Advanced Functions – Table of Contents

Table of Contents

Unit 1: Prerequisite Skills

1.1 Slope and Rate of Change ...... 1

1.2 Cost vs. Time Functions ...... 4 1.3 Distance vs. Time Functions ...... 8 1.4 Common Factoring ...... 11 1.5 Factoring Perfect Square Trinomials ...... 12 1.6 Factoring Differences of Squares ...... 15 1.7 Factoring Trinomials ...... 17 1.8 Graphing Systems of Inequalities ...... 21 Review ...... 25

Unit 2: Functions

2.1 Introduction to Functions ...... 30 2.2 Function Notation and Evaluation ...... 31 2.3 Domain of a Function and Interval Notation ...... 34 2.4 Range of a Function and Interval Notation ...... 38 2.5 Adding and Subtracting Functions ...... 41 2.6 Multiplying and Dividing Functions ...... 44 2.7 Composition of Functions ...... 48 2.8 Inverse Functions ...... 51 2.9 Composition and Inverses ...... 57 Review ...... 61

Unit 3: Polynomials

3.1 Introduction to Polynomials ...... 67 3.2 Finding a Polynomial Given the Roots ...... 68 3.3 Dividing Polynomials Using Long Division ...... 72 3.4 Dividing Polynomials Using Synthetic Division ...... 74 3.5 Remainder Theorem ...... 77 3.6 Factor Theorem ...... 79 3.7 Factoring Using Sum or Difference of Cubes ...... 81 Review ...... 84

Unit 4: Polynomials II

4.1 Location Principle and Multiplicity of Zeros ...... 90 4.2 Rational Root Theorem ...... 94 4.3 Complex Conjugate Root Theorem ...... 99 4.4 Fundamental Theorem of Algebra ...... 103 4.5 Graphing the Cubic Function ...... 108 4.6 Graphing Polynomials ...... 111 4.7 Tangent and Secant Lines ...... 118 Review ...... 124

Unit 5: Rational Functions

5.1 Introduction to Rational Functions ...... 132 5.2 Simplifying Rational Expressions ...... 133 5.3 Adding and Subtracting Rational Expressions ...... 138 5.4 Multiplying Rational Expressions ...... 145

5.5 Dividing Rational Expressions ...... 149 5.6 Complex Fractions ...... 153 5.7 Solving Rational Equations ...... 155 5.8 Graphing Rational Functions ...... 160 5.9 Graphing Rational Functions Part II ...... 168 Review ...... 172

Unit 6: Properties of Logarithms

6.1 Introduction to Exponential and Logarithmic Properties ...... 179 6.2 Logarithmic Functions ...... 182 6.3 Evaluating Logarithmic Functions ...... 184 6.4 Product Property of Logarithms ...... 188 6.5 Quotient Property of Logarithms ...... 192 6.6 Power Property of Logarithms ...... 196 6.7 The Exponential-Logarithmic Inverse Property ...... 200 Review ...... 205

Unit 7: Exponential and Logarithmic Functions

7.1 Graphing Exponential Functions ...... 212 7.2 Applications of Logarithms ...... 216 7.3 The Natural Exponential Function ...... 222 7.4 The Natural Logarithm ...... 225 7.5 Solving Exponential Equations ...... 229 7.6 Solving Logarithmic Equations ...... 234 7.7 Exponential and Logarithmic Models ...... 238 Review ...... 242

Unit 8: Trigonometric Functions

8.1 Introduction to Trigonometric Functions ...... 249 8.2 Radian Measure ...... 252 8.3 Standard Position and Reference Angles ...... 256 8.4 Special Triangles and Exact Ratios ...... 261 8.5 Graphing the Sine and Cosine Functions ...... 269 8.6 Graphing the Tangent and Cotangent Functions ...... 276 8.7 Graphing the Secant and Cosecant Functions ...... 280 8.8 Applications of Trigonometric Functions ...... 284 8.9 Modelling Trigonometric Functions ...... 288 8.10 Inverse Trigonometric Functions ...... 294 Review ...... 300

Unit 9: Analytic Trigonometry

9.1 Introduction to Analytic Trigonometry ...... 312 9.2 Cofunction, Periodicity and Negative Angle Identities ...... 316 9.3 Addition and Subtraction Identities ...... 323 9.4 Double and Half-Angle Identities ...... 331 9.5 Product to Sum Identities ...... 337 9.6 Solving Trigonometric Equations Algebraically ...... 342 9.7 Solving Trigonometric Equations with Identities ...... 348 Review ...... 352

Glossary ...... 359