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Real Numbers: Algebra Essentials

Use the Fundamental Theorem of Algebra

Introduction to Real Numbers: Algebra Essentials

Use the Linear Factorization Theorem to find polynomials with given zeros

Classifying a Real Number

Use Descartes’ Rule of Signs

Performing Calculations Using the Order of Operations

Solve real-world applications of polynomial equations

Using Properties of Real Numbers

Key Concepts & Glossary

Evaluating Algebraic Expressions

Section Exercises

Formulas

Solutions

Simplifying Algebraic Expressions

Rational Functions

Key Concepts & Glossary

Introduction to Rational Functions

Section Exercises

Use arrow notation

Solutions

Solve applied problems involving rational functions

Factoring Polynomials

Find the domains of rational functions

Introduction to Factoring Polynomials

Graphs of Exponential Functions

Factoring the Greatest Common Factor of a Polynomial

Introduction to Graphs of Exponential Functions

Factoring a Trinomial with Leading Coefficient 1

Graph exponential functions

Factoring by Grouping

Graph exponential functions using transformations

Special Products

Key Concepts

Factoring Expressions with Fractional or Negative Exponents

Section Exercises

Using Factoring to Solve Polynomial Equations

Solutions

Key Concepts & Glossary

Logarithmic Functions

Section Exercises

Introduction to Logarithmic Functions

Solutions

Convert from logarithmic to exponential form

The Rectangular Coordinate Systems and Graphs

Convert from exponential to logarithmic form

Introduction to The Rectangular Coordinate Systems and Graphs

Evaluate logarithms

Plotting Ordered Pairs in the Cartesian Coordinate System

Use common logarithms

Graphing Equations by Plotting Points

Use natural logarithms

Graphing Equations with a Graphing Utility

Key Concepts & Glossary

Finding x-intercepts and y-intercepts

Section Exercises

Using the Distance Formula

Solutions

Using the Midpoint Formula

Logarithmic Properties

Circles

Introduction to Logarithmic Properties

Key Concepts & Glossary

Use the product rule for logarithms

Section Exercises

Use the quotient and power rules for logarithms

Solutions

Expand logarithmic expressions

Linear Equations in One Variable

Condense logarithmic expressions

Introduction to Linear Equations in One Variable

Use the change-of-base formula for logarithms

Solving Linear Equations in One Variable

Key Terms & Glossary

Solving a Rational Equation

Section Exercises

Finding a Linear Equation

Solutions

Determining Whether Graphs of Lines are Parallel or Perpendicular

Exponential and Logarithmic Equations

Key Concepts & Glossary

Introduction to Exponential and Logarithmic Equations

Section Exercises

Use like bases to solve exponential equations

Solutions

Use logarithms to solve exponential equations

Solving Quadratic Equations

Use the definition of a logarithm to solve logarithmic equations

Introduction: Quadratic Equations

Use the one-to-one property of logarithms to solve logarithmic equations

Solving Quadratic Equations by Factoring

Solve applied problems involving exponential and logarithmic equations

Complex Numbers

Key Concepts & Glossary

Using the Square Root Property

Section Exercises

Completing the Square

Solutions

Using the Quadratic Formula

Practice Tests

Key Concepts & Glossary

Functions Practice Test

Section Exercises

Solutions: Functions Practice Test

Solutions

Linear Functions Practice Test

Other Types of Equations

Solutions: Linear Functions Practice Test

Introduction: Other Types of Equations

Polynomial and Rational Functions Practice Test

Solving Equations Involving Rational Exponents

Solutions: Polynomial and Rational Functions Practice Test

Solving Equations Using Factoring

Exponential and Logarithmic Functions Practice Test

Solving Radical Equations

Solutions: Exponential and Logarithmic Functions Practice Test

Solving an Absolute Value Equation

Systems of Linear Equations: Two Variables

Solving Other Types of Equations

Introduction to Systems of Linear Equations: Two Variables

Key Concepts & Glossary

Solving Systems of Equations by Graphing

Section Exercises

Solving Systems of Equations by Substitution

Solutions to Selected Exercises

Solving Systems of Equations in Two Variables by the Addition Method

Functions and Function Notation

Identifying and Expressing Solutions to Systems of Equations

Introduction to Functions and Function Notation

Using Systems of Equations to Investigate Profits

Determine whether a relation represents a function

Key Concepts & Glossary

Find the input and output values of a function

Section Exercises

Determine whether a function is one-to-one

Solutions

Use the vertical line test to identify functions

Systems of Linear Equations: Three Variables

Graph the functions listed in the library of functions

Introduction to Systems of Linear Equations: Three Variables

Section Exercises

Solving Systems of Three Equations in Three Variables

Solutions

Inconsistent and Dependent Systems in Three Variables

Domain and Range

Key Concepts & Glossary

Introduction to Domain and Range

Section Exercises

Find domain and range from graphs

Solutions

Find domains and ranges of the toolkit functions

Systems of Nonlinear Equations and Inequalities: Two Variables

Graph piecewise-defined functions

Introduction to Systems of Nonlinear Equations and Inequalities: Two Variables

Key Concepts & Glossary

Solving a System of Nonlinear Equations Using Substitution

Section Exercises

Solving a System of Nonlinear Equations Using Elimination

Solutions

Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities

Rates of Change and Behavior of Graphs

Key Concepts & Glossary

Introduction to Rates of Change and Behaviors of Graphs

Section Exercises

Find the average rate of change of a function

Solutions

Use a graph to determine where a function is increasing, decreasing, or constant

Partial Fractions

Use a graph to locate the absolute maximum and absolute minimum

Introduction to Partial Fractions

Key Concepts & Glossary

Decomposing P(x) / Q(x), Where Q(x) Has Only Nonrepeated Linear Factors

Section Exercises

Decomposing P(x)/ Q(x), Where Q(x) Has Repeated Linear Factors

Solutions

Decomposing P(x) / Q(x), Where Q(x) Has a Nonrepeated Irreducible Quadratic Factor

Composition of Functions

Decomposing P(x) / Q(x), When Q(x) Has a Repeated Irreducible Quadratic Factor

Combine functions using algebraic operations

Key Concepts & Glossary

Create a new function by composition of functions

Section Exercises

Evaluate composite functions

Solutions

Find the domain of a composite function

Matrices and Matrix Operations

Decompose a composite function into its component functions

Introduction: Matrices and Matrix Operations

Key Concepts & Glossary

Finding the Sum and Difference of Two Matrices

Section Exercises

Finding Scalar Multiples of a Matrix

Solutions

Finding the Product of Two Matrices

Transformation of Functions

Key Concepts & Glossary

Introduction to Transformation of Functions

Section Exercises

Graph functions using vertical and horizontal shifts

Solutions

Graph functions using reflections about the x-axis and the y-axis

Solving Systems with Gaussian Elimination

Determine whether a function is even, odd, or neither from its graph

Introduction to Solving Systems with Gaussian Elimination

Graph functions using compressions and stretches

The Augmented Matrix of a System of Equations

Combine vertical and horizontal shifts

Performing Row Operations on a Matrix

Key Concepts & Glossary

Solving a System of Linear Equations Using Matrices

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Absolute Value Functions

Solutions

Introduction to Absolute Value Functions

Solving Systems with Inverses

Graph an absolute value function

Introduction to Solving Systems with Inverses

Solve an absolute value equation

Finding the Inverse of a Matrix

Solve an absolute value inequality

Solving a System of Linear Equations Using the Inverse of a Matrix

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Inverse Functions

Solving Systems with Cramer's Rule

Introduction to Inverse Functions

Introduction to Solving Systems with Cramer's Rule

Verify inverse functions

Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

Determine the domain and range of an inverse function

Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Find or evaluate the inverse of a function

Understanding Properties of Determinants

Use the graph of a function to graph its inverse

Key Concepts & Glossary

Key Concepts & Glossary

Solutions

Section Exercises

Section Exercises

Solutions

The Ellipse

Linear Functions

Introduction to The Ellipse

Represent a linear function

Writing Equations of Ellipses in Standard Form

Introduction to Linear Functions

Deriving the Equation of an Ellipse Centered at the Origin

Determine whether a linear function is increasing, decreasing, or constant

Writing Equations of Ellipses Not Centered at the Origin

Calculate and interpret slope

Graphing Ellipses

Write the point-slope form of an equation

Solving Applied Problems Involving Ellipses

Write and interpret a linear function

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

The Hyperbola

Graphs of Linear Functions

Introduction to the Hyperbola

Introduction to Graphs of Linear Functions

Locating the Vertices and Foci of a Hyperbola

Graph linear functions

Deriving the Equation of a Hyperbola Centered at the Origin

Write the equation for a linear function from the graph of a line

Writing Equations of Hyperbolas in Standard Form

Given the equations of two lines, determine whether their graphs are parallel or perpendicular

Graphing Hyperbolas

Write the equation of a line parallel or perpendicular to a given line

Solving Applied Problems Involving Hyperbolas

Solve a system of linear equations

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

The Parabola

Solutions

Introduction to The Parabola

Modeling with Linear Functions

Graphing Parabolas with Vertices at the Origin

Introduction to Modeling With Linear Functions

Writing Equations of Parabolas in Standard Form

Identify steps for modeling and solving

Graphing Parabolas with Vertices Not at the Origin

Build linear models

Solving Applied Problems Involving Parabolas

Build systems of linear models

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Fitting Linear Models to Data

Rotation of Axes

Introduction to Fitting Linear Models to Data

Introduction to Rotation of Axes

Draw and interpret scatter plots

Identifying Nondegenerate Conics in General Form

Find the line of best fit

Finding a New Representation of the Given Equation after Rotating through a Given Angle

Distinguish between linear and nonlinear relations

Writing Equations of Rotated Conics in Standard Form

Use a linear model to make predictions

Identifying Conics without Rotating Axes

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Quadratic Functions

Arithmetic Sequences

Introduction to Quadratic Functions

Introduction to Arithmetic Sequences

Recognize characteristics of parabolas

Finding Common Differences

Understand how the graph of a parabola is related to its quadratic function

Using Formulas for Arithmetic Sequences

Determine a quadratic function’s minimum or maximum value

Finding the Number of Terms in a Finite Arithmetic Sequence

Solve problems involving a quadratic function’s minimum or maximum value

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Geometric Sequences

Power Functions and Polynomial Functions

Introduction to Geometric Sequences

Introduction to Power Functions and Polynomial Functions

Finding Common Ratios

Identify power functions

Writing Terms of Geometric Sequences

Identify end behavior of power functions

Solving Application Problems with Geometric Sequences

Identify polynomial functions

Key Concepts & Glossary

Identify the degree and leading coefficient of polynomial functions

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Series and Their Notations

Solutions

Introduction to Series and Their Notations

Graphs of Polynomial Functions

Using Summation Notation

Introduction to Graphs of Polynomial Functions

Using the Formula for Arithmetic Series

Recognize characteristics of graphs of polynomial functions

Using the Formula for Geometric Series

Use factoring to ﬁnd zeros of polynomial functions

Finding Sums of Infinite Series

Identify zeros and their multiplicities

Solving Annuity Problems

Determine end behavior

Key Concepts & Glossary

Understand the relationship between degree and turning points

Section Exercises

Graph polynomial functions

Solutions

Solving Polynomial Inequalities

Binomial Theorem

Use the Intermediate Value Theorem

Introduction to Binomial Theorem

Key Concepts & Glossary

Identifying Binomial Coefficients

Section Exercises

Using the Binomial Theorem

Solutions

Using the Binomial Theorem to Find a Single Term

Zeros of Polynomial Functions

Key Concepts & Glossary

Introduction to Zeros of Polynomials

Section Exercises

Evaluate a polynomial using the Remainder Theorem

Solutions

Use the Factor Theorem to solve a polynomial equation

Appendix

Use the Rational Zero Theorem to find rational zeros

Appendix

Find zeros of a polynomial function