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ISBN-13: 978-0-321-69397-6 ISBN-10: 0-321-69397-3 www.net Contents Preface xi R Basic Concepts of Algebra 1 R.1 The Real-Number System 2 Real Numbers / Interval Notation / Properties of the Real Numbers / Absolute Value R.2 Integer Exponents, Scientific Notation, and Order of Operations 8 Integers as Exponents / Scientific Notation / Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials 17 Polynomials / Addition and Subtraction / Multiplication R.4 Factoring 22 Terms with Common Factors / Factoring by Grouping / Trinomials of the Type x 2 + bx + c / Trinomials of the Type ax 2 + bx + c, a Z 1 / Special Factorizations R.5 The Basics of Equation Solving 31 Linear Equations and Quadratic Equations / Formulas R.6 Rational Expressions 36 The Domain of a Rational Expression / Simplifying, Multiplying, and Dividing Rational Expressions / Adding and Subtracting Rational Expressions / Complex Rational Expressions R.7 Radical Notation and Rational Exponents 45 Simplifying Radical Expressions / An Application / Rationalizing Denominators or Numerators / Rational Exponents Study Guide 53 Review Exercises 54 Chapter Test 56 1 Graphs, Functions, and Models 59 1.1 Introduction to Graphing 60 Graphs / Solutions of Equations / Graphs of Equations / The Distance Formula / Midpoints of Segments / Circles VISUALIZING THE GRAPH 71 1.2 Functions and Graphs 75 Functions / Notation for Functions / Graphs of Functions / Finding Domains of Functions / Visualizing Domain and Range / Applications of Functions CONTENTS iii www.net iv CONTENTS 1.3 Linear Functions, Slope, and Applications 91 Linear Functions / The Linear Function f1x2 = mx + b and Slope / Applications of Slope / Slope–Intercept Equations of Lines / Graphing f1x2 = mx + b Using m and b / Applications of Linear Functions VISUALIZING THE GRAPH 102 MID-CHAPTER MIXED REVIEW 107 1.4 Equations of Lines and Modeling 109 Slope–Intercept Equations of Lines / Point–Slope Equations of Lines / Parallel Lines / Perpendicular Lines / Mathematical Models / Curve Fitting 1.5 Linear Equations, Functions, Zeros, and Applications 121 Linear Equations / Applications Using Linear Models / Zeros of Linear Functions 1.6 Solving Linear Inequalities 138 Linear Inequalities / Compound Inequalities / An Application Study Guide 144 Review Exercises 150 Chapter Test 154 2 More on Functions 157 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 158 Increasing, Decreasing, and Constant Functions / Relative Maximum and Minimum Values / Applications of Functions / Functions Defined Piecewise 2.2 The Algebra of Functions 171 The Algebra of Functions: Sums, Differences, Products, and Quotients / Difference Quotients 2.3 The Composition of Functions 179 The Composition of Functions / Decomposing a Function as a Composition MID-CHAPTER MIXED REVIEW 187 2.4 Symmetry and Transformations 189 Symmetry / Even Functions and Odd Functions / Transformations of Functions / Vertical Translations and Horizontal Translations / Reflections / Vertical and Horizontal Stretchings and Shrinkings VISUALIZING THE GRAPH 205 2.5 Variation and Applications 211 Direct Variation / Inverse Variation / Combined Variation Study Guide 220 Review Exercises 227 Chapter Test 231 www.net CONTENTS v 3 Quadratic Functions and Equations; Inequalities 233 3.1 The Complex Numbers 234 The Complex-Number System / Addition and Subtraction / Multiplication / Conjugates and Division 3.2 Quadratic Equations, Functions, Zeros, and Models 241 Quadratic Equations and Quadratic Functions / Completing the Square / Using the Quadratic Formula / The Discriminant / Equations Reducible to Quadratic / Applications 3.3 Analyzing Graphs of Quadratic Functions 257 Graphing Quadratic Functions of the Type f1x2 = a1x - h22 + k / Graphing Quadratic Functions of the Type f1x2 = ax 2 + bx + c, a Z 0 / Applications VISUALIZING THE GRAPH 266 MID-CHAPTER MIXED REVIEW 270 3.4 Solving Rational Equations and Radical Equations 272 Rational Equations / Radical Equations 3.5 Solving Equations and Inequalities with Absolute Value 280 Equations with Absolute Value / Inequalities with Absolute Value Study Guide 285 Review Exercises 289 Chapter Test 293 4 Polynomial Functions and Rational Functions 295 4.1 Polynomial Functions and Models 296 The Leading-Term Test / Finding Zeros of Factored Polynomial Functions / Polynomial Models 4.2 Graphing Polynomial Functions 309 Graphing Polynomial Functions / The Intermediate Value Theorem VISUALIZING THE GRAPH 317 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem 320 Division and Factors / The Remainder Theorem and Synthetic Division / Finding Factors of Polynomials MID-CHAPTER MIXED REVIEW 328 4.4 Theorems about Zeros of Polynomial Functions 330 The Fundamental Theorem of Algebra / Finding Polynomials with Given Zeros / Zeros of Polynomial Functions with Real Coefficients / Rational Coefficients / Integer Coefficients and the Rational Zeros Theorem / Descartes’ Rule of Signs www.net vi CONTENTS 4.5 Rational Functions 340 The Domain of a Rational Function / Asymptotes / Applications VISUALIZING THE GRAPH 356 4.6 Polynomial Inequalities and Rational Inequalities 360 Polynomial Inequalities / Rational Inequalities Study Guide 372 Review Exercises 381 Chapter Test 385 5 Exponential Functions and Logarithmic Functions 387 5.1 Inverse Functions 388 Inverses / Inverses and One-to-One Functions / Finding Formulas for Inverses / Inverse Functions and Composition / Restricting a Domain 5.2 Exponential Functions and Graphs 400 Graphing Exponential Functions / Applications / The Number e / Graphs of Exponential Functions, Base e 5.3 Logarithmic Functions and Graphs 413 Logarithmic Functions / Finding Certain Logarithms / Converting Between Exponential Equations and Logarithmic Equations / Finding Logarithms on a Calculator / Natural Logarithms / Changing Logarithmic Bases / Graphs of Logarithmic Functions / Applications VISUALIZING THE GRAPH 425 MID-CHAPTER MIXED REVIEW 429 5.4 Properties of Logarithmic Functions 430 Logarithms of Products / Logarithms of Powers / Logarithms of Quotients / Applying the Properties / Simplifying Expressions of the Type log a ax and aloga x 5.5 Solving Exponential Equations and Logarithmic Equations 439 Solving Exponential Equations / Solving Logarithmic Equations 5.6 Applications and Models: Growth and Decay; Compound Interest 450 Population Growth / Interest Compounded Continuously / Models of Limited Growth / Exponential Decay Study Guide 465 Review Exercises 471 Chapter Test 475 6 The Trigonometric Functions 477 6.1 Trigonometric Functions of Acute Angles 478 The Trigonometric Ratios / The Six Functions Related / Function Values of 30°, 45°, and 60° / Function Values of Any Acute Angle / Cofunctions and Complements www.net CONTENTS vii 6.2 Applications of Right Triangles 490 Solving Right Triangles / Applications 6.3 Trigonometric Functions of Any Angle 503 Angles, Rotations, and Degree Measure / Trigonometric Functions of Angles or Rotations / The Six Functions Related / Terminal Side on an Axis / Reference Angles: 30°, 45°, and 60° / Function Values for Any Angle MID-CHAPTER MIXED REVIEW 519 6.4 Radians, Arc Length, and Angular Speed 520 Distances on the Unit Circle / Radian Measure / Arc Length and Central Angles / Linear Speed and Angular Speed 6.5 Circular Functions: Graphs and Properties 536 Reflections on the Unit Circle / Finding Function Values / Graphs of the Sine and Cosine Functions / Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 6.6 Graphs of Transformed Sine and Cosine Functions 551 Variations of Basic Graphs / Graphs of Sums: Addition of Ordinates / Damped Oscillation: Multiplication of Ordinates VISUALIZING THE GRAPH 565 Study Guide 569 Review Exercises 578 Chapter Test 581 Trigonometric Identities, Inverse Functions, 7 and Equations 583 7.1 Identities: Pythagorean and Sum and Difference 584 Pythagorean Identities / Simplifying Trigonometric Expressions / Sum and Difference Identities 7.2 Identities: Cofunction, Double-Angle, and Half-Angle 597 Cofunction Identities / Double-Angle Identities / Half-Angle Identities 7.3 Proving Trigonometric Identities 605 The Logic of Proving Identities / Proving Identities / Product-to-Sum and Sum-to-Product Identities MID-CHAPTER MIXED REVIEW 613 7.4 Inverses of the Trigonometric Functions 614 Restricting Ranges to Define Inverse Functions / Composition of Trigonometric Functions and Their Inverses 7.5 Solving Trigonometric Equations 625 VISUALIZING THE GRAPH 635 Study Guide 639 Review Exercises 645 Chapter Test 648 www.net viii CONTENTS 8 Applications of Trigonometry 649 8.1 The Law of Sines 650 Solving Oblique Triangles / The Law of Sines / Solving Triangles (AAS and ASA) / Solving Triangles (SSA) / The Area of a Triangle 8.2 The Law of Cosines 663 The Law of Cosines / Solving Triangles (SAS) / Solving Triangles (SSS) 8.3 Complex Numbers: Trigonometric Form 673 Graphical Representation / Trigonometric Notation for Complex Numbers / Multiplication and Division with Trigonometric Notation / Powers of Complex Numbers / Roots of Complex Numbers MID-CHAPTER MIXED REVIEW 684 8.4 Polar Coordinates and Graphs 686 Polar Coordinates / Polar Equations and Rectangular Equations / Graphing Polar Equations VISUALIZING THE GRAPH 694 8.5 Vectors and Applications 697 Vectors / Vector Addition / Applications / Components 8.6 Vector Operations 705 Position Vectors / Operations on Vectors / Unit Vectors / Direction Angles / Angle Between Vectors / Forces in Equilibrium Study Guide 720 Review Exercises 731 Chapter Test 735 9 Systems of Equations and Matrices 737 9.1 Systems of Equations in Two Variables 738 Solving Systems of Equations Graphically / The Substitution Method / The Elimination Method / Applications VISUALIZING THE GRAPH 749 9.
