net bar51950_fm_i-xviii.qxd 2/1/08 6:08 PM Page i Pinnacle 110:MHIA064:mhbar3:SE:fm: THIRD EDITION College Algebra GRAPHS AND MODELS Raymond A. Barnett Merritt College Michael R. Ziegler Marquette University Karl E. Byleen Marquette University Dave Sobecki Miami University Hamilton www.net bar51950_fm_i-xviii.qxd 2/1/08 6:08 PM Page ii Pinnacle 110:MHIA064:mhbar3:SE:fm: COLLEGE ALGEBRA: GRAPHS AND MODELS, THIRD EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020.
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw- Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States.
This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 9 8 ISBN 978–0–07–305195–6 MHID 0–07–305195–0 ISBN 978–0–07–334187–3 (Annotated Instructor’s Edition) MHID 0–07–334187–8 Editorial Director: Stewart K. Mattson Sponsoring Editor: Dawn R. Bercier Vice-President New Product Launches: Michael Lange Developmental Editor: Katie White Senior Marketing Manager: John Osgood Senior Project Manager: Sheila M.
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25 PhotoDisc/Getty; CO 7: © Corbis RF; CO 8: © BrandX RF/Superstock; p. Library of Congress Cataloging-in-Publication Data College algebra : graphs and models. ISBN 978–0–07–305195–6 — ISBN 0–07–305195–0 (hard copy : alk. Algebra––Graphic methods––Textbooks.net bar51950_fm_i-xviii.qxd 2/1/08 6:08 PM Page iii Pinnacle 110:MHIA064:mhbar3:SE:fm: About the Authors Raymond A.
Barnett, a native of and educated in California, received his B. in mathematical statistics from the University of California at Berkeley and his M. in mathematics from the University of Southern California. He has been a member of the Merritt College Mathematics Department and was chairman of the department for four years.
Associated with four different publishers, Raymond Barnett has authored or co- authored 18 textbooks in mathematics, most of which are still in use. In addition to international English editions, a number of the books have been translated into Span- ish. Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, North- ern Kentucky University; Charles Burke, City College of San Francisco; John Fujii, Merritt College; and Karl Byleen, Marquette University. Ziegler received his B.
from Shippensburg State College and his M. from the University of Delaware. After completing postdoctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he currently holds the rank of Professor in the Department of Mathematics, Statistics, and Computer Science. Ziegler has published more than a dozen research articles in complex analysis and has co-authored more than a dozen undergraduate mathemat- ics textbooks with Raymond Barnett and Karl Byleen.
Byleen received his B. degrees in mathematics from the University of Nebraska. He is currently an Associate Professor in the Department of Mathematics, Statistics, and Computer Science of Marquette University. He has pub- lished a dozen research articles on the algebraic theory of semigroups and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler.
Dave Sobecki earned a B. in math education from Bowling Green State University, then went on to earn an M. in mathematics from Bowling Green. He is an Associate Professor in the Department of Mathematics and Statistics at Miami Uni- versity in Hamilton, Ohio.
He has written or co-authored five journal articles, eleven books and five interactive CD-ROMs. Dave lives in Fairfield, Ohio, with his wife (Cat) and dogs (Large Coney and Macleod). His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, and home improvement projects.net bar51950_fm_i-xviii.qxd 2/1/08 6:08 PM Page iv Pinnacle 110:MHIA064:mhbar3:SE:fm: Barnett, Ziegler, Byleen and Sobecki’s Precalculus Series College Algebra, Eighth Edition This book is the same as Precalculus without the three chapters on trigonometry. ISBN 0-07-286738-8, ISBN 978-0-07-286738-1 Precalculus, Sixth Edition This book is the same as College Algebra with three chapters of trigonometry added.
The trigonometry functions are introduced by a unit circle approach. ISBN 0-07-286739-6, ISBN 978-0-07-286739-8 College Algebra with Trigonometry, Eighth Edition This book differs from Precalculus in that College Algebra with Trigonometry uses right triangle trigonometry to introduce the trigonometric functions. ISBN 0-07-331264-9, ISBN 978-0-07-331264-4 College Algebra: Graphs and Models, Third Edition This book is the same as Precalculus: Graphs and Models without the three chapters on trigonometry. This text assumes the use of a graphing calculator.
ISBN 0-07-305195-0, ISBN 978-0-07-305195-6 Precalculus: Graphs and Models, Third Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry. The trigonometric functions are introduced by a unit circle approach. This text assumes the use of a graphing calculator. ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3 College Algebra with Trigonometry: Graphs and Models This book is the same as Precalculus: Graphs and Models except that the trigono- metric functions are introduced by right triangle trigonometry.
This text assumes the use of a graphing calculator. ISBN 0-07-291699-0, ISBN 978-0-07-291699-7 www.net bar51950_fm_i-xviii.qxd 2/1/08 6:08 PM Page v Pinnacle 110:MHIA064:mhbar3:SE:fm: Contents Preface vii CHAPTER 4 Exponential and Logarithmic Application Index xvi Functions 381 4-1 Exponential Functions 382 CHAPTER 1 Functions, Graphs, 4-2 Exponential Models 399 and Models 1 4-3 Logarithmic Functions 416 1-1 Using Graphing Calculators 2 4-4 Logarithmic Models 430 1-2 Functions 21 4-5 Exponential and Logarithmic Equations 440 1-3 Functions: Graphs and Properties 46 Chapter 4 Review 462 1-4 Functions: Graphs and Transformations 67 Chapter 4 Group Activity: Comparing Regression 1-5 Operations on Functions; Composition 84 Models 456 1-6 Inverse Functions 99 Cumulative Review Exercises Chapter 1 Review 119 Chapters 3–4 457 Chapter 1 Group Activity: Mathematical Modeling: Choosing a Cell Phone Provider 126 CHAPTER 2 Modeling with Linear and CHAPTER 5 Modeling with Systems of Quadratic Functions 127 Equations and Inequalities 463 2-1 Linear Functions 128 5-1 Systems of Linear Equations in Two 2-2 Linear Equations and Models 151 Variables 464 2-3 Quadratic Functions 172 5-2 Systems of Linear Equations in Three 2-4 Complex Numbers 190 Variables 482 2-5 Quadratic Equations and Models 206 5-3 Systems of Linear Inequalities 495 2-6 Additional Equation-Solving Techniques 226 5-4 Linear Programming 510 2-7 Solving Inequalities 241 Chapter 5 Review 522 Chapter 2 Review 258 Chapter 5 Group Activity: Heat Conduction 526 Chapter 2 Group Activity: Mathematical Modeling in Population Studies 265 Cumulative Review Exercises Chapters 1–2 267 CHAPTER 6 Matrices and Determinants 527 6-1 Systems of Linear Equations: Gauss–Jordan Elimination 528 CHAPTER 3 Polynomial and Rational 6-2 Matrix Operations 547 Functions 273 6-3 Inverse of a Square Matrix 565 3-1 Polynomial Functions and Models 274 6-4 Matrix Equations and Systems of Linear 3-2 Polynomial Division 291 Equations 578 3-3 Real Zeros and Polynomial Inequalities 303 6-5 Determinants 588 3-4 Complex Zeros and Rational Zeros of 6-6 Properties of Determinants 597 Polynomials 320 6-7 Determinants and Cramer’s Rule 604 3-5 Rational Functions and Inequalities 336 Chapter 6 Review 610 3-6 Variation and Modeling 361 Chapter 6 Group Activity: Using Matrices to Chapter 3 Review 371 Find Cost, Revenue, and Profit 616 Chapter 3 Group Activity: Interpolating Polynomials 378 Cumulative Review Exercises Chapters 5–6 618 V www.net bar51950_fm_i-xviii.qxd 01/03/2008 12:01 AM Page vi pinnacle 110:MHIA064:mhbar3:SE:fm: CHAPTER 7 Sequences, Induction, and APPENDIX A Basic Algebra Review A-1 Probability 621 A-1 Algebra and Real Numbers A-2 7-1 Sequences and Series 622 A-2 Exponents A-13 7-2 Mathematical Induction 633 A-3 Radicals A-27 7-3 Arithmetic and Geometric Sequences 644 A-4 Polynomials: Basic Operations A-36 7-4 The Multiplication Principle, Permutations, and A-5 Polynomials: Factoring A-47 Combinations 659 A-6 Rational Expressions: Basic Operations A-58 7-5 Sample Spaces and Probability 676 Appendix A Review * 7-6 The Binomial Formula 696 Appendix A Group Activity: Rational Chapter 7 Review 704 and Irrational Numbers * Chapter 7 Group Activity: Sequences Specified *Available online at www.com/barnett by Recursion Formulas 709 APPENDIX B Review of Equations and Graphing A-69 CHAPTER 8 Additional Topics in Analytic B-1 Linear Equations and Inequalities A-70 Geometry 711 B-2 Cartesian Coordinate System A-82 8-1 Conic Sections; Parabola 712 B-3 Basic Formulas in Analytic Geometry A-91 8-2 Ellipse 723 8-3 Hyperbola 735 APPENDIX C Special Topics A-105 8-4 Systems of Nonlinear Equations 750 C-1 Significant Digits A-106 Chapter 8 Review 761 C-2 Partial Fractions A-109 Chapter 8 Group Activity: Focal Chords 765 C-3 Descartes’ Rule of Signs A-118 C-4 Parametric Equations * Cumulative Review Exercises *Available online at www.com/barnett Chapters 7–8 766 APPENDIX D Geometric Formulas A-123 Student Answers SA-1 VI www.net bar51950_fm_i-xviii.qxd 2/1/08 6:09 PM Page vii Pinnacle 110:MHIA064:mhbar3:SE:fm: Preface Enhancing a Tradition of Success We take great satisfaction from the fact that more than 100,000 students have learned col- lege algebra from a Barnett Series textbook. Ray Barnett is one of the masters of college textbook writing. His central approach is proven and remains effective for today’s students.
The third edition of College Algebra: Graphs and Models has benefited greatly from the numerous contributions of new coauthor Dave Sobecki of Miami University Hamilton. Dave brings a fresh approach to the material and many good suggestions for improving student accessibility. Every aspect of the revision focuses on making the text more relevant to students, while retaining the precise presentation of the math- ematics for which the Barnett name is renowned. Specifically we concentrated on the areas of writing, worked examples, exercises, technology, and design.
Based on numerous reviews, advice from expert consultants, and direct correspondence with many users of previous editions, we feel that this edi- tion is more relevant than ever before. We hope you will agree. Writing Without sacrificing breadth or depth or coverage, we have rewritten expla- nations to make them clearer and more direct. As in previous editions, the text empha- sizes computational skills, real-world data analysis and modeling, and problem solv- ing rather than theory.
Examples In the new edition, even more solved examples in the book provide graph- ical solutions side-by-side with algebraic solutions. By seeing the same answer result from their symbol manipulations and from graphical approaches, students gain insight into the power of algebra and make important conceptual and visual connections. Likewise, we added expanded color annotations to many examples, explaining the solution steps in words. Each example is then followed by a similar matched problem for the student to solve.
Answers to the matched problems are located at the end of each section for easy reference. This active involvement in learning while reading helps students develop a more thorough understanding of concepts and processes. Exercises With an eye to improving student performance and to make the book more useful for instructors, we have extensively revised the exercise sets. We added hundreds of new writing questions as well as exercises at the easy to moderate level and expanded the variety of problem types to ensure a gradual increase in difficulty level throughout each exercise set.
Technology Although technology is employed throughout, we strive to balance algebraic skill development with the use of technology as an aid to learning and prob- lem solving. We assume that students using the book will have access to one of the various graphing calculators or computer programs that are available to perform the following operations: • Simultaneously display multiple graphs in a user-selected viewing window • Explore graphs using trace and zoom VII www.net bar51950_fm_i-xviii.