CONTENTS
¥ PROLOGUE: Principles of Problem Solving 1
CHAPTER P PREREQUISITES 3
P.1 Modeling the Real World with Algebra 3
P.2 Real Numbers 4
P.3 Integer Exponents and Scientific Notation 9
P.4 Rational Exponents and Radicals 14
P.5 Algebraic Expressions 18
P.6 Factoring 22
P.7 Rational Expressions 27
P.8 Solving Basic Equations 34
P.9 Modeling with Equations 39
Chapter P Review 45
Chapter P Test 51
¥ FOCUS ON MODELING: Making the Best Decisions 54
CHAPTER 1 EQUATIONS AND GRAPHS 57
1.1 The Coordinate Plane 57
1.2 Graphs of Equations in Two Variables; Circles 65
1.3 Lines 79
1.4 Solving Quadratic Equations 90
1.5 Complex Numbers 98
1.6 Solving Other Types of Equations 101
1.7 Solving Inequalities 110
1.8 Solving Absolute Value Equations and Inequalities 129
1.9 Solving Equations and Inequalities Graphically 131
1.10 Modeling Variation 139
Chapter 1 Review 143
Chapter 1 Test 161
iii
,iv Contents
¥ FOCUS ON MODELING: Fitting Lines to Data 165
CHAPTER 2 FUNCTIONS 169
2.1 Functions 169
2.2 Graphs of Functions 178
2.3 Getting Information from the Graph of a Function 190
2.4 Average Rate of Change of a Function 201
2.5 Linear Functions and Models 206
2.6 Transformations of Functions 212
2.7 Combining Functions 226
2.8 One-to-One Functions and Their Inverses 234
Chapter 2 Review 243
Chapter 2 Test 255
¥ FOCUS ON MODELING: Modeling with Functions 259
CHAPTER 3 POLYNOMIAL AND RATIONAL FUNCTIONS 267
3.1 Quadratic Functions and Models 267
3.2 Polynomial Functions and Their Graphs 276
3.3 Dividing Polynomials 291
3.4 Real Zeros of Polynomials 301
3.5 Complex Zeros and the Fundamental Theorem of Algebra 334
3.6 Rational Functions 344
Chapter 3 Review 377
Chapter 3 Test 395
¥ FOCUS ON MODELING: Fitting Polynomial Curves to Data 398
CHAPTER 4 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 401
4.1 Exponential Functions 401
4.2 The Natural Exponential Function 409
4.3 Logarithmic Functions 414
4.4 Laws of Logarithms 422
4.5 Exponential and Logarithmic Equations 426
, Contents v
4.6 Modeling with Exponential Functions 433
4.7 Logarithmic Scales 438
Chapter 4 Review 440
Chapter 4 Test 448
¥ FOCUS ON MODELING: Fitting Exponential and Power Curves to Data 450
CHAPTER 5 TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE APPROACH 455
5.1 Angle Measure 455
5.2 Trigonometry of Right Triangles 459
5.3 Trigonometric Functions of Angles 464
5.4 Inverse Trigonometric Functions and Right Triangles 468
5.5 The Law of Sines 471
5.6 The Law of Cosines 476
Chapter 5 Review 481
Chapter 5 Test 486
¥ FOCUS ON MODELING: Surveying 536
CHAPTER 6 TRIGONOMETRIC FUNCTIONS: UNIT CIRCLE APPROACH 491
6.1 The Unit Circle 491
6.2 Trigonometric Functions of Real Numbers 495
6.3 Trigonometric Graphs 500
6.4 More Trigonometric Graphs 511
6.5 Inverse Trigonometric Functions and Their Graphs 519
6.6 Modeling Harmonic Motion 521
Chapter 6 Review 527
Chapter 6 Test 534
¥ FOCUS ON MODELING: Fitting Sinusoidal Curves to Data 487
CHAPTER 7 ANALYTIC TRIGONOMETRY 541
7.1 Trigonometric Identities 541
7.2 Addition and Subtraction Formulas 549
7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 556
, vi Contents
7.4 Basic Trigonometric Equations 567
7.5 More Trigonometric Equations 571
Chapter 7 Review 578
Chapter 7 Test 584
¥ FOCUS ON MODELING: Traveling and Standing Waves 586
CHAPTER 8 POLAR COORDINATES AND PARAMETRIC EQUATIONS 589
8.1 Polar Coordinates 589
8.2 Graphs of Polar Equations 593
8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 600
8.4 Plane Curves and Parametric Equations 612
Chapter 8 Review 623
Chapter 8 Test 630
¥ FOCUS ON MODELING: The Path of a Projectile 631
CHAPTER 9 VECTORS IN TWO AND THREE DIMENSIONS 635
9.1 Vectors in Two Dimensions 635
9.2 The Dot Product 641
9.3 Three-Dimensional Coordinate Geometry 644
9.4 Vectors in Three Dimensions 646
9.5 The Cross Product 649
9.6 Equations of Lines and Planes 652
Chapter 9 Review 654
Chapter 9 Test 658
¥ FOCUS ON MODELING: Vector Fields 659
CHAPTER 10 SYSTEMS OF EQUATIONS AND INEQUALITIES 663
10.1 Systems of Linear Equations in Two Variables 663
10.2 Systems of Linear Equations in Several Variables 670
10.3 Partial Fractions 678
10.4 Systems of Nonlinear Equations 689
10.5 Systems of Inequalities 696
¥ PROLOGUE: Principles of Problem Solving 1
CHAPTER P PREREQUISITES 3
P.1 Modeling the Real World with Algebra 3
P.2 Real Numbers 4
P.3 Integer Exponents and Scientific Notation 9
P.4 Rational Exponents and Radicals 14
P.5 Algebraic Expressions 18
P.6 Factoring 22
P.7 Rational Expressions 27
P.8 Solving Basic Equations 34
P.9 Modeling with Equations 39
Chapter P Review 45
Chapter P Test 51
¥ FOCUS ON MODELING: Making the Best Decisions 54
CHAPTER 1 EQUATIONS AND GRAPHS 57
1.1 The Coordinate Plane 57
1.2 Graphs of Equations in Two Variables; Circles 65
1.3 Lines 79
1.4 Solving Quadratic Equations 90
1.5 Complex Numbers 98
1.6 Solving Other Types of Equations 101
1.7 Solving Inequalities 110
1.8 Solving Absolute Value Equations and Inequalities 129
1.9 Solving Equations and Inequalities Graphically 131
1.10 Modeling Variation 139
Chapter 1 Review 143
Chapter 1 Test 161
iii
,iv Contents
¥ FOCUS ON MODELING: Fitting Lines to Data 165
CHAPTER 2 FUNCTIONS 169
2.1 Functions 169
2.2 Graphs of Functions 178
2.3 Getting Information from the Graph of a Function 190
2.4 Average Rate of Change of a Function 201
2.5 Linear Functions and Models 206
2.6 Transformations of Functions 212
2.7 Combining Functions 226
2.8 One-to-One Functions and Their Inverses 234
Chapter 2 Review 243
Chapter 2 Test 255
¥ FOCUS ON MODELING: Modeling with Functions 259
CHAPTER 3 POLYNOMIAL AND RATIONAL FUNCTIONS 267
3.1 Quadratic Functions and Models 267
3.2 Polynomial Functions and Their Graphs 276
3.3 Dividing Polynomials 291
3.4 Real Zeros of Polynomials 301
3.5 Complex Zeros and the Fundamental Theorem of Algebra 334
3.6 Rational Functions 344
Chapter 3 Review 377
Chapter 3 Test 395
¥ FOCUS ON MODELING: Fitting Polynomial Curves to Data 398
CHAPTER 4 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 401
4.1 Exponential Functions 401
4.2 The Natural Exponential Function 409
4.3 Logarithmic Functions 414
4.4 Laws of Logarithms 422
4.5 Exponential and Logarithmic Equations 426
, Contents v
4.6 Modeling with Exponential Functions 433
4.7 Logarithmic Scales 438
Chapter 4 Review 440
Chapter 4 Test 448
¥ FOCUS ON MODELING: Fitting Exponential and Power Curves to Data 450
CHAPTER 5 TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE APPROACH 455
5.1 Angle Measure 455
5.2 Trigonometry of Right Triangles 459
5.3 Trigonometric Functions of Angles 464
5.4 Inverse Trigonometric Functions and Right Triangles 468
5.5 The Law of Sines 471
5.6 The Law of Cosines 476
Chapter 5 Review 481
Chapter 5 Test 486
¥ FOCUS ON MODELING: Surveying 536
CHAPTER 6 TRIGONOMETRIC FUNCTIONS: UNIT CIRCLE APPROACH 491
6.1 The Unit Circle 491
6.2 Trigonometric Functions of Real Numbers 495
6.3 Trigonometric Graphs 500
6.4 More Trigonometric Graphs 511
6.5 Inverse Trigonometric Functions and Their Graphs 519
6.6 Modeling Harmonic Motion 521
Chapter 6 Review 527
Chapter 6 Test 534
¥ FOCUS ON MODELING: Fitting Sinusoidal Curves to Data 487
CHAPTER 7 ANALYTIC TRIGONOMETRY 541
7.1 Trigonometric Identities 541
7.2 Addition and Subtraction Formulas 549
7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 556
, vi Contents
7.4 Basic Trigonometric Equations 567
7.5 More Trigonometric Equations 571
Chapter 7 Review 578
Chapter 7 Test 584
¥ FOCUS ON MODELING: Traveling and Standing Waves 586
CHAPTER 8 POLAR COORDINATES AND PARAMETRIC EQUATIONS 589
8.1 Polar Coordinates 589
8.2 Graphs of Polar Equations 593
8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 600
8.4 Plane Curves and Parametric Equations 612
Chapter 8 Review 623
Chapter 8 Test 630
¥ FOCUS ON MODELING: The Path of a Projectile 631
CHAPTER 9 VECTORS IN TWO AND THREE DIMENSIONS 635
9.1 Vectors in Two Dimensions 635
9.2 The Dot Product 641
9.3 Three-Dimensional Coordinate Geometry 644
9.4 Vectors in Three Dimensions 646
9.5 The Cross Product 649
9.6 Equations of Lines and Planes 652
Chapter 9 Review 654
Chapter 9 Test 658
¥ FOCUS ON MODELING: Vector Fields 659
CHAPTER 10 SYSTEMS OF EQUATIONS AND INEQUALITIES 663
10.1 Systems of Linear Equations in Two Variables 663
10.2 Systems of Linear Equations in Several Variables 670
10.3 Partial Fractions 678
10.4 Systems of Nonlinear Equations 689
10.5 Systems of Inequalities 696
