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Copyright, ii
Preface, xiii
Supplements Guide, xviii
R. Review of Basic Concepts, 1
R.1. Sets, 2
Basic Definitions, 2
Operations on Sets, 4
R.2. Real Numbers and Their Properties, 7
Sets of Numbers and the Number Line, 7
Exponents, 9
Order of Operations, 10
Properties of Real Numbers, 12
Order on the Number Line, 14
Absolute Value, 14
R.3. Polynomials, 22
Rules for Exponents, 22
Polynomials, 24
Addition and Subtraction, 25
Multiplication, 26
Division, 29
R.4. Factoring Polynomials, 34
Factoring Out the Greatest Common Factor, 34
Factoring by Grouping, 35
Factoring Trinomials, 36
Factoring Binomials, 38
Factoring by Substitution, 39
R.5. Rational Expressions, 43
Rational Expressions, 43
Lowest Terms of a Rational Expression, 44
Multiplication and Division, 45
Addition and Subtraction, 46
Complex Fractions, 48
R.6. Rational Exponents, 52
Negative Exponents and the Quotient Rule, 52
Rational Exponents, 55
Complex Fractions Revisited, 58
R.7. Radical Expressions, 62
Radical Notation, 62
Simplified Radicals, 65
Operations with Radicals, 66
Rationalizing Denominators, 68
Summary, 73
Review Exercises, 77
Test, 81
Quantitative Reasoning, 82
1. Equations and Inequalities, 83
1.1. Linear Equations, 84
Basic Terminology of Equations, 84
Solving Linear Equations, 84
Identities, Conditional Equations, and Contradictions, 86
Solving for a Specified Variable (Literal Equations), 87
1.2. Applications and Modeling with Linear Equations, 90
Solving Applied Problems, 90
Geometry Problems, 91
Motion Problems, 92
Mixture Problems, 93
Modeling with Linear Equations, 95
1.3. Complex Numbers, 103
Basic Concepts of Complex Numbers, 103
Operations on Complex Numbers, 104
1.4. Quadratic Equations, 111
Solving a Quadratic Equation, 111
Completing the Square, 112
The Quadratic Formula, 114
Solving for a Specified Variable, 116
The Discriminant, 117
Chapter 1 Quiz (Sections 1.1–1.4), 121
1.5. Applications and Modeling with Quadratic Equations, 121
Geometry Problems, 122
Using the Pythagorean Theorem, 123
Height of a Projected Object, 124
Modeling with Quadratic Equations, 125
1.6. Other Types of Equations and Applications, 133
Rational Equations, 133
Work Rate Problems, 135
Equations with Radicals, 137
Equations Quadratic in Form, 140
Summary Exercises on Solving Equations, 145
1.7. Inequalities, 145
Linear Inequalities, 146
Three-Part Inequalities, 148
Quadratic Inequalities, 148
Rational Inequalities, 152
1.8. Absolute Value Equations and Inequalities, 158
Absolute Value Equations, 159
Absolute Value Inequalities, 160
Special Cases, 161
Absolute Value Models for Distance and Tolerance, 161
Summary, 166
Review Exercises, 170
Test, 177
Quantitative Reasoning, 179
2. Graphs and Functions, 181
2.1. Rectangular Coordinates and Graphs, 182
Ordered Pairs, 182
The Rectangular Coordinate System, 182
The Distance Formula, 183
The Midpoint Formula, 185
Graphing Equations, 187
2.2. Circles, 193
Center-Radius Form, 193
General Form, 195
An Application, 197
2.3. Functions, 201
Relations and Functions, 201
Domain and Range, 203
Determining Functions from Graphs or Equations, 205
Function Notation, 208
Increasing, Decreasing, and Constant Functions, 211
2.4. Linear Functions, 217
Graphing Linear Functions, 217
Standard Form Ax + By = C, 219
Slope, 220
Average Rate of Change, 223
Linear Models, 224
Chapter 2 Quiz (Sections 2.1–2.4), 231
2.5. Equations of Lines; Curve Fitting, 232
Point-Slope Form, 232
Slope-Intercept Form, 233
Vertical and Horizontal Lines, 235
Parallel and Perpendicular Lines, 235
Modeling Data, 238
Solving Linear Equations in One Variable by Graphing, 241
Summary Exercises on Graphs, Circles, Functions, and Equations, 247
2.6. Graphs of Basic Functions, 248
Continuity, 248
The Identity, Squaring, and Cubing Functions, 249
The Square Root and Cube Root Functions, 250
The Absolute Value Function, 251
Piecewise-Defined Functions, 252
The Relation x = y², 254
2.7. Graphing Techniques, 259
Stretching and Shrinking, 259
Reflecting, 261
Symmetry, 262
Even and Odd Functions, 264
Translations, 265
Chapter 2 Quiz (Sections 2.5–2.7), 273
2.8. Function Operations and Composition, 274
Arithmetic Operations on Functions, 274
The Difference Quotient, 277
Composition of Functions and Domain, 278
Summary, 288
Review Exercises, 292
Test, 297
Quantitative Reasoning, 299
3. Polynomial and Rational Functions, 301
3.1. Quadratic Functions and Models, 302
Quadratic Functions, 302
Graphing Techniques, 303
Completing the Square, 305
The Vertex Formula, 307
Quadratic Models and Curve Fitting, 308
3.2. Synthetic Division, 321
Synthetic Division, 321
Evaluating Polynomial Functions Using the Remainder Theorem, 324
Testing Potential Zeros, 325
3.3. Zeros of Polynomial Functions, 328
Factor Theorem, 328
Rational Zeros Theorem, 329
Number of Zeros, 331
Conjugate Zeros Theorem, 333
Finding Zeros of a Polynomial Function, 335
Descartes’ Rule of Signs, 335
3.4. Polynomial Functions: Graphs, Applications, and Models, 340
Graphs of f (x) = axⁿ, 340
Graphs of General Polynomial Functions, 341
Turning Points and End Behavior, 342
Graphing Techniques, 344
Intermediate Value and Boundedness Theorems, 347
Approximating Real Zeros, 349
Polynomial Models and Curve Fitting, 350
Summary Exercises on Polynomial Functions, Zeros, and Graphs, 358
3.5. Rational Functions: Graphs, Applications, and Models, 359
The Reciprocal Function f (x) = 1/x, 359
The Function f (x) = 1/x², 361
Asymptotes, 363
Steps for Graphing Rational Functions, 365
Rational Function Models, 370
Chapter 3 Quiz (Sections 3.1–3.5), 379
3.6. Variation, 379
Direct Variation, 379
Inverse Variation, 381
Combined and Joint Variation, 382
Summary, 388
Review Exercises, 393
Test, 398
Quantitative Reasoning, 400
4. Inverse, Exponential, and Logarithmic Functions, 401
4.1. Inverse Functions, 402
Inverse Operations, 402
One-to-One Functions, 402
Inverse Functions, 404
Equations of Inverses, 407
An Application of Inverse Functions to Cryptography, 410
4.2. Exponential Functions, 415
Exponents and Properties, 415
Exponential Functions, 417
Exponential Equations, 420
Compound Interest, 421
The Number e and Continuous Compounding, 423
Exponential Models and Curve Fitting, 425
4.3. Logarithmic Functions, 432
Logarithms, 432
Logarithmic Equations, 433
Logarithmic Functions, 434
Properties of Logarithms, 437
Summary Exercises on Inverse, Exponential, and Logarithmic Functions, 445
4.4. Evaluating Logarithms and the Change-of-Base Theorem, 447
Common Logarithms, 447
Applications and Modeling with Common Logarithms, 447
Natural Logarithms, 449
Applications and Modeling with Natural Logarithms, 449
Logarithms with Other Bases, 451
Chapter 4 Quiz (Sections 4.1–4.4), 458
4.5. Exponential and Logarithmic Equations, 458
Exponential Equations, 458
Logarithmic Equations, 460
Applications and Modeling, 462
4.6. Applications and Models of Exponential Growth and Decay, 468
The Exponential Growth or Decay Function, 468
Growth Function Models, 469
Decay Function Models, 471
Summary Exercises on Functions: Domains and Defining Equations, 481
Summary, 484
Review Exercises, 487
Test, 490
Quantitative Reasoning, 492
5. Trigonometric Functions, 493
5.1. Angles, 494
Basic Terminology, 494
Degree Measure, 494
Standard Position, 497
Coterminal Angles, 498
5.2. Trigonometric Functions, 503
Trigonometric Functions, 503
Quadrantal Angles, 505
Reciprocal Identities, 507
Signs and Ranges of Function Values, 508
Pythagorean Identities, 510
Quotient Identities, 511
5.3. Evaluating Trigonometric Functions, 515
Right-Triangle-Based Definitions of the Trigonometric Functions, 515
Cofunctions, 516
Trigonometric Function Values of Special Angles, 517
Reference Angles, 519
Special Angles as Reference Angles, 519
Finding Function Values Using a Calculator, 521
Finding Angle Measures, 522
Chapter 5 Quiz (Sections 5.1–5.3), 529
5.4. Solving Right Triangles, 531
Significant Digits, 531
Solving Triangles, 532
Angles of Elevation or Depression, 533
Bearing, 534
Further Applications, 536
Summary, 546
Review Exercises, 549
Test, 553
Quantitative Reasoning, 555
6. The Circular Functions and Their Graphs, 557
6.1. Radian Measure, 558
Radian Measure, 558
Converting Between Degrees and Radians, 558
Arc Length of a Circle, 561
Area of a Sector of a Circle, 564
6.2. The Unit Circle and Circular Functions, 571
Circular Functions, 571
Finding Values of Circular Functions, 574
Determining a Number with a Given Circular Function Value, 576
Linear and Angular Speed, 578
6.3. Graphs of the Sine and Cosine Functions, 585
Periodic Functions, 585
Graph of the Sine Function, 585
Graph of the Cosine Function, 587
Graphing Techniques, Amplitude, and Period, 588
Using a Trigonometric Model, 592
6.4. Translations of the Graphs of the Sine and Cosine Functions, 598
Horizontal Translations, 598
Vertical Translations, 600
Combinations of Translations, 601
Determining a Trigonometric Model Using Curve Fitting, 603
Chapter 6 Quiz (Sections 6.1–6.4), 607
6.5. Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions, 608
Graphs of the Tangent and Cotangent Functions, 608
Graphs of the Secant and Cosecant Functions, 613
Connecting Graphs with Equations, 618
Addition of Ordinates, 619
Summary Exercises on Graphing Circular Functions, 623
6.6. Harmonic Motion, 624
Simple Harmonic Motion, 624
Damped Oscillatory Motion, 626
Summary, 629
Review Exercises, 632
Test, 638
Quantitative Reasoning, 640
7. Trigonometric Identities and Equations, 641
7.1. Fundamental Identities, 642
Fundamental Identities, 642
Using the Fundamental Identities, 643
7.2. Verifying Trigonometric Identities, 648
Verifying Identities by Working with One Side, 649
Verifying Identities by Working with Both Sides, 651
7.3. Sum and Difference Identities, 657
Cosine Sum and Difference Identities, 657
Cofunction Identities, 659
Sine and Tangent Sum and Difference Identities, 662
Chapter 7 Quiz (Sections 7.1–7.3), 671
7.4. Double-Angle and Half-Angle Identities, 672
Double-Angle Identities, 672
Product-to-Sum and Sum-to-Product Identities, 676
Half-Angle Identities, 677
Summary Exercises on Verifying Trigonometric Identities, 685
7.5. Inverse Circular Functions, 686
Inverse Functions, 686
Inverse Sine Function, 687
Inverse Cosine Function, 689
Inverse Tangent Function, 690
Remaining Inverse Circular Functions, 691
Inverse Function Values, 692
7.6. Trigonometric Equations, 700
Solving by Linear Methods, 700
Solving by Factoring, 701
Solving by Quadratic Methods, 701
Solving by Using Trigonometric Identities, 703
Equations with Half-Angles, 704
Equations with Multiple Angles, 705
Applications, 706
Chapter 7 Quiz (Sections 7.5–7.6), 712
7.7. Equations Involving Inverse Trigonometric Functions, 713
Solving for x in Terms of y Using Inverse Functions, 713
Solving Inverse Trigonometric Equations, 713
Summary, 719
Review Exercises, 722
Test, 727
Quantitative Reasoning, 729
8. Applications of Trigonometry, 731
8.1. The Law of Sines, 732
Congruency and Oblique Triangles, 732
Derivation of the Law of Sines, 733
Using the Law of Sines, 734
Ambiguous Case, 735
Area of a Triangle, 739
8.2. The Law of Cosines, 748
Derivation of the Law of Cosines, 748
Using the Law of Cosines, 749
Heron’s Formula for the Area of a Triangle, 752
Chapter 8 Quiz (Sections 8.1–8.2), 762
8.3. Vectors, Operations, and the Dot Product, 763
Basic Terminology, 763
Algebraic Interpretation of Vectors, 764
Operations with Vectors, 766
Dot Product and the Angle Between Vectors, 768
8.4. Applications of Vectors, 773
The Equilibrant, 773
Incline Applications, 774
Navigation Applications, 775
Summary Exercises on Applications of Trigonometry and Vectors, 779
8.5. Trigonometric (Polar) Form of Complex Numbers; Products and Quotients, 781
The Complex Plane and Vector Representation, 781
Trigonometric (Polar) Form, 782
Fractals, 784
Products of Complex Numbers in Trigonometric Form, 785
Quotients of Complex Numbers in Trigonometric Form, 787
8.6. De Moivre’s Theorem; Powers and Roots of Complex Numbers, 791
Powers of Complex Numbers (De Moivre’s Theorem), 791
Roots of Complex Numbers, 792
Chapter 8 Quiz (Sections 8.3–8.6), 799
8.7. Polar Equations and Graphs, 799
Polar Coordinate System, 799
Graphs of Polar Equations, 802
Converting from Polar to Rectangular Equations, 806
Classifying Polar Equations, 807
8.8. Parametric Equations, Graphs, and Applications, 812
Basic Concepts, 812
Parametric Graphs and Their Rectangular Equivalents, 813
The Cycloid, 815
Applications of Parametric Equations, 816
Summary, 822
Review Exercises, 827
Test, 833
Quantitative Reasoning, 835
9. Systems and Matrices, 837
9.1. Systems of Linear Equations, 838
Linear Systems, 838
Substitution Method, 838
Elimination Method, 839
Special Systems, 840
Applying Systems of Equations, 842
Solving Linear Systems with Three Unknowns (Variables), 843
Using Systems of Equations to Model Data, 845
9.2. Matrix Solution of Linear Systems, 855
The Gauss-Jordan Method, 855
Special Systems, 859
9.3. Determinant Solution of Linear Systems, 867
Determinants, 867
Cofactors, 868
Evaluating n × n Determinants, 869
Cramer’s Rule, 871
9.4. Partial Fractions, 879
Decomposition of Rational Expressions, 879
Distinct Linear Factors, 880
Repeated Linear Factors, 881
Distinct Linear and Quadratic Factors, 881
Repeated Quadratic Factors, 883
Chapter 9 Quiz (Sections 9.1–9.4), 885
9.5. Nonlinear Systems of Equations, 886
Solving Nonlinear Systems with Real Solutions, 886
Solving Nonlinear Systems with Nonreal Complex Solutions, 890
Applying Nonlinear Systems, 891
Summary Exercises on Systems of Equations, 897
9.6. Systems of Inequalities and Linear Programming, 898
Solving Linear Inequalities, 898
Solving Systems of Inequalities, 899
Linear Programming, 901
9.7. Properties of Matrices, 909
Basic Definitions, 909
Adding Matrices, 910
Special Matrices, 911
Subtracting Matrices, 911
Multiplying Matrices, 912
Applying Matrix Algebra, 916
9.8. Matrix Inverses, 923
Identity Matrices, 923
Multiplicative Inverses, 925
Solving Systems Using Inverse Matrices, 929
Summary, 936
Review Exercises, 941
Test, 946
Quantitative Reasoning, 948
10. Analytic Geometry, 949
10.1. Parabolas, 950
Conic Sections, 950
Horizontal Parabolas, 950
Geometric Definition and Equations of Parabolas, 952
An Application of Parabolas, 956
10.2. Ellipses, 960
Equations and Graphs of Ellipses, 960
Translated Ellipses, 964
Eccentricity, 965
Applications of Ellipses, 966
Chapter 10 Quiz (Sections 10.1–10.2), 971
10.3. Hyperbolas, 971
Equations and Graphs of Hyperbolas, 971
Translated Hyperbolas, 975
Eccentricity, 975
10.4. Summary of the Conic Sections, 980
Characteristics, 980
Identifying Conic Sections, 982
Geometric Definition of Conic Sections, 984
Summary, 987
Review Exercises, 989
Test, 991
Quantitative Reasoning, 993
11. Further Topics in Algebra, 995
11.1. Sequences and Series, 996
Sequences, 996
Series and Summation Notation, 1000
Summation Properties, 1002
11.2. Arithmetic Sequences and Series, 1007
Arithmetic Sequences, 1007
Arithmetic Series, 1011
11.3. Geometric Sequences and Series, 1016
Geometric Sequences, 1016
Geometric Series, 1018
Infinite Geometric Series, 1020
Annuities, 1022
Summary Exercises on Sequences and Series, 1027
11.4. The Binomial Theorem, 1028
A Binomial Expansion Pattern, 1028
Pascal’s Triangle, 1029
n-Factorial, 1029
Binomial Coefficients, 1030
The Binomial Theorem, 1031
kth Term of a Binomial Expansion, 1033
11.5. Mathematical Induction, 1036
Proof by Mathematical Induction, 1036
Proving Statements, 1037
Generalized Principle of Mathematical Induction, 1038
Proof of the Binomial Theorem, 1040
Chapter 11 Quiz (Sections 11.1–11.5), 1042
11.6. Counting Theory, 1043
Fundamental Principle of Counting, 1043
Permutations, 1044
Combinations, 1046
Distinguishing Between Permutations and Combinations, 1048
11.7. Basics of Probability, 1054
Basic Concepts, 1054
Complements and Venn Diagrams, 1055
Odds, 1056
Union of Two Events, 1057
Binomial Probability, 1059
Summary, 1065
Review Exercises, 1069
Test, 1072
Quantitative Reasoning, 1073
Appendices, 1075
Appendix A. Polar Form of Conic Sections, 1075
Appendix B. Rotation of Axes, 1079
Appendix C. Geometry Formulas, 1084
Glossary, 1085
Solutions to Selected Exercises, S-1
Answers to Selected Exercises, A-1
Index of Applications, I-1
Index, I-7
Photo Credits, P-1

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