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Preface | |

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Linear Functions, Equations, and Inequalities | |

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Real Numbers and the Rectangular Coordinate System | |

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Sets of Real Numbers | |

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The Rectangular Coordinate System | |

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Viewing Windows | |

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Roots | |

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Distance and Midpoint Formulas | |

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Introduction to Relations and Functions | |

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Set-Builder Notation and Interval Notation | |

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Relations, Domain, and Range | |

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Functions | |

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Tables | |

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Function Notation | |

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Reviewing Basic Concepts (Sections 1.1 and 1.2) | |

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Linear Functions | |

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Basic Concepts about Linear Functions | |

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Slope of a Line | |

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Slope-Intercept Form of the Equation of a Line | |

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Equations of Lines and Linear Models | |

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Point-Slope Form of the Equation of a Line | |

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Standard Form of the Equation of a Line | |

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Parallel and Perpendicular Lines | |

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Linear Models and Regression | |

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Reviewing Basic Concepts (Sections 1.3 and 1.4) | |

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Linear Equations and Inequalities | |

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Solving Linear Equations | |

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Graphical Approaches to Solving Linear Equations | |

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Identities and Contradictions | |

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Solving Linear Inequalities | |

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Graphical Approaches to Solving Linear Inequalities | |

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Three-Part Inequalities | |

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Applications of Linear Functions | |

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Problem-Solving Strategies | |

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Applications of Linear Equations | |

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Break-Even Analysis | |

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Direct Variation | |

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Formulas | |

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Reviewing Basic Concepts (Sections 1.5 and 1.6) | |

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Chapter 1 Summary | |

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Chapter 1 Review Exercises | |

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Chapter 1 Test | |

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Chapter 1 Project: Predicting Heights and Weights of Athletes | |

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Analysis of Graphs of Functions | |

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Graphs of Basic Functions and Relations; Symmetry | |

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Continuity | |

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Increasing and Decreasing Functions | |

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The Identity Function | |

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The Squaring Function and Symmetry with Respect to the y-Axis | |

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The Cubing Function and Symmetry with Respect to the Origin | |

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The Square Root and Cube Root Functions | |

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The Absolute Value Function | |

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The Relation x - y[superscript 2] and Symmetry with Respect to the x-Axis | |

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Even and Odd Functions | |

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Vertical and Horizontal Shifts of Graphs | |

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Vertical Shifts | |

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Horizontal Shifts | |

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Combinations of Vertical and Horizontal Shifts | |

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Effects of Shifts on Domain and Range | |

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Horizontal Shifts Applied to Equations for Modeling | |

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Stretching, Shrinking, and Reflecting Graphs | |

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Vertical Stretching | |

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Vertical Shrinking | |

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Horizontal Stretching and Shrinking | |

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Reflecting across an Axis | |

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Combining Transformations of Graphs | |

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Reviewing Basic Concepts (Sections 2.1-2.3) | |

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Absolute Value Functions: Graphs, Equations, Inequalities, and Applications | |

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The Graph of y = [vertical bar]f(x)[vertical bar] | |

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Properties of Absolute Value | |

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Equations and Inequalities Involving Absolute Value | |

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An Application Involving Absolute Value | |

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Piecewise-Defined Functions | |

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Graphing Piecewise-Defined Functions | |

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The Greatest Integer Function | |

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Applications of Piecewise-Defined Functions | |

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Operations and Composition | |

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Operations on Functions | |

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The Difference Quotient | |

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Composition of Functions | |

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Applications of Operations and Composition | |

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Reviewing Basic Concepts (Sections 2.4-2.6) | |

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Chapter 2 Summary | |

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Chapter 2 Review Exercises | |

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Chapter 2 Test | |

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Chapter 2 Project: Modeling the Movement of a Cold Front | |

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Polynomial Functions | |

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Complex Numbers | |

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The Number i | |

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Operations with Complex Numbers | |

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Quadratic Functions and Graphs | |

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Completing the Square | |

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Graphs of Quadratic Functions | |

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Vertex Formula | |

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Extreme Values | |

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Applications and Quadratic Models | |

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Quadratic Equations and Inequalities | |

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Zero-Product Property | |

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Solving x[superscript 2] = k | |

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Quadratic Formula and the Discriminant | |

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Solving Quadratic Equations | |

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Solving Quadratic Inequalities | |

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Formulas Involving Quadratics | |

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Another Quadratic Model | |

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Reviewing Basic Concepts (Sections 3.1-3.3) | |

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Further Applications of Quadratic Functions and Models | |

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Applications of Quadratic Functions | |

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Quadratic Models | |

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Higher-Degree Polynomial Functions and Graphs | |

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Cubic Functions | |

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Quartic Functions | |

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Extrema | |

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End Behavior | |

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x-Intercepts (Real Zeros) | |

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Comprehensive Graphs | |

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Curve Fitting and Polynomial Models | |

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Reviewing Basic Concepts (Sections 3.4 and 3.5) | |

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Topics in the Theory of Polynomial Functions (I) | |

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Intermediate Value Theorem | |

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Division of Polynomials and Synthetic Division | |

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Remainder and Factor Theorems | |

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Topics in the Theory of Polynomial Functions (II) | |

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Complex Zeros and the Fundamental Theorem of Algebra | |

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Number of Zeros | |

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Rational Zeros Theorem | |

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Descartes' Rule of Signs | |

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Boundedness Theorem | |

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Polynomial Equations and Inequalities; Further Applications and Models | |

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Polynomial Equations and Inequalities | |

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Complex nth Roots | |

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Applications and Polynomial Models | |

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Reviewing Basic Concepts (Sections 3.6-3.8) | |

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Chapter 3 Summary | |

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Chapter 3 Review Exercises | |

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Chapter 3 Test | |

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Chapter 3 Project: Creating a Social Security Polynomial | |

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Rational, Power, and Root Functions | |

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Rational Functions and Graphs | |

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The Reciprocal Function | |

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The Rational Function Defined by f(x) = 1 / x[superscript 2] | |

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More on Graphs of Rational Functions | |

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Vertical and Horizontal Asymptotes | |

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Graphing Techniques | |

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Oblique Asymptotes | |

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Graphs with Points of Discontinuity | |

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Rational Equations, Inequalities, Applications, and Models | |

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Solving Rational Equations and Inequalities | |

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Applications and Models of Rational Functions | |

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Inverse Variation | |

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Combined and Joint Variation | |

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Reviewing Basic Concepts (Sections 4.1-4.3) | |

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Functions Defined by Powers and Roots | |

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Power and Root Functions | |

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Modeling Using Power Functions | |

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Graphs of f(x) = [characters not reproducible] | |

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Graphing Circles and Horizontal Parabolas Using Root Functions | |

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Equations, Inequalities, and Applications Involving Root Functions | |

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Equations and Inequalities | |

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An Application of Root Functions | |

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Reviewing Basic Concepts (Sections 4.4 and 4.5) | |

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Chapter 4 Summary | |

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Chapter 4 Review Exercises | |

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Chapter 4 Test | |

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Chapter 4 Project: How Rugged Is Your Coastline? | |

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Inverse, Exponential, and Logarithmic Functions | |

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Inverse Functions | |

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Inverse Operations | |

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One-to-One Functions | |

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Inverse Functions and Their Graphs | |

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Equations of Inverse Functions | |

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An Application of Inverse Functions | |

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Exponential Functions | |

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Real-Number Exponents | |

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Graphs of Exponential Functions | |

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Exponential Equations (Type 1) | |

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Compound Interest | |

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The Number e and Continuous Compounding | |

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An Application of Exponential Functions | |

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Logarithms and Their Properties | |

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Definition of Logarithm | |

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Common Logarithms | |

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Natural Logarithms | |

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Properties of Logarithms | |

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Change-of-Base Rule | |

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Reviewing Basic Concepts (Sections 5.1-5.3) | |

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Logarithmic Functions | |

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Graphs of Logarithmic Functions | |

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Applying Earlier Work to Logarithmic Functions | |

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A Logarithmic Model | |

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Exponential and Logarithmic Equations and Inequalities | |

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Exponential Equations and Inequalities (Type 2) | |

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Logarithmic Equations and Inequalities | |

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Equations and Inequalities Involving Both Exponentials and Logarithms | |

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Formulas Involving Exponentials and Logarithms | |

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Reviewing Basic Concepts (Sections 5.4 and 5.5) | |

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Further Applications and Modeling with Exponential and Logarithmic Functions | |

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Physical Science Applications | |

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Financial Applications | |

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Biological and Medical Applications | |

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Modeling Data with Exponential and Logarithmic Functions | |

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Chapter 5 Summary | |

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Chapter 5 Review Exercises | |

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Chapter 5 Test | |

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Chapter 5 Project: Modeling Motor Vehicle Sales in the United States (with a lesson about the careless use of mathematical models) | |

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Analytic Geometry | |

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Circles and Parabolas | |

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Conic Sections | |

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Equations and Graphs of Circles | |

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Equations and Graphs of Parabolas | |

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Translations of Parabolas | |

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An Application of Parabolas | |

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Ellipses and Hyperbolas | |

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Equations and Graphs of Ellipses | |

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Translations of Ellipses | |

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An Application of Ellipses | |

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Equations and Graphs of Hyperbolas | |

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Translations of Hyperbolas | |

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Reviewing Basic Concepts (Sections 6.1 and 6.2) | |

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Summary of the Conic Sections | |

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Characteristics | |

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Identifying Conic Sections | |

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Eccentricity | |

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Parametric Equations | |

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Graphs of Parametric Equations and Their Rectangular Equivalents | |

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Alternative Forms of Parametric Equations | |

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An Application of Parametric Equations | |

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Reviewing Basic Concepts (Sections 6.3 and 6.4) | |

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Chapter 6 Summary | |

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Chapter 6 Review Exercises | |

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Chapter 6 Test | |

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Chapter 6 Project: Modeling the Path of a Bouncing Ball | |

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Systems of Equations and Inequalities; Matrices | |

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Systems of Equations | |

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Linear Systems | |

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Substitution Method | |

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Elimination Method | |

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Special Systems | |

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Nonlinear Systems | |

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Applications of Systems | |

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Solution of Linear Systems in Three Variables | |

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Geometric Considerations | |

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Analyttic Solution of Systems in Three Variables | |

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Applications of Systems | |

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Curve Fitting Using a System | |

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Solution of Linear Systems by Row Transformations | |

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Matrix Row Transformations | |

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Row Echelon Method | |

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Reduced Row Echelon Method | |

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Special Cases | |

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An Application of Matrices | |

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Reviewing Basic Concepts (Sections 7.1-7.3) | |

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Matrix Properties and Operations | |

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Terminology of Matrices | |

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Operations on Matrices | |

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Applying Matrix Algebra | |

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Determinants and Cramer's Rule | |

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Determinants of 2 x 2 Matrices | |

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Determinants of Larger Matrices | |

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Derivation of Cramer's Rule | |

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Using Cramer's Rule to Solve Systems | |

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Solution of Linear Systems by Matrix Inverses | |

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Identity Matrices | |

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Multiplicative Inverses of Square Matrices | |

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Using Determinants to Find Inverses | |

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Solving Linear Systems Using Inverse Matrices | |

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Curve Fitting Using a System | |

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Reviewing Basic Concepts (Sections 7.4-7.6) | |

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Systems of Inequalities and Linear Programming | |

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Solving Linear Inequalities | |

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Solving Systems of Inequalities | |

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Linear Programming | |

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Partial Fractions | |

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Decomposition of Rational Expressions | |

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Distinct Linear Factors | |

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Repeated Linear Factors | |

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Distinct Linear and Quadratic Factors | |

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Repeated Quadratic Factors | |

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Reviewing Basic Concepts (Sections 7.7 and 7.8) | |

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Chapter 7 Summary | |

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Chapter 7 Review Exercises | |

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Chapter 7 Test | |

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Chapter 7 Project: Finding a Polynomial Whose Graph Passes through Any Number of Given Points | |

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Trigonometric Functions and Applications | |

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Angles and Their Measures | |

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Basic Terminology | |

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Degree Measure | |

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Standard Position and Coterminal Angles | |

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Radian Measure | |

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Arc Lengths and Areas of Sectors | |

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Angular and Linear Speed | |

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Trigonometric Functions and Fundamental Identities | |

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Trigonometric Functions | |

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Quadrantal Angles | |

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Reciprocal Identities | |

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Signs and Ranges of Function Values | |

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Pythagorean Identities | |

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Quotient Identities | |

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An Application of Trigonometric Functions | |

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Reviewing Basic Concepts (Sections 8.1 and 8.2) | |

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Evaluating Trigonometric Functions | |

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Definitions of the Trigonometric Functions | |

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Trigonometric Function Values of Special Angles | |

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Cofunction Identities | |

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Reference Angles | |

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Special Angles as Reference Angles | |

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Finding Function Values with a Calculator | |

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Finding Angle Measures | |

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Applications of Right Triangles | |

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Significant Digits | |

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Solving Triangles | |

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Angles of Elevation or Depression | |

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Bearing | |

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Further Applications of Trigonometric Functions | |

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Reviewing Basic Concepts (Sections 8.3 and 8.4) | |

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The Circular Functions | |

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Circular Functions | |

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Applications of Circular Functions | |

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Graphs of the Sine and Cosine Functions | |

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Periodic Functions | |

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Graph of the Sine Function | |

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Graph of the Cosine Function | |

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Graphing Techniques, Amplitude, and Period | |

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Translations | |

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Determining a Trigonometric Model Using Curve Fitting | |

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Reviewing Basic Concepts (Sections 8.5 and 8.6) | |

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Graphs of the Other Circular Functions | |

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Graphs of the Cosecant and Secant Functions | |

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Graphs of the Tangent and Cotangent Functions | |

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Addition of Ordinates | |

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Harmonic Motion | |

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Simple Harmonic Motion | |

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Damped Oscillatory Motion | |

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Reviewing Basic Concepts (Sections 8.7 and 8.8) | |

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Chapter 8 Summary | |

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Chapter 8 Review Exercises | |

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Chapter 8 Test | |

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Chapter 8 Project: Modeling Sunset Times | |

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Trigonometric Identities and Equations | |

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Trigonometric Identities | |

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Fundamental Identities | |

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Using the Fundamental Identities | |

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Verifying Identities | |

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Sum and Difference Identities | |

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Cosine Sum and Difference Identities | |

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Sine and Tangent Sum and Difference Identities | |

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Reviewing Basic Concepts (Sections 9.1 and 9.2) | |

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Further Identities | |

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Double-Number Identities | |

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Product-to-Sum and Sum-to-Product Identities | |

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Half-Number Identities | |

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The Inverse Circular Functions | |

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Review of Inverse Functions | |

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Inverse Sine Function | |

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Inverse Cosine Function | |

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Inverse Tangent Function | |

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Remaining Inverse Trigonometric Functions | |

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Inverse Function Values | |

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Reviewing Basic Concepts (Sections 9.3 and 9.4) | |

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Trigonometric Equations and Inequalities (I) | |

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Equations Solvable by Linear Methods | |

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Equations Solvable by Factoring | |

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Equations Solvable by the Quadratic Formula | |

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Using Trigonometric Identities to Solve Equations | |

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Trigonometric Equations and Inequalities (II) | |

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Equations and Inequalities Involving Multiple-Number Identities | |

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Equations and Inequalities Involving Half-Number Identities | |

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An Application of Trigonometric Equations | |

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Reviewing Basic Concepts (Sections 9.5 and 9.6) | |

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Chapter 9 Summary | |

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Chapter 9 Review Exercises | |

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Chapter 9 Test | |

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Chapter 9 Project: Modeling a Damped Pendulum | |

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Applications of Trigonometry; Vectors | |

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The Law of Sines | |

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Congruency and Oblique Triangles | |

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Derivation of the Law of Sines | |

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Applications of Triangles | |

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Ambiguous Case | |

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The Law of Cosines and Area Formulas | |

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Derivation of the Law of Cosines | |

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Applications of Triangles | |

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Area Formulas | |

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Vectors and Their Applications | |

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Basic Terminology | |

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Algebraic Interpretation of Vectors | |

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Operations with Vectors | |

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Dot Product and the Angle between Vectors | |

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Applications of Vectors | |

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Reviewing Basic Concepts (Sections 10.1-10.3) | |

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Trigonometric (Polar) Form of Complex Numbers | |

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The Complex Plane and Vector Representation | |

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Trigonometric (Polar) Form | |

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Products of Complex Numbers in Trigonometric Form | |

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Quotients of Complex Numbers in Trigonometric Form | |

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Powers and Roots of Complex Numbers | |

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Powers of Complex Numbers (De Moivre's Theorem) | |

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Roots of Complex Numbers | |

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Reviewing Basic Concepts (Sections 10.4 and 10.5) | |

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Polar Equations and Graphs | |

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Polar Coordinate System | |

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Graphs of Polar Equations | |

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Classifying Polar Equations | |

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Converting Equations | |

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More Parametric Equations | |

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Parametric Equations with Trigonometric Functions | |

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The Cycloid | |

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Applications of Parametric Equations | |

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Reviewing Basic Concepts (Sections 10.6 and 10.7) | |

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Chapter 10 Summary | |

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Chapter 10 Review Exercises | |

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Chapter 10 Test | |

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Chapter 10 Project: When Is a Circle Really a Polygon? | |

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Further Topics in Algebra | |

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Sequences and Series | |

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Sequences | |

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Series and Summation Notation | |

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Summation Properties | |

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Arithmetic Sequences and Series | |

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Arithmetic Sequences | |

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Arithmetic Series | |

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Geometric Sequences and Series | |

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Geometric Sequences | |

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Geometric Series | |

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Infinite Geometric Series | |

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Annuities | |

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Reviewing Basic Concepts (Sections 11.1-11.3) | |

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The Binomial Theorem | |

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A Binomial Expansion Pattern | |

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Pascal's Triangle | |

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n-Factorial | |

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Binomial Coefficients | |

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The Binomial Theorem | |

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rth Term of a Binomial Expansion | |

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Mathematical Induction | |

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Proof by Mathematical Induction | |

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Proving Statements | |

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Generalized Principle of Mathematical Induction | |

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Proof of the Binomial Theorem | |

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Reviewing Basic Concepts (Sections 11.4 and 11.5) | |

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Counting Theory | |

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Fundamental Principle of Counting | |

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Permutations | |

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Combinations | |

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Distinguishing between Permutations and Combinations | |

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Probability | |

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Basic Concepts | |

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Complements and Venn Diagrams | |

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Odds | |

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Union of Two Events | |

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Binomial Probability | |

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Reviewing Basic Concepts (Sections 11.6 and 11.7) | |

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Chapter 11 Summary | |

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Chapter 11 Review Exercises | |

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Chapter 11 Test | |

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Chapter 11 Project: Using Experimental Probabilities to Simulate Family Makeup | |

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Reference: Basic Algebraic Concepts | |

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Review of Exponents and Polynomials | |

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Rules for Exponents | |

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Terminology for Polynomials | |

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Adding and Subtracting Polynomials | |

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Multiplying Polynomials | |

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Review of Factoring | |

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Factoring Out the Greatest Common Factor | |

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Factoring by Grouping | |

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Factoring Trinomials | |

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Factoring Special Products | |

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Factoring by Substitution | |

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Review of Rational Expressions | |

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Domain of a Rational Expression | |

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Lowest Terms of a Rational Expression | |

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Multipling and Dividing Rational Expressions | |

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Adding and Subtracting Rational Expressions | |

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Complex Fractions | |

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Review of Negative and Rational Exponents | |

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Negative Exponents and the Quotient Rule | |

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Rational Exponents | |

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Review of Radicals | |

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Radical Notation | |

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Rules for Radicals | |

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Simplifying Radicals | |

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Operations with Radicals | |

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Rationalizing Denominators | |

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Chapter R Test | |

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Geometry Formulas | |

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Deciding Which Model Best Fits a Set of Data | |

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Vectors in Space | |

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Polar Form of Conic Sections | |

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Rotation of Axes | |

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Answers to Selected Exercises | |

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Index of Applications | |

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Index | |