Algebra & Trig Textbook

1055 Pages·2013·27.82 MB·English

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Sixth Edition
(cid:141) Ron Larson
(cid:141) Robert P. Hostetler
ThePennsylvaniaState University
TheBehrendCollege
(cid:141) With the assistance ofDavid C. Falvo
ThePennsylvaniaState University
TheBehrend College
VALENCIA HIGH SCHOOL
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Copyright© 2004 by HoughtonMifflin Company.All rightsreserved.
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LibraryofCongressCatalogCardNumber:2002115582
ISBN: 0-618-31782-1
4567-DOW-070605
Contents
AWord from the Authors (Preface) vii
Textbook Highlights xii
Chapter P Prerequisites i
p.i Review of RealNumbers and TheirProperties 2
p.2 Exponents and Radicals 12
P.3 Polynomialsand Special Products 24
P.4 Factoring 33
P.5 Rational Expressions 41
P.6 Errorsand the Algebra of Calculus 51
P.7 GraphicalRepresentation of Data 58
Chapter Summary 68 Review Exercises 69
ChapterTest 72 Proofs in Mathematics 73
P.S.Problem Solving 74
Chapter 1 Equations and Inequalities 76
1.1 Graphsof Equations 78
1.2 LinearEquations in OneVariable 88
1.3 Modeling with LinearEquations 97
1.4 QuadraticEquations 109
1.5 Complex Numbers 123
1.5 Other Types of Equations 130
1.7 LinearInequalities in OneVariable 141
1.8 Other Types of Inequalities 151
Chapter Summary 161 Review Exercises 162
Chapter Test 166 Proofs in Mathematics 167
P.S.Problem Solving 168
Chapter 2 Functions and Their Graphs
170
2.1 LinearEquations in Two Variables 172
2.2 Functions 187
2.3 AnalyzingGraphsof Functions 201
2.4 ALibrary of Functions 211
2.5 Shifting, Reflecting, and Stretching Graphs 219
2.6 Combinations of Functions 229
2.7 Inverse Functions 237
Chapter Summary 247 Review Exercises 248
ChapterTest 252 Cumulative Test: Chapters P-2 253
Proofs in Mathematics 255 P.S. Problem Solving 256
iii
IV Contents (cid:141)
Chapter 3 Polynomial Functions 258
3.1 Quadratic Functions 260
3.2 PolynomialFunctions of Higher Degree 271
3.3 Polynomialand Synthetic Division 284
3.4 Zeros of Polynomial Functions 293
3.5 Mathematical Modeling 308
Chapter Summary 320 Review Exercises 321
Chapter Test 325 Proofs in Mathematics 326
P.S.Problem Solving 328
Chapter 4 Rational Functions and Conies 330
4.1 Rational Functions and Asymptotes 332
4.2 Graphsof Rational Functions 341
4.3 Partial Fractions 350
4.4 Conies 358
4.5 Translations of Conies 371
Chapter Summary 379 Review Exercises 380
ChapterTest 384 Proofs in Mathematics 385
P.S.Problem Solving 386
Chapter 5 Exponential and Logarithmic Functions 388
5.1 Exponential Functionsand TheirGraphs 390
5.2 Logarithmic Functionsand TheirGraphs 401
5.3 Properties of Logarithms 411
5.4 Exponential and Logarithmic Equations 417
5.5 Exponential and Logarithmic Models 428
Chapter Summary 441 Review Exercises 442
ChapterTest 446 CumulativeTest:Chapters 3-5
447
Proofsin Mathematics 449 P.S. ProblemSolving 450
Chapter 6 Trigonometry 452
6.1 Angles and Their Measure 454
6.2 RightTriangle Trigonometry 465
6.3 Trigonometric Functions ofAny Angle 476
6.4 Graphs ofSine and Cosine Functions 488
6.5 Graphsof Other Trigonometric Functions 499
6.6 Inverse Trigonometric Functions 510
6.7 Applications and Models 520
Chapter Summary 531 ReviewExercises 532
Chapter Test 536 Proofs in Mathematics
537
P.S. Problem Solving 538
(cid:141) Contents
Chapter 7 Analytic Trigonometry 540
7.1 Using Fundamental Identities 542
7.2 Verifying TrigonometricIdentities 550
7.3 Solving Trigonometric Equations 557
7.4 Sum and DiiTerence Formulas 568
7.5 Multiple-Angle and Product-to-Sum Formulas 575
Chapter Summary 586 Review Exercises 587
Chapter Test 590 Proofs in Mathematics 591
P.S. Problem Solving 594
Chapter 8 Additional Topics in Trigonometry 596
8.1 Law of Sines 598
8.2 Law of Cosines 607
8.3 Vectors in the Plane 615
8.4 Vectors and Dot Products 628
8.5 Trigonometric Form of a Complex Number 637
Chapter Summary 648 Review Exercises 649
ChapterTest 653 Cumulative Test; Chapters 6-8 654
Proofs in Mathematics 656 P.S,Problem Solving 660
Chapter 9 Systems of Equations and Inequalities 662
9.1 SolvingSystems of Equations 664
9.2 Two-Variable LinearSystems 675
9.3 Multivariable LinearSystems 687
9.4 Systems of Inequalities 700
9.5 Linear Programming 711
Chapter Summary 721 ReviewExercises 722
Chapter Test 726 Proofs in Mathematics 727
P.S.Problem Solving 728
Chapter 10 Matrices and Determinants 730
10.1 Matricesand Systems of Equations 732
10.2 Operations with Matrices 747
10.3 The Inverse of a Square Matrix 761
10.4 The Determinant of a Square Matrix 770
10.5 Applications of Matricesand Determinants 778
Chapter Summary 790 ReviewExercises 791
ChapterTest 796 Proofs in Mathematics 797
P.S.Problem Solving 798
vi Contents (cid:141)
Chapter 11 ^quences. Series, and Probability soo
11.1 Sequences and Series 802
11.2 Arithmetic Sequences and Partial Sums 813
11.3 Geometric Sequences and Series 822
11.4 Mathematical Induction 832
11.5 The Binomial Theorem 842
11.6 Counting Principles 850
11.7 Probability 860
Chapter Summary 873 Review Exercises 874
ChapterTest 878 CumulativeTest:Chapters 9-11 879
Proofsin Mathematics 881 P.S. ProblemSolving 884
Answers to Odd-Numbered Exercises and Tests A1
Index of Applications A131
Index A136
A Word from the Authors
Welcome to Algebra and Trigonometry, Sixth Edition. In this revision we
continue to focus on promoting student success, while providing an accessible
textthat offers flexible teaching and learning options.
Inkeepingwithourphilosophythat studentslearn bestwhen theyknow what
they are expected to learn, we have retained the thematic study thread from the
Fifth Edition. We first introduce this study thread in the Chapter Opener. Each
chapterbegins with a study guide that contains acomprehensiveoverview of the
chapterconcepts{Whatyou shouldlearn), alist ofImportantVocabulary integral
tolearningthe chapterconcepts, a list of additional chapter-specificStudy Tools,
and additional text-specific resources. The study guide allows students to get
organized and prepare for the chapter. Then, each section opens with a a set of
learningobjectivesoutliningtheconceptsandskillsstudentsareexpectedtolearn
(Whatyoushould learn), followed by aninteresting real-life application used to
illustrate whyitisimportantto learntheconcepts inthat section {Whyyoushould
learn it).Study Tipsat point-of-use provide supportas students read through the
section. And finally, toprovide study support and acomprehensive review ofthe
chapter, each chapter concludes with a chaptersummai^ {Whatdid you learn?),
which reinforces the section objectives, and chapterReviewExercises, which are
con-elatedto the chaptersummary.
In addition to providing in-text study support, we have taken care to write a
text for the student. We paid careful attention to the presentation, using precise
mathematical language and clear writing, to create an effective learning tool.We
arecommittedtoprovidingatext that makes the mathematics within it accessible
toaUstudents. IntheSixthEdition,wehaverevised andimproveduponmanytext
featuresdesigned forthispurpose.The Technology,Exploration featureshavebeen
expanded. ChapterTests,which gave students an opportunity for self-assessment,
are included in every chapter. We have retained the Synthesis exercises, which
check students' conceptual understanding, and the Review exercises, which
reinforceskills learnedinprevioussectionswithineach sectionexercise set.Also,
students have access to several media resources that offer additional text-specific
resources to enhancethe learningprocess.
From the time we first began writing in the early 1970s, we have always
viewed part of our authoring role as that ofproviding instructors with flexible
teaching programs. The optional features within the text allow instructors with
differentpedagogicalapproachestodesigntheircoursetomeet both theirinstruc
tional needs and the needs oftheir students. Instructors who stress applications
and problem solving, or exploration and technology, and more traditional meth
ods willbe abletouse this textsuccessfully.Wehopeyou enjoytheSixth Edition.
Ron Larson
RobertR Hostetler
VII
Acknowledgments
We would like to thankthe manypeople who helped us at various stages ofthis
project.Theirencouragement,criticisms,and suggestionshavebeeninvaluableto
us.
Sixth Edition Reviewers
Ahmad Abusaid, Southern Polytechnic University; Catherine Banks, Texas
Woman'sCollege;JaredBurch,Collegeofthe Sequoias;Dr. MichelleR. DeDeo,
University ofNorth Florida; Brian Hickey, East Central College; Gangadhar R,
Hiremath, Miles College; Erick Hofacker, University ofWisconsin-River Falls;
Dr.Kevin W.Hopkins, SouthwestBaptistUniversity; CharlesW.Johnson, South
Georgia College; Gary S. Kersting, North Central Michigan College; Namyong
Lee, MinnesotaState University;MaryLeeseberg,ManateeCommunityCollege;
TristanLondre, BlueRiverCommunity College; BruceN. Lundberg, University
of Southern Colorado; Dr. Carl V. Lutzer, Rochester Institute of Technology;
Rudy Maglio, Oakton Community College; James Miller, West Virginia
University; Steve O'Donnell, Rogue Community College; Armando I. Perez,
Laredo Community College; Rita Randolfi, Brevard Community College; David
Ray, TheUniversityofTennesseeatMartin;Miguel San MiguelGonzalez,Texas
A&M International University; Scott Satake, North Idaho College; Jed Soifer,
AtlanticCape CommunityCollege;Dr.Roy N.Tucker, PaloAlto CollegeandThe
University of Texas at San Antonio; Karen Villarreal, Xavier University of
Louisiana; Carol Walker, Hinds Community College; J. Lewis Walston,
Methodist College; Jun Wang, Alabama State University; Ibrahim Wazir,
AmericanInternational School; RobertWylie, Carl AlbertState College.
Previous Edition Reviewers
James Alsobrook, Southern Union State Community College; Sherry Biggers,
Clemson University; Charles Biles, Humboldt State University; Randall Boan,
Aims Community College; Jeremy Carr, Pensacola Junior College; D. J. Clark,
Portland Community College; Donald Clayton, Madisonville Community
College; Linda Crabtree, Metropolitan Community College; David DeLatte,
University of North Texas; Gregory Dlabach, Northeastern Oklahoma A&M
College; JosephLloydHarris, GulfCoastCommunity College; JeffHeiking, St.
PetersburgJuniorCollege; CelesteHernandez, Richland College; HeidiHoward,
Florida Community College at Jacksonville; Wanda Long, St. Charles County
Community College; Wayne F. Mackey, University of Arkansas; Rhonda
MacLeod, Florida State University; M. Maheswaran, University of
Wisconsin-MarathonCounty;ValerieMiller, GeorgiaState University;Katharine
MuUer,Cisco Junior College; Bonnie Oppenheimer, Mississippi University for
Women; James PoM, FloridaAtlantic University; Hari Pulapaka, Valdosta State
University; Michael Russo, Suffolk County Community College; CynthiaFloyd
Sikes, Georgia Southern University; Susan Schindler, Baruch College-CUNY;
Stanley Smith, BlackHills State University.
VIII
(cid:141) Acknowledgments Ix
We would like to extend a special thanks to all ofthe instructors who took time
to participate inour phoneinterviews.
We would like to thank the staffofLarson Texts, Inc. and the staffofMeridian
Creative Group, who assisted in proofreading the manuscript, preparing and
proofreading the art package, and typesetting the supplements.
We are grateful to our wives, Deanna Gilbert Larson and Eloise Hostetler, for
their love, patience, and support. Also, aspecial thanks goes to R. Scott O'Neil.
If you have suggestions for improving this text, please feel free to write to us.
Over theyears wehave receivedmany useful commentsfrom both instructorsand
students, and we value these comments very much.
Ron Larson
Robert R Hostetler
How can this book help you
Support for Student Success EH
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absence.
Options for Students and
Instructors
Graph each ofthe functions
• Concepts are presentedthrough examples,
with a graphing utility.
applications, technology, orexplorationsto
Determinewhetherthefunction
adaptthe course to the curriculumneeds or
is even, odd, orneither.
studentlearning styles.
• A variety of exercises that increasein difficulty
allows professorsthe flexibility to assign home
Technology
work to studentswith various learning styles.
Exercise options include skills, technology, Youcan use agraphing
critical thinking, writing, appUcations,modeling utilityto determinethedomain
data, true/false, proofs, and theoreticalques ofacompositionoffunctions.
tions. Forthecomposition in
• TheP.S. Problem Solving section at the end of Example5,enterthefunction
every chapteroffers more challengingexercises
for advanced students.
ProofsInMathematfcs
• This text provides a solidmathematicalfounda
tion by foreshadowing concepts that will be Proofs in Mathematics
used in future courses. Topics that wiUbe espe
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withan "AlgebraofCalculus" tf icon. alsousepreviouslyprovedtheoremsinyourproof.Forinsiance,iheDistance
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differentproofmethods,whichyouwillseeinlaterchapters.