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Printed: August 12, 2015 iii Contents www.ck12.org Contents 1 Optional Topics (Algebra Review)1 1.1 Algebraic Expressions Basics.................................... 2 1.2 Exponents, Radicals and Rational Exponents............................ 5 1.3 Basics of Polynomials........................................ 12 1.4 Factoring............................................... 16 1.5 Rational Expression Basics..................................... 21 1.6 Equations .............................................. 26 1.7 Inequalities ............................................. 30 2 Relations, Functions and Their Graphs 35 2.1 Relations and Basics of Functions ................................. 36 2.2 Piecewise Functions......................................... 50 2.3 Information From Graphs...................................... 56 2.4 Rates of Change........................................... 66 2.5 Transformations of Functions.................................... 73 2.6 Algebra of Functions ........................................ 80 2.7 Composition of Functions...................................... 84 2.8 Inverse Functions .......................................... 90 3 Polynomials and Rational Functions 97 3.1 Complex numbers.......................................... 98 3.2 Quadratic Functions.........................................104 3.3 Characteristics of Polynomials ...................................110 3.4 Pascals Triangle and Binomial Expansion .............................114 3.5 Graphs of Polynomials, Zeros and Their Multiplicities.......................120 3.6 Factoring Review (Optional) ....................................127 3.7 Dividing Polynomials........................................133 3.8 Zeros of Polynomials ........................................139 3.9 Rational Expressions ........................................144 3.10 Rational Functions..........................................151 3.11 Rational Equations .........................................158 3.12 Polynomial and Rational Inequalities................................164 3.13 References..............................................170 4 Exponential and Logarithmic Functions 171 4.1 Exponential Functions........................................172 4.2 Logarithmic Functions .......................................178 4.3 Properties of Logarithms ......................................183 4.4 Exponential and Logarithmic Equations ..............................187 5 Basic Triangle Trigonometry 193 5.1 Angles in Radians and Degrees...................................194 iv www.ck12.org Contents 5.2 Circular Motion and Dimensional Analysis.............................198 5.3 Special Right Triangles.......................................203 5.4 Right Triangle Trigonometry ....................................212 5.5 The Unit Circle ...........................................219 5.6 Solving Simple Trig equations ...................................225 5.7 Law of Cosines ...........................................231 5.8 Law of Sines.............................................239 5.9 Applications of Basic Triangle Trigonometry............................247 5.10 References..............................................255 6 Graphs of Trigonometric Functions 256 6.1 Graphs of Sine, Cosine, Tangent and Vertical stretches.......................257 6.2 Graphs of Cosecant, Secant and Cotangent and Vertical Stretches . 261 6.3 Transformations of Trigonometric Functions Horizontal Stretch and Compression . 265 6.4 Transformations 2: Vertical Shift, Horizontal Shift (Phase Shift) . 268 6.5 Tranformations 3: Stretching and Shifting (Putting Them Together) . 272 6.6 Graphs of arcsin, arccos, and arctan (Inverse Trigonometric Functions) . 275 6.7 Equations of Trigonometric Graphs.................................282 6.8 Applications of Trigonometric Functions (Harmonic-Sinusoidal Motion) . 287 7 Analytic Trigonometry 294 7.1 Verifying with Trigonometric Identities...............................295 7.2 Sum and Difference Formulas....................................302 7.3 Double Angle Formulas.......................................312 7.4 Solving Trigonometric Equations..................................317 8 Polars and Complex Numbers 329 8.1 Polar and Rectangular Coordinates.................................330 8.2 Graphing Polar Equations......................................338 8.3 Complex Numbers in Polar Form..................................346 8.4 Parametric Equations ........................................352 9 Vectors 359 9.1 Basic Properties of Vectors .....................................360 9.2 Operations with Vectors.......................................370 9.3 Resolution of Vectors into Components...............................379 9.4 Dot Product and Angle Between Two Vectors ...........................387 9.5 Vector Projection ..........................................394 9.6 References..............................................402 10 Conic Sections 403 10.1 General Form of a Conic ......................................404 10.2 Parabolas (Optional).........................................412 10.3 Circles................................................419 10.4 Ellipses ...............................................426 10.5 Hyperbolas .............................................434 10.6 Degenerate Conics..........................................442 10.7 References..............................................447 11 Sequences and Series 448 11.1 Recursion ..............................................449 11.2 Arithmetic and Geometric Sequences................................453 v Contents www.ck12.org 11.3 Sigma Notation ...........................................459 11.4 Arithmetic Series ..........................................463 11.5 Geometric Series ..........................................467 11.6 Counting with Permutations and Combinations...........................472 11.7 Basic Probability ..........................................477 11.8 Binomial Theorem..........................................482 11.9 Induction Proofs...........................................486 11.10 References..............................................492 12 Concepts of Calculus 493 12.1 Limit Notation............................................494 12.2 Graphs to Find Limits........................................497 12.3 Tables to Find Limits ........................................504 12.4 Substitution to Find Limits .....................................509 12.5 Rationalization to Find Limits....................................512 12.6 One Sided Limits and Continuity..................................517 12.7 Intermediate and Extreme Value Theorems.............................523 12.8 Instantaneous Rate of Change....................................529 12.9 Area Under a Curve.........................................535 12.10 References..............................................544 vi www.ck12.org Chapter 1. Optional Topics (Algebra Review) CHAPTER 1 Optional Topics (Algebra Review) Chapter Outline 1.1 ALGEBRAIC EXPRESSIONS BASICS 1.2 EXPONENTS,RADICALS AND RATIONAL EXPONENTS 1.3 BASICS OF POLYNOMIALS 1.4 FACTORING 1.5 RATIONAL EXPRESSION BASICS 1.6 EQUATIONS 1.7 INEQUALITIES 1 1.1. Algebraic Expressions Basics www.ck12.org 1.1 Algebraic Expressions Basics TEKS 1. P.1.F Lesson Objectives In this section you will review: 1. Evaluating algebraic expressions. 2. Properties of real numbers. 3. Simplifying algebraic expressions. Introduction When an expression looks complex or overcomplicated there are ways to simplify them. In the following examples you will learn skills associated with simplifying overcomplicated expressions. Vocabulary algebraic expression, evaluate, real number, commutative property, associative property, distributive property, identity property, inverse property, like terms 1. Evaluating Algebraic