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Table of Contents


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1 Real Numbers and Their Properties

  • 1.1 Real Numbers
    • 1.1.1 Ordering and Classifying Real Numbers
    • 1.1.2 Evaluating an Algebraic Expression
    • 1.1.3 Using Properties of Real Numbers
    • 1.1.4 Inequalities and Interval Notation
    • 1.1.5 Interval Notation: Another Example
    • 1.1.6 Set Notation
    • 1.1.7 Evaluating Absolute Value Expressions
  • 1.2 Integer Exponents
    • 1.2.1 An Introduction to Exponents
    • 1.2.2 Product of Powers Property
    • 1.2.3 Power of a Power Property
    • 1.2.4 Power of a Product Property
    • 1.2.5 Quotient Properties and Negative Exponents
    • 1.2.6 Scientific Notation
  • 1.3 Rational Exponents and Radicals
    • 1.3.1 Finding Real Roots
    • 1.3.2 Simplifying Radical Expressions
    • 1.3.3 Adding and Subtracting Radical Expressions
    • 1.3.4 Rationalizing a Denominator
    • 1.3.5 Converting Rational Exponents and Radicals
    • 1.3.6 Simplifying Expressions with Rational Exponents
    • 1.3.7 Simplifying Radical Expressions with Variables
  • 1.4 Polynomials
    • 1.4.1 Introduction to Polynomials
    • 1.4.2 Adding, Subtracting, and Multiplying Polynomials
    • 1.4.3 Multiplying Polynomials by Polynomials
    • 1.4.4 Special Products of Binomials: Squares and Cubes
  • 1.5 Factoring
    • 1.5.1 Factoring Using the Greatest Common Factor
    • 1.5.2 Factoring Polynomials by Grouping
    • 1.5.3 Factoring Trinomials Using Trial and Error
    • 1.5.4 Factoring Trinomials Using the Product-and-Sum Method
    • 1.5.5 Factoring Perfect-Square Trinomials
    • 1.5.6 Factoring the Difference of Two Squares
    • 1.5.7 Factoring Sums and Differences of Cubes
  • 1.6 Rational Expressions
    • 1.6.1 Rational Expressions and Domain
    • 1.6.2 Simplifying Rational Expressions
    • 1.6.3 Factoring -1 from a Rational Expression to Simplify
    • 1.6.4 Multiplying and Dividing Rational Expressions
    • 1.6.5 Adding and Subtracting Rational Expressions
    • 1.6.6 Rewriting Complex Fractions

2 Equations and Inequalities

  • 2.1 Solving Linear Equations
    • 2.1.1 An Introduction to Solving Equations
    • 2.1.2 Solving a Linear Equation
    • 2.1.3 Solving a Linear Equation with Rationals
    • 2.1.4 Solving Literal Equations
  • 2.2 Modeling with Linear Equations
    • 2.2.1 Modeling with Linear Equations
    • 2.2.2 Modeling Averages with Equations
    • 2.2.3 Equations for Consecutive Numbers
    • 2.2.4 Equations with Percents
    • 2.2.5 Modeling Interest with Equations
    • 2.2.6 Modeling a Geometric Situation
    • 2.2.7 Equations Using Proportional Reasoning with Similar Triangles
    • 2.2.8 Equations Using Distance, Rate, and Time
    • 2.2.9 Modeling Mixtures with Equations
    • 2.2.10 Modeling a Work Situation
  • 2.3 Quadratic Equations
    • 2.3.1 Solving Quadratics by Factoring
    • 2.3.2 Solving Quadratic Equations by Using Square Roots
    • 2.3.3 Introducing Complex Numbers
    • 2.3.4 Solving Quadratic Equations with Complex Solutions
    • 2.3.5 Solving Quadratics by Completing the Square
    • 2.3.6 Completing the Square: Another Example
    • 2.3.7 Proving the Quadratic Formula
    • 2.3.8 Using the Quadratic Formula
    • 2.3.9 Predicting the Type of Solutions Using the Discriminant
    • 2.3.10 Modeling with Quadratic Geometric Equations
    • 2.3.11 The Pythagorean Theorem
    • 2.3.12 Modeling Projectiles with Quadratic Equations
    • 2.3.13 Modeling with Other Quadratic Equations
  • 2.4 Other Types of Equations
    • 2.4.1 Solving a Polynomial Equation by Factoring
    • 2.4.2 Using Quadratic Techniques to Solve Literal Equations
    • 2.4.3 Solving an Equation Containing a Radical
    • 2.4.4 Solving an Equation with Two Radicals
    • 2.4.5 Solving an Equation with Rational Exponents
    • 2.4.6 Solving Rational Equations
    • 2.4.7 Modeling with Rational Equations
    • 2.4.8 Solving Equations of Quadratic Type
    • 2.4.9 Solving Absolute Value Equations
    • 2.4.10 Solving Equations with Two Absolute Value Expressions
  • 2.5 Inequalities
    • 2.5.1 An Introduction to Solving Inequalities
    • 2.5.2 Modeling with Inequalities
    • 2.5.3 Writing Compound Inequalities
    • 2.5.4 Solving Compound Inequalities
    • 2.5.5 Solving Absolute Value Inequalities
    • 2.5.6 Solving Absolute Value Inequalities: Another Example
    • 2.5.7 Solving Quadratic Inequalities
    • 2.5.8 Solving Quadratic Inequalities: Another Example
    • 2.5.9 Solving a Polynomial Inequality
    • 2.5.10 Solving Rational Inequalities
    • 2.5.11 Solving Rational Inequalities: Another Example
    • 2.5.12 Determining the Domains of Expressions with Radicals

3 Functions and Their Graphs

  • 3.1 Coordinates and Graphs
    • 3.1.1 Using the Cartesian System
    • 3.1.2 The Distance and Midpoint Formulas
    • 3.1.3 Finding the Second Endpoint of a Segment
    • 3.1.4 Collinearity and Distance
    • 3.1.5 Triangles and Distance
    • 3.1.6 Graphing Equations by Plotting Points
    • 3.1.7 Finding the x- and y-Intercepts of an Equation
    • 3.1.8 Introduction to the Equation of a Circle
    • 3.1.9 Graphing a Circle
    • 3.1.10 Writing the Equation of a Circle
    • 3.1.11 Writing the Equation of a Circle: Another Example
    • 3.1.12 Testing for Symmetry
    • 3.1.13 Using Symmetry as a Sketching Aid
  • 3.2 Slope and the Equation of a Line
    • 3.2.1 Finding the Slope of a Line from Two Points
    • 3.2.2 Graphing a Line Using a Point and the Slope
    • 3.2.3 Introduction to Slope-Intercept Form
    • 3.2.4 Writing the Equation of a Line
    • 3.2.5 Introduction to Point-Slope Form
    • 3.2.6 Vertical and Horizontal Lines
    • 3.2.7 Graphing a Standard Form Linear Equation
    • 3.2.8 Equations of Parallel and Perpendicular Lines
    • 3.2.9 Modeling Rate of Change with Linear Equations
  • 3.3 Variation
    • 3.3.1 Direct Variation
    • 3.3.2 Modeling with Direct Variation
    • 3.3.3 Inverse Variation
    • 3.3.4 Modeling with Inverse Variation
    • 3.3.5 Joint Variation
    • 3.3.6 Modeling with Combined Variation
  • 3.4 Functions
    • 3.4.1 Functions and the Vertical Line Test
    • 3.4.2 Identifying Functions Algebraically
    • 3.4.3 Function Notation and Finding Function Values
    • 3.4.4 Evaluating Piecewise-Defined Functions
    • 3.4.5 Finding Specific Function Values
    • 3.4.6 Modeling with Functions
    • 3.4.7 Satisfying the Domain of a Function
  • 3.5 Graphs of Functions
    • 3.5.1 Finding the Domain and Range
    • 3.5.2 Graphing Some Important Functions
    • 3.5.3 Graphing Piecewise-Defined Functions
    • 3.5.4 Using a Table to Graph Piecewise-Defined Functions
    • 3.5.5 Modeling with Piecewise-Defined Functions
    • 3.5.6 The Greatest Integer Function
    • 3.5.7 Graphing the Greatest Integer Function
    • 3.5.8 Finding Zeros of a Function
    • 3.5.9 Determining Intervals Over Which a Function Is Increasing
    • 3.5.10 Relative Minimums and Maximums
  • 3.6 Transformations of Functions
    • 3.6.1 Translating Functions
    • 3.6.2 Reflecting Functions
    • 3.6.3 Stretching Functions
    • 3.6.4 Using Patterns to Graph Functions
    • 3.6.5 Even and Odd Functions
  • 3.7 Combining Functions
    • 3.7.1 Using Operations with Functions
    • 3.7.2 The Difference Quotient
    • 3.7.3 Composition of Functions
    • 3.7.4 Composition of Functions: Another Example
    • 3.7.5 Finding Functions That Form a Given Composite
    • 3.7.6 Modeling with Composite Functions
  • 3.8 Inverse Functions
    • 3.8.1 Introduction to Inverse Functions
    • 3.8.2 The Horizontal Line Test
    • 3.8.3 Verifying That Functions Are Inverses
    • 3.8.4 Finding the Inverse of a Function Graphically
    • 3.8.5 Finding the Inverse of a Function Algebraically

4 Polynomial Functions

  • 4.1 Quadratic Functions and Models
    • 4.1.1 Reflecting, Stretching, and Compressing Quadratic Functions
    • 4.1.2 Identifying the Vertex and Axis of Symmetry
    • 4.1.3 Finding the Vertex by Completing the Square
    • 4.1.4 Translations of Quadratic Functions
    • 4.1.5 Relating the Discriminant to the Graph of a Quadratic Function
    • 4.1.6 Graphing Quadratic Functions
    • 4.1.7 Writing the Equation of a Quadratic Function
    • 4.1.8 Finding the Maximum or Minimum of a Quadratic Function
    • 4.1.9 Modeling with Quadratic Functions
  • 4.2 Polynomial Functions and Their Graphs
    • 4.2.1 End Behavior of Graphs of Polynomial Functions
    • 4.2.2 Reflecting, Stretching, and Translating Polynomial Functions
    • 4.2.3 Finding Zeros and Their Multiplicities for a Polynomial
    • 4.2.4 Graphing Polynomial Functions
    • 4.2.5 Intermediate Value Theorem and Local Extrema
  • 4.3 Dividing Polynomials
    • 4.3.1 Using Long Division with Polynomials
    • 4.3.2 Using Synthetic Division with Polynomials
    • 4.3.3 The Remainder Theorem
    • 4.3.4 The Factor Theorem
  • 4.4 Real Zeros of Polynomials
    • 4.4.1 Factoring a Polynomial Given a Zero
    • 4.4.2 Using Zeros to Write a Polynomial Function
    • 4.4.3 Using Zeros, Degree, and a Point to Write a Polynomial Function
    • 4.4.4 The Rational Zero Theorem
    • 4.4.5 Finding the Zeros of a Polynomial from Start to Finish
    • 4.4.6 Using Descartes' Rule of Signs
    • 4.4.7 Upper and Lower Bounds
  • 4.5 Complex Zeros and the Fundamental Theorem of Algebra
    • 4.5.1 Rewriting Powers of i
    • 4.5.2 Adding and Subtracting Complex Numbers
    • 4.5.3 Multiplying Complex Numbers
    • 4.5.4 Dividing Complex Numbers
    • 4.5.5 The Fundamental Theorem of Algebra
    • 4.5.6 Finding All Solutions of a Polynomial Equation
    • 4.5.7 Finding All Solutions of a Polynomial Equation: Another Example
    • 4.5.8 The Conjugate Pair Theorem

5 Rational Functions and Conics

  • 5.1 Graphing Rational Functions
    • 5.1.1 Graphing Basic Rational Functions
    • 5.1.2 Finding the Vertical Asymptotes of a Rational Function
    • 5.1.3 Graphing Rational Functions with Vertical Asymptotes
    • 5.1.4 Graphing Rational Functions with Vertical and Horizontal Asymptotes
    • 5.1.5 Oblique Asymptotes
    • 5.1.6 Oblique Asymptotes: Another Example
  • 5.2 Parabolas
    • 5.2.1 Introduction to Conic Sections
    • 5.2.2 Graphing Parabolas
    • 5.2.3 Writing the Equation of a Parabola
  • 5.3 Ellipses
    • 5.3.1 Writing the Equation of an Ellipse
    • 5.3.2 Graphing Ellipses
    • 5.3.3 The Eccentricity of an Ellipse
  • 5.4 Hyperbolas
    • 5.4.1 Writing the Equation of a Hyperbola
    • 5.4.2 Writing the Equation of a Hyperbola: Another Example
    • 5.4.3 Graphing Hyperbolas
    • 5.4.4 Applying Hyperbolas: Navigation
  • 5.5 Translations of Conics
    • 5.5.1 Translations of Parabolas
    • 5.5.2 Translations of Ellipses
    • 5.5.3 Translations of Hyperbolas
    • 5.5.4 Identifying a Conic
    • 5.5.5 Using the Discriminant and Coefficients to Identify a Conic

6 Exponential and Logarithmic Functions

  • 6.1 Exponential Functions
    • 6.1.1 An Introduction to Exponential Functions
    • 6.1.2 An Introduction to Graphing Exponential Functions
    • 6.1.3 Transformations of Exponential Functions
    • 6.1.4 Graphing Exponential Functions: Another Example
    • 6.1.5 Finding Present Value and Future Value
    • 6.1.6 Finding an Interest Rate to Match Given Goals
    • 6.1.7 Evaluating and Graphing a Natural Exponential Function
    • 6.1.8 Applying Natural Exponential Functions
  • 6.2 Logarithmic Functions
    • 6.2.1 An Introduction to Logarithmic Functions
    • 6.2.2 Converting between Exponential and Logarithmic Functions
    • 6.2.3 Evaluating Logarithms
    • 6.2.4 Using Properties to Evaluate Logarithms
    • 6.2.5 Graphing Logarithmic Functions
    • 6.2.6 Matching Logarithmic Functions with Their Graphs
    • 6.2.7 Common Logs and Natural Logs
    • 6.2.8 Evaluating Common Logs and Natural Logs Using a Calculator
    • 6.2.9 Evaluating Logarithmic Models
    • 6.2.10 Domain of a Natural Log Function
  • 6.3 Properties of Logarithms
    • 6.3.1 Properties of Logarithms
    • 6.3.2 Expanding Logarithmic Expressions
    • 6.3.3 Combining Logarithmic Expressions
    • 6.3.4 Using the Change of Base Formula
  • 6.4 Exponential and Logarithmic Equations
    • 6.4.1 Using the One-to-One Property to Solve Exponential Equations
    • 6.4.2 Solving Exponential Equations Using Logs
    • 6.4.3 Solving Natural Exponential Equations
    • 6.4.4 Solving Exponential Equations of Quadratic Type
    • 6.4.5 Using Exponential Form to Solve Logarithmic Equations
    • 6.4.6 The Distance Modulus Formula
    • 6.4.7 Solving Logarithmic Equations
    • 6.4.8 Compound Interest
  • 6.5 Exponential and Logarithmic Models
    • 6.5.1 Predicting Change with Logarithmic Models
    • 6.5.2 Exponential Growth and Decay
    • 6.5.3 Half-Life
    • 6.5.4 Newton's Law of Cooling
    • 6.5.5 Continuously Compounded Interest

7 Systems of Equations and Inequalities

  • 7.1 Solving Systems of Two Linear Equations in Two Variables
    • 7.1.1 An Introduction to Linear Systems
    • 7.1.2 Solving Systems by Graphing
    • 7.1.3 The Substitution Method
    • 7.1.4 The Elimination Method: Adding
    • 7.1.5 The Elimination Method: Subtracting
    • 7.1.6 Solving Systems by Elimination
    • 7.1.7 Three Cases for Linear Systems
  • 7.2 Nonlinear Systems
    • 7.2.1 Solving Nonlinear Systems by Graphing
    • 7.2.2 Solving Nonlinear Systems with Substitution
    • 7.2.3 Solving Nonlinear Systems with Substitution: Another Example
    • 7.2.4 Solving Nonlinear Systems with Elimination
    • 7.2.5 Solving Nonlinear Systems with Elimination: Another Example
  • 7.3 Modeling with Systems
    • 7.3.1 Applying Linear Systems: Investments
    • 7.3.2 Applying Linear Systems: Distance, Rate, and Time
    • 7.3.3 Applying Linear Systems: Mixtures
    • 7.3.4 Applying Nonlinear Systems: Physics
    • 7.3.5 Applying Nonlinear Systems: Paths of Objects
  • 7.4 Multivariable Linear Systems
    • 7.4.1 An Introduction to Linear Systems in Three Variables
    • 7.4.2 Solving a Triangular System Using Back-Substitution
    • 7.4.3 Using Gaussian Elimination to Solve a System
    • 7.4.4 Gaussian Elimination: Special Cases
    • 7.4.5 Nonsquare Systems
    • 7.4.6 Modeling with Multivariable Linear Systems
  • 7.5 Partial Fractions
    • 7.5.1 Partial Fraction Decomposition
    • 7.5.2 Repeated Linear Factors
    • 7.5.3 Distinct Linear and Quadratic Factors
    • 7.5.4 Repeated Quadratic Factors
  • 7.6 Systems of Inequalities and Linear Programming
    • 7.6.1 An Introduction to Graphing Linear Inequalities
    • 7.6.2 Graphing Linear and Nonlinear Inequalities
    • 7.6.3 Graphing the Solution Set of a System of Inequalities
    • 7.6.4 Solving for Maxima-Minima
    • 7.6.5 Applying Linear Programming

8 Matrices and Determinants

  • 8.1 Matrices and Systems of Equations
    • 8.1.1 An Introduction to Matrices
    • 8.1.2 Augmented Matrices
    • 8.1.3 Elementary Row Operations
    • 8.1.4 Gauss-Jordan Elimination
    • 8.1.5 Gaussian Elimination
    • 8.1.6 Inconsistent and Dependent Systems
  • 8.2 Operations with Matrices
    • 8.2.1 Equality of Matrices
    • 8.2.2 The Arithmetic of Matrices
    • 8.2.3 Multiplying Matrices by a Scalar
    • 8.2.4 Solving a Matrix Equation
    • 8.2.5 Multiplying Matrices
  • 8.3 Determinants and Cramer's Rule
    • 8.3.1 Evaluating 2 x 2 Determinants
    • 8.3.2 Finding a Determinant Using Expanding by Cofactors
    • 8.3.3 Evaluating a Determinant Using Elementary Row Operations
    • 8.3.4 Applying Determinants
    • 8.3.5 Using Cramer's Rule
    • 8.3.6 Using Cramer's Rule in a 3 x 3 Matrix
  • 8.4 Inverses of Matrices
    • 8.4.1 Finding the Inverse of a 2 x 2 Matrix
    • 8.4.2 Finding the Inverse of a 2 x 2 Matrix: Another Example
    • 8.4.3 Finding the Inverse of an n x n Matrix
    • 8.4.4 Finding the Inverse of an n x n Matrix Using Row Operations
    • 8.4.5 Solving a System of Equations with Inverses

9 Sequences, Series, and Probability

  • 9.1 Sequences and Series
    • 9.1.1 Introduction to Sequences
    • 9.1.2 Finding the nth Term of a Sequence
    • 9.1.3 Recursive Sequences
    • 9.1.4 Summation Notation and Finite Series
  • 9.2 Arithmetic Sequences
    • 9.2.1 Introduction to Arithmetic Sequences
    • 9.2.2 Finding Terms of an Arithmetic Sequence
    • 9.2.3 Using Two Terms to Find an Arithmetic Sequence
    • 9.2.4 Finding the Sum of an Arithmetic Sequence
  • 9.3 Geometric Sequences
    • 9.3.1 Introduction to Geometric Sequences
    • 9.3.2 Finding the Sum of a Geometric Sequence
    • 9.3.3 Finding the Sum of an Infinite Geometric Sequence
    • 9.3.4 Writing a Repeated Decimal as a Fraction
  • 9.4 Mathematical Induction
    • 9.4.1 Introduction to Proof by Induction
    • 9.4.2 Proving with Induction
    • 9.4.3 Proving with Induction: Another Example
  • 9.5 Counting Principles
    • 9.5.1 Using the Fundamental Counting Principle
    • 9.5.2 Permutations
    • 9.5.3 Distinguishable Permutations
    • 9.5.4 Combinations
  • 9.6 Probability
    • 9.6.1 The Probability of an Event
    • 9.6.2 The Probability of an Event: Another Example
    • 9.6.3 Calculating Probability by Counting
    • 9.6.4 Independent Events
    • 9.6.5 Inclusive Events
    • 9.6.6 Inclusive Events: Another Example
    • 9.6.7 Mutually Exclusive Events
    • 9.6.8 Using the Complement
  • 9.7 The Binomial Theorem
    • 9.7.1 Using the Binomial Theorem
    • 9.7.2 Binomial Coefficients
    • 9.7.3 Finding a Term of a Binomial Expansion