Algebra and Trigonometry

1238 Pages·2009·80.533 MB·English

by  Robert F. Blitzer

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A BRIEF GUIDE TO GETTING THE MOST FROM THIS BOOK
(cid:1)
Read the Book
Feature Description Benefit Page
Section-Opening Every section opens with a scenario presenting Realizing that algebra and trigonometry 101
Scenarios a unique application of algebra or trigonometry are everywhere will help motivate your
in your life outside the classroom. learning.
Detailed Worked-Out Examples are clearly written and provide The blue annotations will help you 612
Examples step-by-step solutions.No steps are omitted, understand the solutions by providing
and each step is thoroughly explained to the the reason why every algebraic and
right of the mathematics. trigonometric step is true.
Applications Using Interesting applications from nearly every Ever wondered how you’ll use algebra 238
Real-World Data discipline,supported by up-to-date real-world and trigonometry? This feature will
data,are included in every section. show you how algebra and trigonometry
can solve real problems.
Study Tips The book’s study tip boxes offer suggestions By seeing common mistakes,you’ll 524
for problem solving,point out common errors be able to avoid them.
to avoid, and provide informal hints and
suggestions.
Explanatory Voice Voice balloons help to demystify algebra and Does math ever look foreign to you? 184
Balloons trigonometry. They translate mathematical This feature often translates math into
language into plain English, clarify problem- everyday English.
solving procedures, and present alternative
ways of understanding.
Learning Objectives Every section begins with a list of objectives. The objectives focus your reading by 571
Each objective is restated in the margin where emphasizing what is most important and
the objective is covered. where to find it.
Technology The screens displayed in the technology boxes Even if you are not using a graphing 105
show how graphing utilities verify and visualize utility in the course,this feature will help
algebraic and trigonometric results. you understand different approaches to
problem solving.
(cid:2)
Work the Problems
Feature Description Benefit Page
Check Point Each example is followed by a similar You learn best by doing. You’ll solidify 671
Examples matched problem,called a Check Point,that your understanding of worked examples
offers you the opportunity to work a similar if you try a similar problem right away to
exercise. The answers to the Check Points be sure you understand what you’ve
are provided in the answer section. just read.
Extensive and Varied An abundant collection of exercises is The parallel order of the Practice 554-557
Exercise Sets included in an exercise set at the end of each Exercises lets you refer to the worked
section. Exercises are organized within examples and use them as models for
categories. Your instructor will usually solving these problems.
provide guidance on which exercises to NEW to This EditionThe “make sense” 136
work. The exercises in the first category, exercises ask you to determine whether
Practice Exercises,follow the same order as statements are sensible,and to explain
the section’s worked examples. why or why not. This enables you to
test your conceptual understanding of
the section's objectives.
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A BRIEF GUIDE TO GETTING THE MOST FROM THIS BOOK
Feature Description Benefit Page
Practice Plus This category of exercises contains more It is important to dig in and develop 362
Problems challenging problems that often require you your problem-solving skills. Practice
to combine several skills or concepts. Plus Exercises provide you with ample
opportunity to do so.
Preview Problems NEW to This Edition Each exercise set These exercises let you review 597
concludes with three problems to help you previously covered material that you'll
prepare for the next section. need to be successful for the
forthcoming section. Some of these
problems will get you thinking about
concepts you'll soon encounter.
(cid:3)
Review for Quizzes and Tests
Feature Description Benefit Page
Mid-Chapter At approximately the midway point in the By combining exercises from the first 705
Check Points chapter,an integrated set of review exercises half of the chapter, the Mid-Chapter
allows you to review the skills and concepts Check Points give a comprehensive
you learned separately over several sections. review before you move on to the
material in the remainder of the
chapter.
Chapter Review Grids Each chapter contains a review chart that Review this chart and you’ll know the 598-600
summarizes the definitions and concepts in most important material in the chapter!
every section of the chapter. Examples that
illustrate these key concepts are also
included in the chart.
Chapter Review A comprehensive collection of review exercises Practice makes perfect. These 601-603
Exercises for each of the chapter’s sections follows the exercises contain the most significant
grid. problems for each of the chapter’s
sections.
Chapter Tests Each chapter contains a practice test with You can use the chapter test to 603-604
approximately 25 problems that cover the determine whether you have
important concepts in the chapter.Take the mastered the material covered in the
practice test, check your answers, and then chapter.
watch the Chapter Test Prep Video CD to see
worked-out solutions for any exercises you
miss.
Chapter Test Prep These video CDs found at the back of your The videos let you review any 603-604
Video CDs text contain worked-out solutions to every exercises you miss on the chapter
exercise in each chapter test. test.
Cumulative Review Beginning with Chapter 2, each chapter Ever forget what you’ve learned? 604
Exercises concludes with a comprehensive collection These exercises ensure that you are
of mixed cumulative review exercises.These not forgetting anything as you move
exercises combine problems from previous forward.
chapters and the present chapter, providing
an ongoing cumulative review.
A-BLTZMFMT_i-xviii-NASTA-hr 3-11-2008 15:33 Page iii
Algebra
and Trigonometry
A-BLTZMFMT_i-xviii-NASTA-hr 3-11-2008 15:33 Page v
Algebra
and Trigonometry
4e
Robert Blitzer
Miami Dade College
Upper Saddle River,NJ 07458
A-BLTZMFMT_i-xviii-NASTA-hr2 4-11-2008 17:40 Page vi
Library of Congress Cataloging-in-Publication Data
Blitzer,Robert.
Algebra and trigonometry / Robert Blitzer.—4th ed.
p. cm.
Includes index.
ISBN 0-321-55985-1
1. Algebra—Textbooks. 2. Trigonometry—Textboks I. Title.
QA152.3.B63 2010
512.13—dc22 2008043253
Editorial Director,Mathematics:Christine Hoag
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Compositor:Prepare,Inc.
© 2010,2007,2004,2001 Pearson Education,Inc.
Pearson Prentice Hall
Pearson Education,Inc.
Upper Saddle River,New Jersey 07458
All rights reserved.No part of this book may be reproduced,in any form or by any means,without written permission from the publisher.
Pearson Prentice Hall™ is a trademark of Pearson Education,Inc.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
ISBN-13: 978-0-13-136218-5 (High School binding)
ISBN-10: 0-13-136218-6 (High School binding)
ISBN-13: 978-0-321-55985-2 (College Edition)
ISBN-10: 0-321-55985-1 (College Edition)
Pearson Education Ltd.,London
Pearson Education Singapore,Pte.Ltd.
Pearson Education Canada,Inc.
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Pearson Education Australia PTY,Limited
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A-BLTZMFMT_i-xviii-NASTA-hr 3-11-2008 15:33 Page vii
Preface xi CONTENTS
Acknowledgments xiv
To the Student xv
About the Author xvi P Prerequisites: Fundamental Concepts of Algebra 1
P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 2
P.2 Exponents and Scientific Notation 19
P.3 Radicals and Rational Exponents 33
P.4 Polynomials 49
Mid-Chapter Check Point 61
P.5 Factoring Polynomials 62
P.6 Rational Expressions 73
SUMMARY, REVIEW, AND TEST 85
REVIEW EXERCISES 86
CHAPTER P TEST 88
1 Equations and Inequalities 89
1.1 Graphs and Graphing Utilities 90
1.2 Linear Equations and Rational Equations 101
1.3 Models and Applications 116
1.4 Complex Numbers 130
1.5 Quadratic Equations 136
Mid-Chapter Check Point 156
1.6 Other Types of Equations 158
1.7 Linear Inequalities and Absolute Value Inequalities 172
SUMMARY, REVIEW, AND TEST 189
REVIEW EXERCISES 191
CHAPTER 1 TEST 195
vii
A01_BLIT2185_04_SB_FM.QXD 6/8/10 1:53 PM Page viii
viii Contents
2 Functions and Graphs 197 4 Exponential and Logarithmic
Functions 411
2.1 Basics of Functions and Their Graphs 198
2.2 Moreon Functions and Their Graphs 215 4.1 Exponential Functions 412
2.3 Linear Functions and Slope 229 4.2 Logarithmic Functions 424
2.4 More on Slope 243 4.3 Properties of Logarithms 437
Mid-Chapter Check Point 253 Mid-Chapter Check Point 447
2.5 Transformations of Functions 254 4.4 Exponential and Logarithmic Equations 447
2.6 Combinations of Functions; Composite 4.5 Exponential Growth and Decay; Modeling
Functions 270 Data 460
2.7 Inverse Functions 282 SUMMARY, REVIEW, AND TEST 474
2.8 Distance and Midpoint Formulas; Circles 293 REVIEW EXERCISES 475
SUMMARY, REVIEW, AND TEST 302 CHAPTER 4 TEST 479
REVIEW EXERCISES 304 CUMULATIVE REVIEW EXERCISES
(CHAPTERS1–4) 480
CHAPTER 2 TEST 308
CUMULATIVE REVIEW EXERCISES
(CHAPTERS 1–2) 310
3 Polynomial and Rational 5 Trigonometric Functions 481
Functions 311
5.1 Angles and Radian Measure 482
3.1 Quadratic Functions 312 5.2 Right Triangle Trigonometry 497
3.2 Polynomial Functions and Their Graphs 328 5.3 Trigonometric Functions of Any Angle 513
3.3 Dividing Polynomials; Remainder and Factor 5.4 Trigonometric Functions of Real Numbers;
Theorems 342 Periodic Functions 526
3.4 Zeros of Polynomial Functions 352 Mid-Chapter Check Point 535
Mid-Chapter Check Point 365 5.5 Graphs of Sine and Cosine Functions 536
3.5 Rational Functions and Their Graphs 366 5.6 Graphs of Other Trigonometric Functions 558
3.6 Polynomial and Rational Inequalities 384 5.7 Inverse Trigonometric Functions 571
3.7 Modeling Using Variation 394 5.8 Applications of Trigonometric Functions 587
SUMMARY, REVIEW, AND TEST 404 SUMMARY, REVIEW, AND TEST 598
REVIEW EXERCISES 406 REVIEW EXERCISES 601
CHAPTER 3 TEST 409 CHAPTER 5 TEST 603
CUMULATIVE REVIEW EXERCISES CUMULATIVE REVIEW EXERCISES
(CHAPTERS 1–3) 410 (CHAPTERS1–5) 604
A-BLTZMFMT_i-xviii-NASTA-hr 3-11-2008 15:33 Page ix
Contents ix
6 Analytic Trigonometry 605 8 Systems of Equations and
Inequalities 747
6.1 Verifying Trigonometric Identities 606
6.2 Sum and Difference Formulas 616 8.1 Systems of Linear Equations in Two
Variables 748
6.3 Double-Angle, Power-Reducing, and
Half-Angle Formulas 627 8.2 Systems of Linear Equations in Three
Variables 764
Mid-Chapter Check Point 638
8.3 Partial Fractions 772
6.4 Product-to-Sum and Sum-to-Product
Formulas 639 8.4 Systems of Nonlinear Equations in Two
Variables 783
6.5 Trigonometric Equations 646
Mid-Chapter Check Point 793
SUMMARY, REVIEW, AND TEST 659
8.5 Systems of Inequalities 794
REVIEW EXERCISES 660
8.6 Linear Programming 806
CHAPTER 6 TEST 662
SUMMARY, REVIEW, AND TEST 814
CUMULATIVE REVIEW EXERCISES
(CHAPTERS1–6) 662 REVIEW EXERCISES 815
CHAPTER 8 TEST 818
CUMULATIVE REVIEW EXERCISES
(CHAPTERS1–8) 819
7 Additional Topics in
Trigonometry 663
9 Matrices and Determinants 821
7.1 The Law of Sines 664
7.2 The Law of Cosines 676
9.1 Matrix Solutions to Linear Systems 822
7.3 Polar Coordinates 684
9.2 Inconsistent and Dependent Systems and Their
7.4 Graphs of Polar Equations 695 Applications 834
Mid-Chapter Check Point 705 9.3 Matrix Operations and Their Applications 843
7.5 Complex Numbers in Polar Form; Mid-Chapter Check Point 858
DeMoivre’s Theorem 706
9.4 Multiplicative Inverses of Matrices and Matrix
7.6 Vectors 718 Equations 858
7.7 The Dot Product 733 9.5 Determinants and Cramer’s Rule 872
SUMMARY, REVIEW, AND TEST 741 SUMMARY, REVIEW, AND TEST 884
REVIEW EXERCISES 743 REVIEW EXERCISES 885
CHAPTER 7 TEST 745 CHAPTER 9 TEST 887
CUMULATIVE REVIEW EXERCISES CUMULATIVE REVIEW EXERCISES
(CHAPTERS1–7) 746 (CHAPTERS1–9) 888
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x Contents
10 Conic Sections and Analytic Geometry 889
10.1 The Ellipse 890
10.2 The Hyperbola 902
10.3 The Parabola 916
Mid-Chapter Check Point 928
10.4 Rotation of Axes 929
10.5 Parametric Equations 941
10.6 Conic Sections in Polar Coordinates 951
SUMMARY, REVIEW, AND TEST 960
REVIEW EXERCISES 962
CHAPTER 10 TEST 964
CUMULATIVE REVIEW EXERCISES (CHAPTERS1–10) 965
11 Sequences, Induction, and Probability 967
11.1 Sequences and Summation Notation 968
11.2 Arithmetic Sequences 979
11.3 Geometric Sequences and Series 988
Mid-Chapter Check Point 1002
11.4 Mathematical Induction 1003
11.5 The Binomial Theorem 1012
11.6 Counting Principles, Permutations, and Combinations 1019
11.7 Probability 1031
SUMMARY, REVIEW, AND TEST 1045
REVIEW EXERCISES 1047
CHAPTER 11 TEST 1050
CUMULATIVE REVIEW EXERCISES (CHAPTERS1–11) 1051
Appendix A Where Did That Come From?
Selected Proofs A-1
Answers to Selected Exercises AA1
Applications Index AP-1
Subject Index I-1
Photo Credits P-1
A-BLTZMFMT_i-xviii-NASTA-hr1 4-11-2008 13:05 Page xi
Preface
I’ve written Algebra and Trigonometry,Fourth Edition, Although sample answers are provided, students
to help diverse students, with different backgrounds have skills and perspectives that frequently differ
and future goals, to succeed.The book has three funda- from those of math teachers, so answers and
mental goals: explanations may vary. The important part of this
new feature is to let you ask students what they
1. To help students acquire a solid foundation in think about selected statements,determine whether
algebra and trigonometry,preparing them for other they understand the concepts, and give them
courses such as calculus,business calculus,and finite feedback to clarify any misunderstandings.
mathematics.
• New Directions for the True/False Critical Thinking
2. To show students how algebra and trigonometry can
Exercises. The Fourth Edition asks students to
model and solve authentic real-world problems.
determine whether each statement in an itemized
3. To enable students to develop problem-solving skills, list is true or false.If the statement is false,students
while fostering critical thinking,within an interesting are then asked to make the necessary change or
setting. changes to produce a true statement.
• Preview Exercises. Each exercise set concludes
One major obstacle in the way of achieving these
with three problems to help students prepare for
goals is the fact that very few students actually read their
the next section. Some of these problems review
textbook.This has been a regular source of frustration
previously covered material that students
for me and for my colleagues in the classroom.
will need to be successful in the forthcoming
Anecdotal evidence gathered over years highlights two
section. Other problems are designed to get
basic reasons that students do not take advantage of
students thinking about concepts they will soon
their textbook:
encounter.
• “I’ll never use this information.”
• More Detailed Directions When Comparing
• “I can’t follow the explanations.” Mathematical Models with Actual Data. The
Fourth Edition asks students if values obtained
I’ve written every page of the Fourth Edition with the from mathematical models understimate or over-
intent of eliminating these two objections.The ideas and estimate data displayed by graphs, and, if so, by
tools I’ve used to do so are described for the student in “A how much.
Brief Guide to Getting the Most from This Book,”which
• Increased Study Tip Boxes. The book’s Study Tip
appears in the front of this book.
boxes offer suggestions for problem solving, point
out common errors to avoid, and provide informal
hints and suggestions.These invaluable hints appear
What’s New in the Fourth Edition?
in greater abundance in the Fourth Edition.
• New Chapter-Opening and Section-Opening
• New Applications and Real-World Data. I’m on a
Scenarios. Every chapter and every section open
constant search for real-world data that can be used to
with a scenario based on an application, the
illustrate unique algebraic and trigonometric
majority of which are unique to the Fourth Edition.
applications. I researched hundreds of books,
These scenarios are revisited in the course of the
magazines,newspapers,almanacs,and online sites to
chapter or section in one of the book’s new
prepare the Fourth Edition. Among the 263
examples, exercises, or discussions. The often-
worked-out examples and application exercises based
humorous tone of these openers is intended to help
on new data sets, you’ll find models that describe
fearful and reluctant students overcome their
changing attitudes of college students over time.
negative perceptions about math.
• “Make Sense?” Classroom Discussion Exercises.
Each exercise set contains four Critical Thinking • 840 New Examples and Exercises. The Fourth
exercises intended for classroom discussion in order Edition contains 24 detailed worked-out examples
to engage participation in the learning process. involving new data, 239 new application exercises,
These items test conceptual understanding by 308 “make sense”discussion exercises,228 preview
asking students to determine whether statements exercises, and 41 new exercises that appear in the
are sensible, and to explain why or why not. various other categories of the exercise sets.
xi