5 2 ln x and y2 4 in the same viewing window. Use the intersect feature or the zoom and trace features to approximate the intersection point, as shown in Figure 5.29. So, the solution is x 0.607. 6 Exponentiate each side. x e12 Inverse Property x 0.607 Use a calculator. y2 = 4 y1 = 5 + 2 ln x 0 1 0 FIGURE 5.29 Now try Exercise 93. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 412 Chapter 5 Exponential and Logarithmic Functions Example 8 Solving a Logarithmic Equation Solve 2 log5 3x 4. Solution 2 log5 3x 4 Write original equation. log5 3x 2 Divide each side by 2. 5 log5 3x 52 Exponentiate each side base 5 . 3x 25 x Notice in Example 9 that the logarithmic part of the equation is condensed into a single logarithm before exponentiating each side of the equation. Example 9 Inverse Property 25 3 Divide each side by 3. The solution is x 25 3 . Check this in the original equation. Now try Exercise 97. Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. Checking for Extraneous Solutions Solve log 5x logx 1 2. Graphical Solution Algebraic Solution log 5x logx 1 2 log 5xx 1 2 10 log5x 2 5x 102 5x 2 5x 100 x x 20 0 2 x 5x 4 0 x50 x5 x40 x 4 Write original equation. Product Property of Logarithms Exponentiate each side base 10 . Inverse Property Write in general form. Use a graphing utility to graph y1 log 5x logx 1 and y2 2 in the same viewing window. From the graph shown in Figure 5.30, it appears that the graphs intersect at one point. Use the intersect feature or the zoom and trace features to determine that the graphs intersect at approximately 5, 2. So, the solution is x 5. Verify that 5 is an exact solution algebraically. Factor. 5 y1 = log 5x + log x 1 Set 1st factor equal to 0. Solution y2 = 2 Set 2nd factor equal to 0. 0 Solution The solutions appear to be x 5 and x 4. However, when you check these in the original equation, you can see that x 5 is the only solution. 9 1 FIGURE 5.30 Now try Exercise 109. In Example 9, the domain of log 5x is x 0 and the domain of logx 1 is x 1, so the domain of the original equation is x 1. Because the domain is all real numbers greater than 1, the solution x 4 is extraneous. The graph in Figure 5.30 verifies this conclusion. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.4 Exponential and Logarithmic Equations 413 Applications Example 10 Doubling an Investment You have deposited 500 in an account that pays 6.75 interest, compounded continuously. How long will it take your money to double Solution Using the formula for continuous compounding, you can find that the balance in the account is A Pe rt A 500e 0.0675t. To find the time required for the balance to double, let A 1000 and solve the resulting equation for t. 500e 0.0675t 1000 e 0.0675t 0.0675t ln e Let A 1000. 2 Divide each side by 500. ln 2 Take natural log of each side. 0.0675t ln 2 t Inverse Property ln 2 0.0675 Divide each side by 0.0675. t 10.27 Use a calculator. The balance in the account will double after approximately 10.27 years. This result is demonstrated graphically in Figure 5.31. Doubling an Investment A Account balance in dollars 1100 ES AT ES STAT D D ST ITE ITE UN E E UN TH TH 900 C4 OF OF 10.27, 1000 ICA ICA ER ER AM AM N, D.C. INGTO WASH 1 C 31 1 IES SER 1993 A 1 N A ON GT SHI W 1 700 500 A = 500e 0.0675t 0, 500 300 100 t 2 4 6 8 10 Time in years FIGURE 5.31 Now try Exercise 117. In Example 10, an approximate answer of 10.27 years is given. Within the context of the problem, the exact solution, ln 20.0675 years, does not make sense as an answer. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 414 Chapter 5 Exponential and Logarithmic Functions Retail Sales of e Commerce Companies Example 11 y The retail sales y in billions of e commerce companies in the United States from 2002 through 2007 can be modeled by 180 Sales in billions Retail Sales 160 y 549 236.7 ln t, 12 t 17 140 120 where t represents the year, with t 12 corresponding to 2002 see Figure 5.32 . During which year did the sales reach 108 billion Source: U.S. Census Bureau 100 80 Solution 60 40 20 t 12 13 14 15 16 Year 12 2002 FIGURE 5.32 17 549 236.7 ln t y Write original equation. 549 236.7 ln t 108 Substitute 108 for y. 236.7 ln t 657 ln t Add 549 to each side. 657 236.7 Divide each side by 236.7. e ln t e657236.7 t e657236.7 t 16 Exponentiate each side. Inverse Property Use a calculator. The solution is t 16. Because t 12 represents 2002, it follows that the sales reached 108 billion in 2006. Now try Exercise 133. CLASSROOM DISCUSSION Analyzing Relationships Numerically Use a calculator to fill in the table row byrow. Discuss the resulting pattern. What can you conclude Find two equations that summarize the relationships you discovered. x 1 2 1 2 10 25 50 ex lne x ln x e ln x www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.4 5.4 EXERCISES 415 Exponential and Logarithmic Equations See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. To ________ an equation in x means to find all values of x for which the equation is true. 2. To solve exponential and logarithmic equations, you can use the following One to One and Inverse Properties. a ax ay if and only if ________. b loga x loga y if and only if ________. log x a c a ________ d loga a x ________ 3. To solve exponential and logarithmic equations, you can use the following strategies. a Rewrite the original equation in a form that allows the use of the ________ Properties of exponential or logarithmic functions. b Rewrite an exponential equation in ________ form and apply the Inverse Property of ________ functions. c Rewrite a logarithmic equation in ________ form and apply the Inverse Property of ________ functions. 4. An ________ solution does not satisfy the original equation. SKILLS AND APPLICATIONS In Exercises 512, determine whether each x value is a solution or an approximate solution of the equation. 5. 42x7 64 a x 5 b x 2 7. 3e x2 75 a x 2 e25 b x 2 ln 25 c x 1.219 9. log43x 3 a x 21.333 b x 4 64 c x 3 6. 23x1 32 a x 1 b x 2 8. 4ex1 60 a x 1 ln 15 b x 3.7081 c x ln 16 10. log2x 3 10 a x 1021 b x 17 c x 102 3 25. f x 2x gx 8 y 13. 4x 16 1 x 15. 2 32 17. ln x ln 2 0 19. e x 2 21. ln x 1 23. log4 x 3 14. 3x 243 1 x 16. 4 64 18. ln x ln 5 0 20. e x 4 22. log x 2 1 24. log5 x 2 In Exercises 2528, approximate the point of intersection of the graphs of f and g. Then solve the equation f x gx algebraically to verify your approximation. y 12 12 g 8 4 8 f 4 x 4 4 g f 4 8 8 27. f x log3 x gx 2 4 x 4 4 8 28. f x lnx 4 gx 0 y y 12 4 8 g 11. ln2x 3 5.8 12. lnx 1 3.8 1 a x 23 ln 5.8 a x 1 e3.8 1 b x 2 3 e5.8 b x 45.701 c x 163.650 c x 1 ln 3.8 In Exercises 1324, solve for x. 26. f x 27x gx 9 4 f 4 x 8 f g 12 x 8 4 12 In Exercises 2970, solve the exponential equation algebraically. Approximate the result to three decimal places. 29. 31. 33. 35. 37. 39. 41. 43. 45. e x e x 2 2 e x 3 e x2 43x 20 2e x 10 ex 9 19 32x 80 5t2 0.20 3x1 27 23x 565 www.elsolucionario.net 2 30. 32. 34. 36. 38. 40. 42. 44. 46. e2x e x 8 2 2 ex e x 2x 25x 32 4e x 91 6x 10 47 65x 3000 43t 0.10 2x3 32 82x 431 2 http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 416 Chapter 5 Exponential and Logarithmic Functions 47. 49. 51. 53. 55. 57. 59. 61. 8103x 12 35x1 21 e3x 12 500ex 300 7 2e x 5 623x1 7 9 e 2x 4e x 5 0 e2x 3ex 4 0 63. 500 20 100 e x2 65. 3000 2 2 e2x 1 365 0.10 69. 1 12 67. 0.065 510 x6 7 836x 40 e2x 50 1000e4x 75 14 3e x 11 8462x 13 41 e2x 5e x 6 0 e2x 9e x 36 0 400 64. 350 1 ex 48. 50. 52. 54. 56. 58. 60. 62. 66. 365t 12t 4 2 119 7 e 6x 14 21 4 2.471 40 0.878 70. 16 30 26 9t 68. 3t In Exercises 7180, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 71. 73. 75. 77. 79. 7 2x 6e1x 25 3e3x2 962 e0.09t 3 e 0.125t 8 0 72. 74. 76. 78. 80. 5x 212 4ex1 15 0 8e2x3 11 e 1.8x 7 0 e 2.724x 29 In Exercises 81112, solve the logarithmic equation algebraically. Approximate the result to three decimal places. 81. ln x 3 83. 85. 87. 89. 91. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. ln x 7 0 ln 2x 2.4 log x 6 3ln 5x 10 lnx 2 1 7 3 ln x 5 2 6 ln x 10 82. ln x 1.6 84. 86. 88. 90. 92. ln x 1 0 2.1 ln 6x log 3z 2 2 ln x 7 lnx 8 5 2 2 ln 3x 17 2 3 ln x 12 6 log30.5x 11 4 logx 6 11 ln x lnx 1 2 ln x lnx 1 1 ln x lnx 2 1 ln x lnx 3 1 lnx 5 lnx 1 lnx 1 104. 105. 106. 107. 108. 109. 110. 111. 112. lnx 1 lnx 2 ln x log22x 3 log2x 4 log3x 4 logx 10 logx 4 log x logx 2 log2 x log2x 2 log2x 6 log4 x log4x 1 12 log3 x log3x 8 2 log 8x log1 x 2 log 4x log12 x 2 In Exercises 113116, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 113. 3 ln x 0 115. 2 lnx 3 3 114. 10 4 lnx 2 0 116. lnx 1 2 ln x COMPOUND INTEREST In Exercises 117120, 2500 is invested in an account at interest rate r, compounded continuously. Find the time required for the amount to a double and b triple. 117. r 0.05 119. r 0.025 118. r 0.045 120. r 0.0375 In Exercises 121128, solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. 121. 2x2e2x 2xe2x 0 123. xex ex 0 122. x2ex 2xex 0 124. e2x 2xe2x 0 125. 2x ln x x 0 126. 127. 1 ln x 0 2 1 ln x 0 x2 128. 2x ln 1x x 0 129. DEMAND The demand equation for a limited edition coin set is p 1000 1 5 . 5 e0.001x Find the demand x for a price of a p 139.50 and b p 99.99. 130. DEMAND The demand equation for a hand held electronic organizer is p 5000 1 4 . 4 e0.002x Find the demand x for a price of a p 600 and b p 400. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.4 y 2875 2635.11 , 1 14.215e0.8038t 0 t 7 where t represents the year, with t 0 corresponding to 2000. Source: Verispan a Use a graphing utility to graph the model. b Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded 3600. 135. AVERAGE HEIGHTS The percent m of American males between the ages of 18 and 24 who are no more than x inches tall is modeled by mx 100 Percent of population 131. FOREST YIELD The yield V in millions of cubic feet per acre for a forest at age t years is given by V 6.7e48.1t. a Use a graphing utility to graph the function. b Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. c Find the time necessary to obtain a yield of 1.3 million cubic feet. 132. TREES PER ACRE The number N of trees of a given species per acre is approximated by the model N 68100.04x, 5 x 40, where x is the average diameter of the trees in inches 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when N 21. 133. U.S. CURRENCY The values y in billions of dollars of U.S. currency in circulation in the years 2000 through 2007 can be modeled by y 451 444 ln t, 10 t 17, where t represents the year, with t 10 corresponding to 2000. During which year did the value of U.S. currency in circulation exceed 690 billion Source: Board of Governors of the Federal Reserve System 134. MEDICINE The numbers y of freestanding ambulatory care surgery centers in the United States from 2000 through 2007 can be modeled by 80 f x 60 40 m x 20 x 55 65 70 75 b What is the average height of each sex 136. LEARNING CURVE In a group project in learning theory, a mathematical model for the proportion P of correct responses after n trials was found to be P .0.831 e0.2n. a Use a graphing utility to graph the function. b Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. c After how many trials will 60 of the responses be correct 137. AUTOMOBILES Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer gs the crash victims experience. One g is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 gs. In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of gs experienced during deceleration by crash dummies that were permitted to move x meters during impact. The data are shown in the table. A model for the data is given by y 3.00 11.88 ln x 36.94x, where y is the number of gs. 100 1 e0.6114x69.71 100 . 1 e0.66607x64.51 Source: U.S. National Center for Health Statistics a Use the graph to determine any horizontal asymptotes of the graphs of the functions. Interpret the meaning in the context of the problem. 60 Height in inches and the percent f of American females between the ages of 18 and 24 who are no more than x inches tall is modeled by f x 417 Exponential and Logarithmic Equations x gs 0.2 0.4 0.6 0.8 1.0 158 80 53 40 32 a Complete the table using the model. www.elsolucionario.net x 0.2 0.4 0.6 0.8 1.0 y http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 418 Chapter 5 Exponential and Logarithmic Functions b Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare c Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed 30 gs. d Do you think it is practical to lower the number of gs experienced during impact to fewer than 23 Explain your reasoning. 138. DATA ANALYSIS An object at a temperature of 160C was removed from a furnace and placed in a room at 20C. The temperature T of the object was measured each hour h and recorded in the table. A model for the data is given by T 20 1 72h. The graph of this model is shown in the figure. Hour, h Temperature, T 0 1 2 3 4 5 160 90 56 38 29 24 a Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. b Use the model to approximate the time when the temperature of the object was 100C. T Temperature in degrees Celsius 160 140 120 100 80 60 40 20 h 1 2 3 4 5 6 7 8 Hour EXPLORATION TRUE OR FALSE In Exercises 139142, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 139. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 140. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 141. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. 142. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 143. THINK ABOUT IT Is it possible for a logarithmic equation to have more than one extraneous solution Explain. 144. FINANCE You are investing P dollars at an annual interest rate of r, compounded continuously, for t years. Which of the following would result in the highest value of the investment Explain your reasoning. a Double the amount you invest. b Double your interest rate. c Double the number of years. 145. THINK ABOUT IT Are the times required for the investments in Exercises 117120 to quadruple twice as long as the times for them to double Give a reason for your answer and verify your answer algebraically. 146. The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yield for each savings plan when 1000 is deposited in a savings account. Which savings plan has the greatest effective yield Which savings plan will have the highest balance after 5 years a 7 annual interest rate, compounded annually b 7 annual interest rate, compounded continuously c 7 annual interest rate, compounded quarterly d 7.25 annual interest rate, compounded quarterly 147. GRAPHICAL ANALYSIS Let f x loga x and gx ax, where a 1. a Let a 1.2 and use a graphing utility to graph the two functions in the same viewing window. What do you observe Approximate any points of intersection of the two graphs. b Determine the value s of a for which the two graphs have one point of intersection. c Determine the value s of a for which the two graphs have two points of intersection. 148. CAPSTONE Write two or three sentences stating the general guidelines that you follow when solving a exponential equations and b logarithmic equations. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.5 419 Exponential and Logarithmic Models 5.5 EXPONENTIAL AND LOGARITHMIC MODELS What you should learn Recognize the five most common types of models involving exponential and logarithmic functions. Use exponential growth and decay functions to model and solve real life problems. Use Gaussian functions to model and solve real life problems. Use logistic growth functions to model and solve real life problems. Use logarithmic functions to model and solve real life problems. Why you should learn it Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are as follows. 1. Exponential growth model: y ae bx, 2. Exponential decay model: y ae 3. Gaussian model: y ae 4. Logistic growth model: y 5. Logarithmic models: y a b ln x, b 0 , xb 2c a 1 berx y a b log x The basic shapes of the graphs of these functions are shown in Figure 5.33. y Exponential growth and decay models are often used to model the populations of countries. For instance, in Exercise 44 on page 427, you will use exponential growth and decay models to compare the populations of several countries. y 4 4 3 3 y = e x y = ex 2 y 2 y = e x 2 2 1 1 1 x 1 2 3 1 3 2 1 2 x 1 1 2 2 1 y y = 1 + ln x 1 3 y= 1 + e 5x 1 x 1 Gaussian model y 3 1 1 Logistic growth model FIGURE 5.33 1 1 Exponential decay model y x 1 2 Exponential growth model Alan BeckerStoneGetty Images b 0 bx 2 y = 1 + log x 1 1 x x 1 1 1 2 2 Natural logarithmic model 2 Common logarithmic model You can often gain quite a bit of insight into a situation modeled by an exponential or logarithmic function by identifying and interpreting the functions asymptotes. Use the graphs in Figure 5.33 to identify the asymptotes of the graph of each function. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 420 Chapter 5 Exponential and Logarithmic Functions Exponential Growth and Decay Example 1 Online Advertising Estimates of the amounts in billions of dollars of U.S. online advertising spending from 2007 through 2011 are shown in the table. A scatter plot of the data is shown in Figure 5.34. Source: eMarketer Advertising spending 2007 2008 2009 2010 2011 21.1 23.6 25.7 28.5 32.0 S Dollars in billions Year Online Advertising Spending 50 40 30 20 10 t 7 8 9 10 11 Year 7 2007 An exponential growth model that approximates these data is given by S 10.33e0.1022t, 7 t 11, where S is the amount of spending in billions and t 7 represents 2007. Compare the values given by the model with the estimates shown in the table. According to this model, when will the amount of U.S. online advertising spending reach 40 billion FIGURE 5.34 Algebraic Solution Graphical Solution The following table compares the two sets of advertising spending figures. Use a graphing utility to graph the model y 10.33e0.1022x and the data in the same viewing window. You can see in Figure 5.35 that the model appears to fit the data closely. Year 2007 2008 2009 2010 2011 Advertising spending 21.1 23.6 25.7 28.5 32.0 Model 21.1 23.4 25.9 28.7 31.8 50 To find when the amount of U.S. online advertising spending will reach 40 billion, let S 40 in the model and solve for t. 10.33e0.1022t S 10.33e0.1022t 40 e0.1022t 3.8722 ln e0.1022t ln 3.8722 0.1022t 1.3538 t 13.2 Write original model. 0 Substitute 40 for S. 14 6 FIGURE Divide each side by 10.33. Take natural log of each side. Inverse Property Divide each side by 0.1022. According to the model, the amount of U.S. online advertising spending will reach 40 billion in 2013. 5.35 Use the zoom and trace features of the graphing utility to find that the approximate value of x for y 40 is x 13.2. So, according to the model, the amount of U.S. online advertising spending will reach 40 billion in 2013. Now try Exercise 43. T E C H N O LO G Y Some graphing utilities have an exponential regression feature that can be used to find exponential models that represent data. If you have such a graphing utility, try using it to find an exponential model for the data given in Example 1. How does your model compare with the model given in Example 1 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.5 Exponential and Logarithmic Models 421 In Example 1, you were given the exponential growth model. But suppose this model were not given; how could you find such a model One technique for doing this is demonstrated in Example 2. Example 2 Modeling Population Growth In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days Solution Let y be the number of flies at time t. From the given information, you know that y 100 when t 2 and y 300 when t 4. Substituting this information into the model y ae bt produces 100 ae2b and 300 ae 4b. To solve for b, solve for a in the first equation. 100 ae 2b a 100 e2b Solve for a in the first equation. Then substitute the result into the second equation. 300 ae 4b 300 e 100 e Write second equation. 4b Substitute 2b 100 for a. e2b 300 e 2b 100 Divide each side by 100. ln 3 2b Take natural log of each side. 1 ln 3 b 2 Solve for b. Using b 12 ln 3 and the equation you found for a, you can determine that 100 e212 ln 3 Substitute 12 ln 3 for b. 100 e ln 3 Simplify. 100 3 Inverse Property a Fruit Flies y 600 5, 520 Population 500 y = 33.33e 0.5493t 400 33.33. 4, 300 300 So, with a 33.33 and b ln 3 0.5493, the exponential growth model is 200 100 y 33.33e 0.5493t 2, 100 t 1 2 3 4 Time in days FIGURE Simplify. 1 2 5.36 5 as shown in Figure 5.36. This implies that, after 5 days, the population will be y 33.33e 0.54935 520 flies. Now try Exercise 49. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 422 Chapter 5 In living organic material, the ratio of the number of radioactive carbon isotopes carbon 14 to the number of nonradioactive carbon isotopes carbon 12 is about 1 to 1012. When organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14 begins to decay with a half life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon 14 to carbon 12 present at any time t in years . Carbon Dating R 1012 Exponential and Logarithmic Functions t=0 Ratio R = 112 e t8223 10 1 2 t = 5700 1012 R t = 19,000 1013 t 5000 1 t 8223 e 1012 Carbon dating model The graph of R is shown in Figure 5.37. Note that R decreases as t increases. 15,000 Time in years FIGURE 5.37 Example 3 Carbon Dating Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is R 11013. Algebraic Solution Graphical Solution In the carbon dating model, substitute the given value of R to obtain the following. Use a graphing utility to graph the formula for the ratio of carbon 14 to carbon 12 at any time t as 1 t 8223 e R 1012 et 8223 1 13 1012 10 et 8223 ln et 8223 1 10 1 ln 10 t 2.3026 8223 t 18,934 Write original model. Let R 1 . 1013 Multiply each side by 1012. y1 1 x8223 e . 1012 In the same viewing window, graph y2 11013. Use the intersect feature or the zoom and trace features of the graphing utility to estimate that x 18,934 when y 11013, as shown in Figure 5.38. 1012 y1 = Take natural log of each side. 1 ex8223 1012 y2 = Inverse Property Multiply each side by 8223. So, to the nearest thousand years, the age of the fossil is about 19,000 years. 0 1 1013 25,000 0 FIGURE 5.38 So, to the nearest thousand years, the age of the fossil is about 19,000 years. Now try Exercise 51. The value of b in the exponential decay model y aebt determines the decay of radioactive isotopes. For instance, to find how much of an initial 10 grams of 226Ra isotope with a half life of 1599 years is left after 500 years, substitute this information into the model y aebt. 1 10 10eb1599 2 1 ln 1599b 2 1 b ln 2 1599 Using the value of b found above and a 10, the amount left is y 10eln121599500 8.05 grams. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.5 Exponential and Logarithmic Models 423 Gaussian Models As mentioned at the beginning of this section, Gaussian models are of the form y aexb c. 2 This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell shaped curve. Try graphing the normal distribution with a graphing utility. Can you see why it is called a bell shaped curve For standard normal distributions, the model takes the form y 1 2 ex 2. 2 The average value of a population can be found from the bell shaped curve by observing where the maximum y value of the function occurs. The x value corresponding to the maximum y value of the function represents the average value of the independent variablein this case, x. Example 4 SAT Scores In 2008, the Scholastic Aptitude Test SAT math scores for college bound seniors roughly followed the normal distribution given by y 0.0034ex515 26,912, 2 200 x 800 where x is the SAT score for mathematics. Sketch the graph of this function. From the graph, estimate the average SAT score. Source: College Board Solution The graph of the function is shown in Figure 5.39. On this bell shaped curve, the maximum value of the curve represents the average score. From the graph, you can estimate that the average mathematics score for college bound seniors in 2008 was 515. SAT Scores y 50 of population Distribution 0.003 0.002 0.001 x = 515 x 200 400 600 800 Score FIGURE 5.39 Now try Exercise 57. www.elsolucionario.net . http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 424 Chapter 5 Exponential and Logarithmic Functions y Logistic Growth Models Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph in Figure 5.40. One model for describing this type of growth pattern is the logistic curve given by the function Decreasing rate of growth y Increasing rate of growth x FIGURE a 1 ber x where y is the population size and x is the time. An example is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. 5.40 Example 5 Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long lasting flu virus. The spread of the virus is modeled by y 5000 , 1 4999e0.8t t 0 where y is the total number of students infected after t days. The college will cancel classes when 40 or more of the students are infected. a. How many students are infected after 5 days b. After how many days will the college cancel classes Algebraic Solution Graphical Solution a. After 5 days, the number of students infected is a. Use a graphing utility to graph y 5000 5000 54. 1 4999e0.85 1 4999e4 b. Classes are canceled when the number infected is 0.405000 2000. y 2000 5000 1 4999e0.8t y1 1 4999e0.8t 2.5 e0.8t 1.5 4999 ln e0.8t ln 1.5 4999 0.8t ln 1.5 4999 t 5000 . Use 1 4999e0.8x the value feature or the zoom and trace features of the graphing utility to estimate that y 54 when x 5. So, after 5 days, about 54 students will be infected. b. Classes are canceled when the number of infected students is 0.405000 2000. Use a graphing utility to graph 5000 and y2 2000 1 4999e0.8x in the same viewing window. Use the intersect feature or the zoom and trace features of the graphing utility to find the point of intersection of the graphs. In Figure 5.41, you can see that the point of intersection occurs near x 10.1. So, after about 10 days, at least 40 of the students will be infected, and the college will cancel classes. 6000 1 1.5 ln 0.8 4999 y2 = 2000 y1 = 5000 1 + 4999e0.8x t 10.1 So, after about 10 days, at least 40 of the students will be infected, and the college will cancel classes. 0 20 0 FIGURE 5.41 Now try Exercise 59. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.5 Exponential and Logarithmic Models 425 Logarithmic Models Claro Cortes IVReuters Landov Example 6 Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensity I is given by R log On May 12, 2008, an earthquake of magnitude 7.9 struck Eastern Sichuan Province, China. The total economic loss was estimated at 86 billion U.S. dollars. I I0 where I0 1 is the minimum intensity used for comparison. Find the intensity of each earthquake. Intensity is a measure of the wave energy of an earthquake. a. Nevada in 2008: R 6.0 b. Eastern Sichuan, China in 2008: R 7.9 Solution a. Because I0 1 and R 6.0, you have 6.0 log I 1 Substitute 1 for I0 and 6.0 for R. 106.0 10log I I 106.0 1,000,000. Exponentiate each side. Inverse Property b. For R 7.9, you have 7.9 log I 1 Substitute 1 for I0 and 7.9 for R. 107.9 10log I I 10 7.9 79,400,000. Exponentiate each side. Inverse Property Note that an increase of 1.9 units on the Richter scale from 6.0 to 7.9 represents an increase in intensity by a factor of 79,400,000 79.4. 1,000,000 In other words, the intensity of the earthquake in Eastern Sichuan was about 79 times as great as that of the earthquake in Nevada. Now try Exercise 63. t Year Population, P 1 2 3 4 5 6 7 8 9 10 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 92.23 106.02 123.20 132.16 151.33 179.32 203.30 226.54 248.72 281.42 CLASSROOM DISCUSSION Comparing Population Models The populations P in millions of the United States for the census years from 1910 to 2000 are shown in the table at the left. Least squares regression analysis gives the best quadratic model for these data as P 1.0328t 2 9.607t 81.82, and the best exponential model for these data as P 82.677e0.124t. Which model better fits the data Describe how you reached your conclusion. Source: U.S. Census Bureau www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 426 Chapter 5 5.5 Exponential and Logarithmic Functions EXERCISES See www.CalcChat.com for worked out solutions to odd numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. An exponential growth model has the form ________ and an exponential decay model has the form ________. A logarithmic model has the form ________ or ________. Gaussian models are commonly used in probability and statistics to represent populations that are ________ ________. The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum y value of the graph. 5. A logistic growth model has the form ________. 6. A logistic curve is also called a ________ curve. SKILLS AND APPLICATIONS In Exercises 712, match the function with its graph. The graphs are labeled a , b , c , d , e , and f . y a y b 6 COMPOUND INTEREST In Exercises 1522, complete the table for a savings account in which interest is compounded continuously. 8 4 4 2 2 x 2 4 6 2 y c x 4 2 4 6 y d 4 12 2 8 8 x 2 4 2 4 6 4 8 y e y f 4 2 6 12 6 6 11. y lnx 1 2 4 2 12 7. y 2e x4 9. y 6 logx 2 x 2 14. A P 1 r n 7 34 yr 12 yr 4.5 2 1505.00 19,205.00 10,000.00 2000.00 24. r 312, t 15 COMPOUND INTEREST In Exercises 25 and 26, determine the time necessary for 1000 to double if it is invested at interest rate r compounded a annually, b monthly, c daily, and d continuously. r In Exercises 13 and 14, a solve for P and b solve for t. 13. A Pe rt 26. r 6.5 27. COMPOUND INTEREST Complete the table for the time t in years necessary for P dollars to triple if interest is compounded continuously at rate r. 4 1 e2x nt Amount After 10 Years 25. r 10 8. y 6ex4 2 10. y 3ex2 5 12. y Time to Double 10 12 23. r 5, t 10 6 x Annual Rate 3.5 COMPOUND INTEREST In Exercises 23 and 24, determine the principal P that must be invested at rate r, compounded monthly, so that 500,000 will be available for retirement in t years. x 4 15. 16. 17. 18. 19. 20. 21. 22. Initial Investment 1000 750 750 10,000 500 600 2 4 6 8 10 12 t 28. MODELING DATA Draw a scatter plot of the data in Exercise 27. Use the regression feature of a graphing utility to find a model for the data. www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com Section 5.5 29. COMPOUND INTEREST Complete the table for the time t in years necessary for P dollars to triple if interest is compounded annually at rate r. 2 r 4 6 8 10 12 41. x 0 4 y 5 1 30. MODELING DATA Draw a scatter plot of the data in Exercise 29. Use the regression feature of a graphing utility to find a model for the data. 31. COMPARING MODELS If 1 is invested in an account over a 10 year period, the amount in the account, where t represents the time in years, is given by A 1 0.075 t or A e0.07t depending on whether 1 the account pays simple interest at 72 or continuous compound interest at 7. Graph each function on the same set of axes. Which grows at a higher rate Remember that t is the greatest integer function discussed in Section 2.4. 32. COMPARING MODELS If 1 is invested in an account over a 10 year period, the amount in the account, where t represents the time in years, is given by A 1 0.06 t or A 1 0.055365365t depending on whether the account pays simple interest at 6 or compound interest at 512 compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a higher rate 36. 37. 38. 226Ra 14C 239Pu Initial Quantity 10 g 6.5 g 2.1 g 1599 5715 24,100 y 39. 3, 10 10 2g 2g 0.4 g 4, 5 6 6 4 4 2 aebx 8 8 0, 12 2 0, 1 x 1 2 3 4 5 3 y 1 1 4 Year 1970 1980 1990 2000 2007 Population b According to the model, when will the population of Horry County reach 300,000 c Do you think the model is valid for long term predictions of the population Explain. 44. POPULATION The table shows the populations in millions of five countries in 2000 and the projected populations in millions for the year 2015. Source: U.S. Census Bureau Country 2000 2015 Bulgaria Canada China United Kingdom United States 7.8 31.1 1268.9 59.5 282.2 6.9 35.1 1393.4 62.2 325.5 a Find the exponential growth or decay model y ae bt or y aebt for the population of each country by letting t 0 correspond to 2000. Use the model to predict the population of each country in 2030. y 40. 0 where t represents the year, with t 0 corresponding to 1970. Source: U.S. Census Bureau a Use the model to complete the table. Amount After 1000 Years In Exercises 3942, find the exponential model y fits the points shown in the graph or table. x P 18.5 92.2e0.0282t RADIOACTIVE DECAY In Exercises 3338, complete the table for the radioactive isotope. Half life years 1599 5715 24,100 42. 427 43. POPULATION The populations P in thousands of Horry County, South Carolina from 1970 through 2007 can be modeled by t Isotope 33. 226Ra 34. 14C 35. 239Pu Exponential and Logarithmic Models that b You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y ae bt is determined by these different growth rates Discuss the relationship between the different growth rates and the magnitude of the constant. c You can see that the population of China is increasing while the population of Bulgaria is decreasing. What constant in the equation y ae bt reflects this difference Explain. x 1 2 3 4 www.elsolucionario.net http:www.leeydescarga.com mas libros gratis en http:www.leeydescarga.com 428 Chapter 5 Exponential and Logarithmic Functions 45. WEBSITE GROWTH The number y of hits a new search engine website receives each month can be modeled by y 4080e kt, where t represents the number of months the website has been operating. In the websites third month, there were 10,000 hits. Find the value of k, and use this value to predict the number of hits the website will receive after 24 months. 46. VALUE OF A PAINTING The value V in millions of dollars of a famous painting can be modeled by V 10e kt, where t represents the year, with t 0 corresponding to 2000. In 2008, the same painting was sold for 65 million. Find the value of k, and use this value to predict the value of the painting in 2014. 47. POPULATION The populations P in thousands of Reno, Nevada from 2000 through 2007 can be modeled by P 346.8ekt, where t represents the year, with t 0 corresponding to 2000. In 2005, the population of Reno was about 395,000. Source: U.S. Census Bureau a Find the value of k. Is the population increasing or decreasing Explain. b Use the model to find the populations of Reno in 2010 and 2015. Are the results reasonable Explain. c According to the model, during what year will the population reach 500,000 48. POPULATION The populations P in thousands of Orlando, Florida from 2000 through 2007 can be modeled by P