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Copyright, vi
Preface, xvii
Acknowledgments, xxvi
R. Basic Concepts of Algebra, 1
R.1. The Real-Number System, 2
Real Numbers, 2
Interval Notation, 3
Properties of the Real Numbers, 5
Absolute Value, 6
R.2. Integer Exponents, Scientific Notation, and Order of Operations, 9
Integers as Exponents, 9
Scientific Notation, 11
Order of Operations, 12
R.3. Addition, Subtraction, and Multiplication of Polynomials, 17
Polynomials, 17
Addition and Subtraction, 18
Multiplication, 19
R.4. Factoring, 21
Terms with Common Factors, 22
Factoring by Grouping, 22
Trinomials of the Type x² + bx + c, 22
Trinomials of the Type ax² + bx + c, a≠1, 24
Special Factorizations, 27
R.5. Rational Expressions, 30
The Domain of a Rational Expression, 31
Simplifying, Multiplying, and Dividing Rational Expressions, 31
Adding and Subtracting Rational Expressions, 33
Complex Rational Expressions, 35
R.6. Radical Notation and Rational Exponents, 38
Simplifying Radical Expressions, 40
An Application, 42
Rationalizing Denominators or Numerators, 43
Rational Exponents, 43
R.7. The Basics of Equation Solving, 47
Linear and Quadratic Equations, 48
Summary and Review, 51
Test, 54
1. Graphs, Functions, and Models, 57
1.1. Introduction to Graphing, 58
Graphs, 58
Solutions of Equations, 59
Graphs of Equations, 60
The Distance Formula, 64
Midpoints of Segments, 66
Circles, 67
Visualizing the Graph, 69
1.2. Functions and Graphs, 74
Functions, 74
Notation for Functions, 77
Graphs of Functions, 78
Finding Domains of Functions, 81
Visualizing Domain and Range, 82
Applications of Functions, 83
1.3. Linear Functions, Slope, and Applications, 90
Linear Functions, 90
The Linear Function f(x) = mx + b and Slope, 92
Applications of Slope, 95
Applications of Linear Functions, 98
1.4. Equations of Lines and Modeling, 103
Slope–Intercept Equations of Lines, 103
Point–Slope Equations of Lines, 106
Parallel Lines, 107
Perpendicular Lines, 108
Mathematical Models, 110
Curve Fitting, 110
Visualizing the Graph, 114
1.5. More on Functions, 119
Increasing, Decreasing, and Constant Functions, 119
Relative Maximum and Minimum Values, 121
Applications of Functions, 122
Functions Defined Piecewise, 124
1.6. The Algebra of Functions, 134
The Algebra of Functions: Sums, Differences, Products and Quotients, 134
Difference Quotients, 137
The Composition of Functions, 138
Decomposing a Function as a Composition, 142
1.7. Symmetry and Transformations, 146
Symmetry, 146
Even and Odd Functions, 149
Transformations of Functions, 151
Vertical and Horizontal Translations, 152
Reflections, 154
Vertical and Horizontal Stretchings and Shrinkings, 157
Visualizing the Graph, 162
Summary and Review, 168
Test, 173
2. Functions, Equations, and Inequalities, 175
2.1. Linear Equations, Functions, and Models, 176
Linear Equations, 176
Applications Using Linear Models, 178
Zeros of Linear Functions, 184
Formulas, 186
2.2. The Complex Numbers, 193
The Complex-Number System, 193
Addition and Subtraction, 195
Multiplication, 196
Conjugates and Division, 197
2.3. Quadratic Equations, Functions, and Models, 200
Quadratic Equations and Quadratic Functions, 200
Completing the Square, 203
Using the Quadratic Formula, 205
The Discriminant, 208
Equations Reducible to Quadratic, 208
Applications, 210
2.4. Analyzing Graphs of Quadratic Functions, 216
Graphing Quadratic Functions of the Type f(x) = a(x-h)² + k, 216
Graphing Quadratic Functions of the Type f(x) = ax² +bx + c, a≠0, 218
Applications, 221
Visualizing the Graph, 225
2.5. More Equation Solving, 229
Rational Equations, 229
Radical Equations, 232
Equations with Absolute Value, 235
2.6. Solving Linear Inequalities, 239
Linear Inequalities, 239
Compound Inequalities, 240
Inequalities with Absolute Value, 242
An Application, 243
Summary and Review, 247
Test, 250
3. Polynomial and Rational Functions, 253
3.1. Polynomial Functions and Models, 254
The Leading-Term Test, 257
Finding Zeros of Factored Polynomial Functions, 259
Polynomial Models, 262
3.2. Graphing Polynomial Functions, 270
Graphing Polynomial Functions, 270
The Intermediate Value Theorem, 276
Visualizing the Graph, 278
3.3. Polynomial Division; The Remainder and Factor Theorems, 281
Division and Factors, 281
The Remainder Theorem and Synthetic Division, 282
Finding Factors of Polynomials, 285
3.4. Theorems about Zeros of Polynomial Functions, 290
The Fundamental Theorem of Algebra, 290
Finding Polynomials with Given Zeros, 291
Zeros of Polynomial Functions with Real Coefficients, 291
Rational Coefficients, 292
Integer Coefficients and the Rational Zeros Theorem, 292
Descartes' Rule of Signs, 296
3.5. Rational Functions, 300
The Domain of a Rational Function, 301
Asymptotes, 303
Applications, 314
Visualizing the Graph, 315
3.6. Polynomial and Rational Inequalities, 320
Polynomial Inequalities, 320
Rational Inequalities, 324
3.7. Variation and Applications, 330
Direct Variation, 330
Inverse Variation, 332
Combined Variation, 334
Summary and Review, 340
Test, 345
4. Exponential and Logarithmic Functions, 347
4.1. Inverse Functions, 348
Inverses, 348
Inverses and One-to-One Functions, 349
Finding Formulas for Inverses, 353
Inverse Functions and Composition, 355
Restricting a Domain, 356
4.2. Exponential Functions and Graphs, 360
Graphing Exponential Functions, 361
Applications, 366
The Number e, 367
Graphs of Exponential Functions, Base e, 368
4.3. Logarithmic Functions and Graphs, 374
Logarithmic Functions, 374
Finding Certain Logarithms, 377
Converting Between Exponential and Logarithmic Equations, 378
Finding Logarithms on a Calculator, 379
Natural Logarithms, 380
Changing Logarithmic Bases, 381
Graphs of Logarithmic Functions, 382
Applications, 384
Visualizing the Graph, 386
4.4. Properties of Logarithmic Functions, 390
Logarithms of Products, 390
Logarithms of Powers, 391
Logarithms of Quotients, 391
Applying the Properties, 392
Simplifying Expressions of the Type log_aa^x and a^log_ax, 394
4.5. Solving Exponential and Logarithmic Equations, 398
Solving Exponential Equations, 398
Solving Logarithmic Equations, 404
4.6. Applications and Models: Growth and Decay; Compound Interest, 410
Population Growth, 410
Interest Compounded Continuously, 411
Models of Limited Growth, 414
Exponential Decay, 416
Summary and Review, 426
Test, 431
5. The Trigonometric Functions, 433
5.1. Trigonometric Functions of Acute Angles, 434
The Trigonometric Ratios, 434
The Six Functions Related, 437
Function Values of 30°, 45°, and 60°, 437
Function Values of Any Acute Angle, 440
Cofunctions and Complements, 442
5.2. Applications of Right Triangles, 446
Solving Right Triangles, 446
Applications, 448
5.3. Trigonometric Functions of Any Angle, 459
Angles, Rotations, and Degree Measure, 459
Trigonometric Functions of Angles or Rotations, 462
The Six Functions Related, 466
Terminal Side on an Axis, 467
Reference Angles: 30°, 45°, and 60°, 468
Function Values for Any Angle, 470
5.4. Radians, Arc Length, and Angular Speed, 475
Distances on the Unit Circle, 475
Radian Measure, 478
Arc Length and Central Angles, 481
Linear Speed and Angular Speed, 483
5.5. Circular Functions: Graphs and Properties, 491
Reflections on the Unit Circle, 492
Finding Function Values, 493
Graphs of the Sine and Cosine Functions, 496
Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions, 501
5.6. Graphs of Transformed Sine and Cosine Functions, 507
Variations of Basic Graphs, 507
Graphs of Sums: Addition of Ordinates, 518
Damped Oscillation: Multiplication of Ordinates, 520
Visualizing the Graph, 522
Summary and Review, 527
Test, 532
6. Trigonometric Identities, Inverse Functions, and Equations, 533
6.1. Identities: Pythagorean and Sum and Difference, 534
Pythagorean Identities, 534
Simplifying Trigonometric Expressions, 536
Sum and Difference Identities, 539
6.2. Identities: Cofunction, Double-Angle, and Half-Angle, 547
Cofunction Identities, 547
Double-Angle Identities, 549
Half-Angle Identities, 551
6.3. Proving Trigonometric Identities, 556
The Logic of Proving Identities, 556
Proving Identities, 557
Product-to-Sum and Sum-to-Product Identities, 560
6.4. Inverses of the Trigonometric Functions, 565
Restricting Ranges to Define Inverse Functions, 566
Composition of Trigonometric Functions and Their Inverses, 571
6.5. Solving Trigonometric Equations, 577
Visualizing the Graph, 587
Summary and Review, 591
Test, 595
7. Applications of Trigonometry, 597
7.1. The Law of Sines, 598
Solving Oblique Triangles, 598
The Law of Sines, 600
Solving Triangles (AAS and ASA), 601
Solving Triangles (SSA), 602
The Area of a Triangle, 606
7.2. The Law of Cosines, 611
The Law of Cosines, 611
Solving Triangles (SAS), 613
Solving Triangles (SSS), 615
7.3. Complex Numbers: Trigonometric Form, 622
Graphical Representation, 622
Trigonometric Notation for Complex Numbers, 623
Multiplication and Division with Trigonometric Notation, 625
Powers of Complex Numbers, 628
Roots of Complex Numbers, 629
7.4. Polar Coordinates and Graphs, 634
Polar Coordinates, 634
Polar and Rectangular Equations, 637
Graphing Polar Equations, 639
Visualizing the Graph, 643
7.5. Vectors and Applications, 646
Vectors, 647
Vector Addition, 648
Applications, 649
Components, 650
7.6. Vector Operations, 655
Position Vectors, 655
Operations on Vectors, 657
Unit Vectors, 660
Direction Angles, 661
Angle Between Vectors, 664
Forces in Equilibrium, 665
Summary and Review, 669
Test, 674
8. Systems of Equations and Matrices, 675
8.1. Systems of Equations in Two Variables, 676
Solving Systems of Equations Graphically, 676
The Substitution Method, 677
The Elimination Method, 678
Applications, 682
Visualizing the Graph, 687
8.2. Systems of Equations in Three Variables, 693
Solving Systems of Equations in Three Variables, 693
Applications, 696
Mathematical Models and Applications, 697
8.3. Matrices and Systems of Equations, 704
Matrices and Row-Equivalent Operations, 704
Gaussian Elimination with Matrices, 705
Gauss–Jordan Elimination, 708
8.4. Matrix Operations, 712
Matrix Addition and Subtraction, 712
Scalar Multiplication, 715
Products of Matrices, 716
Matrix Equations, 719
8.5. Inverses of Matrices, 724
The Identity Matrix, 724
The Inverse of a Matrix, 725
Solving Systems of Equations, 728
8.6. Determinants and Cramer's Rule, 731
Determinants of Square Matrices, 731
Evaluating Determinants Using Cofactors, 731
Cramer's Rule, 734
8.7. Systems of Inequalities and Linear Programming, 739
Graphs of Linear Inequalities, 739
Systems of Linear Inequalities, 745
Applications: Linear Programming, 745
8.8. Partial Fractions, 753
Partial Fraction Decompositions, 753
Summary and Review, 760
Test, 765
9. Analytic Geometry Topics, 767
9.1. The Parabola, 768
Parabolas, 768
Finding Standard Form by Completing the Square, 770
Applications, 773
9.2. The Circle and the Ellipse, 776
Circles, 776
Ellipses, 778
Applications, 783
9.3. The Hyperbola, 787
Standard Equations of Hyperbolas, 787
Applications, 794
9.4. Nonlinear Systems of Equations and Inequalities, 797
Nonlinear Systems of Equations, 797
Modeling and Problem Solving, 804
Nonlinear Systems of Inequalities, 805
Visualizing the Graph, 807
9.5. Rotation of Axes, 812
Rotation of Axes, 812
The Discriminant, 817
9.6. Polar Equations of Conics, 821
Polar Equations of Conics, 822
Converting from Polar to Rectangular Equations, 826
Finding Polar Equations of Conics, 826
9.7. Parametric Equations, 829
Graphing Parametric Equation, 829
Determining a Rectangular Equation for Given Parametric Equations, 830
Determining Parametric Equations for a Given Rectangular Equation, 832
Applications, 832
Summary and Review, 836
Test, 841
10. Sequences, Series, and Combinatorics, 843
10.1. Sequences and Series, 844
Sequences, 844
Finding the General Term, 846
Sums and Series, 847
Sigma Notation, 848
Recursive Definitions, 849
10.2. Arithmetic Sequences and Series, 853
Arithmetic Sequences, 853
Sum of the first n Terms of an Arithmetic Sequence, 855
Applications, 858
10.3. Geometric Sequences and Series, 863
Geometric Sequences, 863
Sum of the First nTerms of a Geometric Sequence, 865
Infinite Geometric Series, 866
Applications, 868
Visualizing the Graph, 871
10.4. Mathematical Induction, 875
Sequences of Statements, 875
Proving Infinite Sequences of Statements, 876
10.5. Combinatorics: Permutations, 881
Permutations, 881
Factorial Notation, 884
Permutations of n Objects Taken k at a Time, 885
Permutations of Sets with Nondistinguishable Objects, 887
10.6. Combinatorics: Combinations, 891
Combinations, 891
10.7. The Binomial Theorem, 899
Binomial Expansions Using Pascal's Triangle, 899
Binomial Expansion Using Factorial Notation, 902
Finding a Specific Term, 903
Total Number of Subsets, 904
10.8. Probability, 907
Experimental and Theoretical Probability, 907
Computing Experimental Probabilities, 908
Theoretical Probability, 910
Summary and Review, 918
Test, 922
Appendix: Basic Concepts from Geometry, 925
Answers, A-1
Photo Credits, P-1
Index, I-1
Index of Applications, I-15
Visualize Your Success, EP-1
Trigonometry, BM-1
A Library of Functions, BM-3

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