Preface x
To the Student xvii
ARE YO U READY FOR THIS COURSE? xix
Prologue: Principles of Problem Solving P1
chapter P Prerequisites 1
Chapter Overview 1
P.1 Modeling the Real World with Algebra 2
P.2 Real Numbers 6
P.3 Integer Exponents and Scientific Notation 18
P.4 Rational Exponents and Radicals 25
P.5 Algebraic Expressions 32
P.6 Factoring 37
P.7 Rational Expressions 44
P.8 Solving Basic Equations 53
P.9 Modeling with Equations 61
Chapter P Review 74
Chapter P Test 79
¦ FOCUS ON MODELING Making the Best Decisions 81
chapter 1 Equations and Graphs 87
Chapter Overview 87
1.1 The Coordinate Plane 88
1.2 Graphs of Equations in Two Variables; Circles 94
1.3 Lines 104
1.4 Solving Quadratic Equations 115
1.5 Complex Numbers 126
1.6 Solving Other Types of Equations 132
1.7 Solving Inequalities 141
1.8 Solving Absolute Value Equations and Inequalities 150
1.9 Solving Equations and Inequalities Graphically 154
1.10 Modeling Variation 159
Chapter 1 Review 167
Chapter 1 Test 172
¦ FOCUS ON MODELING Fitting Lines to Data 174
v
vi Contents
chapter 2 Functions 183
Chapter Overview 183
2.1 Functions 184
2.2 Graphs of Functions 195
2.3 Getting Information from the Graph of a Function 206
2.4 Average Rate of Change of a Function 219
2.5 Linear Functions and Models 226
2.6 Transformations of Functions 234
2.7 Combining Functions 246
2.8 One-to-One Functions and Their Inverses 255
Chapter 2 Review 265
Chapter 2 Test 271
¦ FOCUS ON MODELING Modeling with Functions 273
chapter 3 Polynomial and Rational Functions 281
Chapter Overview 281
3.1 Quadratic Functions and Models 282
3.2 Polynomial Functions and Their Graphs 290
3.3 Dividing Polynomials 305
3.4 Real Zeros of Polynomials 311
3.5 Complex Zeros and the Fundamental Theorem of Algebra 323
3.6 Rational Functions 331
3.7 Polynomial and Rational Inequalities 347
Chapter 3 Review 353
Chapter 3 Test 359
¦ FOCUS ON MODELING Fitting Polynomial Curves to Data 361
chapter 4 Exponential and Logarithmic Functions 365
Chapter Overview 365
4.1 Exponential Functions 366
4.2 The Natural Exponential Function 374
4.3 Logarithmic Functions 380
4.4 Laws of Logarithms 390
4.5 Exponential and Logarithmic Equations 396
4.6 Modeling with Exponential Functions 406
4.7 Logarithmic Scales 417
Chapter 4 Review 422
Chapter 4 Test 427
¦ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 428
Cumulative Review Test: Chapters 2, 3, and 4 (Website)
Contents vii
chapter 5 T rigonometric Functions: Right Triangle
Approach 437
Chapter Overview 437
5.1 Angle Measure 438
5.2 Trigonometry of Right Triangles 448
5.3 Trigonometric Functions of Angles 457
5.4 Inverse Trigonometric Functions and Right Triangles 467
5.5 The Law of Sines 474
5.6 The Law of Cosines 482
Chapter 5 Review 490
Chapter 5 Test 497
¦ FOCUS ON MODELING Surveying 499
chapter 6 Trigonometric Functions: Unit Circle Approach 503
Chapter Overview 503
6.1 The Unit Circle 504
6.2 Trigonometric Functions of Real Numbers 511
6.3 Trigonometric Graphs 521
6.4 More Trigonometric Graphs 534
6.5 Inverse Trigonometric Functions and Their Graphs 541
6.6 Modeling Harmonic Motion 547
Chapter 6 Review 562
Chapter 6 Test 567
¦ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 568
chapter 7 Analytic Trigonometry 573
Chapter Overview 573
7.1 Trigonometric Identities 574
7.2 Addition and Subtraction Formulas 581
7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 589
7.4 Basic Trigonometric Equations 600
7.5 More Trigonometric Equations 606
Chapter 7 Review 612
Chapter 7 Test 616
¦ FOCUS ON MODELING Traveling and Standing Waves 617
Cumulative Review Test: Chapters 5, 6, and 7 (Website)
viii Contents
chapter 8 Polar Coordinates and Parametric Equations 623
Chapter Overview 623
8.1 Polar Coordinates 624
8.2 Graphs of Polar Equations 630
8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 638
8.4 Plane Curves and Parametric Equations 647
Chapter 8 Review 656
Chapter 8 Test 660
¦ FOCUS ON MODELING The Path of a Projectile 661
chapter 9 Vectors in Two and Three Dimensions 665
Chapter Overview 665
9.1 Vectors in Two Dimensions 666
9.2 The Dot Product 675
9.3 Three-Dimensional Coordinate Geometry 683
9.4 Vectors in Three Dimensions 689
9.5 The Cross Product 695
9.6 Equations of Lines and Planes 702
Chapter 9 Review 706
Chapter 9 Test 711
¦ FOCUS ON MODELING Vector Fields 712
Cumulative Review Test: Chapters 8 and 9 (Website)
chapter 10 Systems of Equations and Inequalities 715
Chapter Overview 715
10.1 Systems of Linear Equations in Two Variables 716
10.2 Systems of Linear Equations in Several Variables 726
10.3 Partial Fractions 735
10.4 Systems of Nonlinear Equations 740
10.5 Systems of Inequalities 745
Chapter 10 Review 755
Chapter 10 Test 759
¦ FOCUS ON MODELING Linear Programming 760
chapter 11 Matrices and Determinants 767
Chapter Overview 767
11.1 Matrices and Systems of Linear Equations 768
11.2 The Algebra of Matrices 781
11.3 Inverses of Matrices and Matrix Equations 793
11.4 Determinants and Cramer’s Rule 803
Chapter 11 Review 814
Chapter 11 Test 819
¦ FOCUS ON MODELING Computer Graphics 820
Contents ix
chapter 12 Conic Sections 825
Chapter Overview 825
12.1 Parabolas 826
12.2 Ellipses 834
12.3 Hyperbolas 843
12.4 Shifted Conics 851
12.5 Rotation of Axes 860
12.6 Polar Equations of Conics 868
Chapter 12 Review 875
Chapter 12 Test 879
¦ FOCUS ON MODELING Conics in Architecture 880
Cumulative Review Test: Chapters 10, 11, and 12 (Website)
chapter 13 Sequences and Series 885
Chapter Overview 885
13.1 Sequences and Summation Notation 886
13.2 Arithmetic Sequences 897
13.3 Geometric Sequences 902
13.4 Mathematics of Finance 911
13.5 Mathematical Induction 917
13.6 The Binomial Theorem 923
Chapter 13 Review 931
Chapter 13 Test 936
¦ FOCUS ON MODELING Modeling with Recursive Sequences 937
chapter 14 Counting and Probability 941
Chapter Overview 941
14.1 Counting 942
14.2 Probability 954
14.3 Binomial Probability 966
14.4 Expected Value 971
Chapter 14 Review 975
Chapter 14 Test 980
¦ FOCUS ON MODELING The Monte Carlo Method 981
Cumulative Review Test: Chapters 13 and 14 (Website)
APPENDIX A Geometry Review 985
APPENDIX B Calculations and Significant Figures 991
APPENDIX C Graphing with a Graphing Calculator 993
APPENDIX D Using the TI-83/84 Graphing Calculator 999
ANSWERS A 1
INDEX I1