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Real Numbers: Algebra Essentials

Solve real-world applications of polynomial equations

Introduction to Real Numbers: Algebra Essentials

Key Concepts & Glossary

Classifying a Real Number

Section Exercises

Performing Calculations Using the Order of Operations

Solutions

Using Properties of Real Numbers

Rational Functions

Evaluating Algebraic Expressions

Introduction to Rational Functions

Formulas

Use arrow notation

Simplifying Algebraic Expressions

Solve applied problems involving rational functions

Key Concepts & Glossary

Find the domains of rational functions

Section Exercises

Identify vertical and horizontal asymptotes

Solutions

Identify horizontal asymptotes

Exponents and Scientific Notation

Graph rational functions

Introduction to Exponents and Scientific Notation

Key Concepts & Glossary

Using the Product Rule of Exponents

Section Exercises

Using the Quotient Rule of Exponents

Solutions

Using the Power Rule of Exponents

Inverses and Radical Functions

Using the Zero Exponent Rule of Exponents

Introduction to Inverses and Radical Functions

Using the Negative Rule of Exponents

Find the inverse of a polynomial function

Finding the Power of a Product and a Quotient

Restrict the domain to find the inverse of a polynomial function

Simplifying Exponential Expressions

Key Concepts & Glossary

Using Scientific Notation

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Modeling Using Variation

Solutions

Introduction to Modeling Using Variation

Radicals and Rational Expressions

Solve direct variation problems

Introduction to Radicals and Rational Expressions

Solve inverse variation problems

Evaluating Square Roots

Solve problems involving joint variation

Simplifying Square Roots

Key Concepts & Glossary

Adding and Subtracting Square Roots

Section Exercises

Rationalizing Denominators

Solutions

Understanding nth Roots

Exponential Functions

Using Rational Exponents

Introduction to Exponential Functions

Key Concepts & Glossary

Evaluate exponential functions

Section Exercises

Find the equation of an exponential function

Solutions

Use compound interest formulas

Polynomials

Evaluate exponential functions with base e

Introduction to Polynomials

Key Concepts & Glossary

Identifying the Degree and Leading Coefficient of Polynomials

Section Exercises

Adding and Subtracting Polynomials

Solutions

Multiplying Polynomials

Graphs of Exponential Functions

Special Products of Polynomials

Introduction to Graphs of Exponential Functions

Performing Operations with Polynomials of Several Variables

Graph exponential functions

Key Concepts & Glossary

Graph exponential functions using transformations

Section Exercises

Key Concepts

Solutions

Section Exercises

Factoring Polynomials

Solutions

Introduction to Factoring Polynomials

Logarithmic Functions

Factoring the Greatest Common Factor of a Polynomial

Introduction to Logarithmic Functions

Factoring a Trinomial with Leading Coefficient 1

Convert from logarithmic to exponential form

Factoring by Grouping

Convert from exponential to logarithmic form

Special Products

Evaluate logarithms

Factoring Expressions with Fractional or Negative Exponents

Use common logarithms

Key Concepts & Glossary

Use natural logarithms

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Rational Expressions

Solutions

Introduction to Rational Expressions

Graphs of Logarithmic Functions

Simplifying Rational Expressions

Introduction to Graphs of Logarithmic Functions

Multiplying and Dividing Rational Expressions

Identify the domain of a logarithmic function

Adding and Subtracting Rational Expressions

Graph logarithmic functions

Simplifying Complex Rational Expressions

Graphing Transformations of Logarithmic Functions

Key Concepts & Glossary

Key Concepts

Section Exercises

Section Exercises

Solutions

Solutions

The Rectangular Coordinate Systems and Graphs

Logarithmic Properties

Introduction to The Rectangular Coordinate Systems and Graphs

Introduction to Logarithmic Properties

Plotting Ordered Pairs in the Cartesian Coordinate System

Use the product rule for logarithms

Graphing Equations by Plotting Points

Use the quotient and power rules for logarithms

Graphing Equations with a Graphing Utility

Expand logarithmic expressions

Finding x-intercepts and y-intercepts

Condense logarithmic expressions

Using the Distance Formula

Use the change-of-base formula for logarithms

Using the Midpoint Formula

Key Terms & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Exponential and Logarithmic Equations

Linear Equations in One Variable

Introduction to Exponential and Logarithmic Equations

Introduction to Linear Equations in One Variable

Use like bases to solve exponential equations

Solving Linear Equations in One Variable

Use logarithms to solve exponential equations

Solving a Rational Equation

Use the definition of a logarithm to solve logarithmic equations

Finding a Linear Equation

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

Determining Whether Graphs of Lines are Parallel or Perpendicular

Solve applied problems involving exponential and logarithmic equations

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Models and Applications

Fitting Exponential Models to Data

Introduction: Models and Applicaitons

Introduction to Fitting Exponential Models to Data

Setting up a Linear Equation to Solve a Real-World Application

Build an exponential model from data

Using a Formula to Solve a Real-World Application

Build a logarithmic model from data

Key Concepts & Glossary

Build a logistic model from data

Section Exercises

Key Concepts

Solutions to Selected Exercises

Section Exercises

Complex Numbers

Solutions

Introduction to Complex Numbers

Exponential and Logarithmic Models

Express square roots of negative numbers as multiples of i

Introduction to Exponential and Logarithmic Models

Plot complex numbers on the complex plane

Model exponential growth and decay

Add and subtract complex numbers

Use Newton’s Law of Cooling

Multiply and divide complex numbers

Use logistic-growth models

Key Concepts & Glossary

Choose an appropriate model for data

Section Exercises

Key Concepts & Glossary

Quadratic Equations

Section Exercises

Introduction: Quadratic Equations

Solutions

Solving Quadratic Equations by Factoring

Systems of Linear Equations: Two Variables

Using the Square Root Property

Introduction to Systems of Linear Equations: Two Variables

Completing the Square

Solving Systems of Equations by Graphing

Using the Quadratic Formula

Solving Systems of Equations by Substitution

Key Concepts & Glossary

Solving Systems of Equations in Two Variables by the Addition Method

Section Exercises

Identifying and Expressing Solutions to Systems of Equations

Solutions

Using Systems of Equations to Investigate Profits

Other Types of Equations

Key Concepts & Glossary

Introduction: Other Types of Equations

Section Exercises

Solving Equations Involving Rational Exponents

Solutions

Solving Equations Using Factoring

Systems of Linear Equations: Three Variables

Solving Radical Equations

Introduction to Systems of Linear Equations: Three Variables

Solving an Absolute Value Equation

Solving Systems of Three Equations in Three Variables

Solving Other Types of Equations

Inconsistent and Dependent Systems in Three Variables

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Linear Inequalities and Absolute Value Inequalities

Systems of Nonlinear Equations and Inequalities: Two Variables

Introduction: Linear Inequalities and Absolute Value Inequalities

Introduction to Systems of Nonlinear Equations and Inequalities: Two Variables

Using Interval Notation

Solving a System of Nonlinear Equations Using Substitution

Using the Properties of Inequalities

Solving a System of Nonlinear Equations Using Elimination

Understanding Compound Inequalities

Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities

Solving Absolute Value Inequalities

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Partial Fractions

Functions and Function Notation

Introduction to Partial Fractions

Introduction to Functions and Function Notation

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

Determine whether a relation represents a function

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

Find the input and output values of a function

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

Determine whether a function is one-to-one

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

Use the vertical line test to identify functions

Key Concepts & Glossary

Graph the functions listed in the library of functions

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Matrices and Matrix Operations

Solutions

Introduction: Matrices and Matrix Operations

Domain and Range

Finding the Sum and Difference of Two Matrices

Introduction to Domain and Range

Finding Scalar Multiples of a Matrix

Find the domain of a function defined by an equation

Finding the Product of Two Matrices

Use notations to specify domain and range

Key Concepts & Glossary

Find domain and range from graphs

Section Exercises

Find domains and ranges of the toolkit functions

Solutions

Graph piecewise-defined functions

Solving Systems with Gaussian Elimination

Key Concepts & Glossary

Introduction to Solving Systems with Gaussian Elimination

Section Exercises

The Augmented Matrix of a System of Equations

Solutions

Performing Row Operations on a Matrix

Rates of Change and Behavior of Graphs

Solving a System of Linear Equations Using Matrices

Introduction to Rates of Change and Behaviors of Graphs

Key Concepts & Glossary

Find the average rate of change of a function

Section Exercises

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

Solutions

Use a graph to locate the absolute maximum and absolute minimum

Solving Systems with Inverses

Key Concepts & Glossary

Introduction to Solving Systems with Inverses

Section Exercises

Finding the Inverse of a Matrix

Solutions

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

Composition of Functions

Key Concepts & Glossary

Introduction to Composition of Functions

Section Exercises

Combine functions using algebraic operations

Solutions

Create a new function by composition of functions

Solving Systems with Cramer's Rule

Evaluate composite functions

Introduction to Solving Systems with Cramer's Rule

Find the domain of a composite function

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

Decompose a composite function into its component functions

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

Key Concepts & Glossary

Understanding Properties of Determinants

Section Exercises

Key Concepts & Glossary

Solutions

Solutions

Transformation of Functions

Section Exercises

Introduction to Transformation of Functions

The Ellipse

Graph functions using vertical and horizontal shifts

Introduction to The Ellipse

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

Writing Equations of Ellipses in Standard Form

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

Deriving the Equation of an Ellipse Centered at the Origin

Graph functions using compressions and stretches

Writing Equations of Ellipses Not Centered at the Origin

Combine vertical and horizontal shifts

Graphing Ellipses

Key Concepts & Glossary

Solving Applied Problems Involving Ellipses

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Absolute Value Functions

Solutions

Introduction to Absolute Value Functions

The Hyperbola

Graph an absolute value function

Introduction to the Hyperbola

Solve an absolute value equation

Locating the Vertices and Foci of a Hyperbola

Solve an absolute value inequality

Deriving the Equation of a Hyperbola Centered at the Origin

Key Concepts & Glossary

Writing Equations of Hyperbolas in Standard Form

Section Exercises

Graphing Hyperbolas

Solutions

Solving Applied Problems Involving Hyperbolas

Inverse Functions

Key Concepts & Glossary

Introduction to Inverse Functions

Section Exercises

Verify inverse functions

Solutions

Determine the domain and range of an inverse function

The Parabola

Find or evaluate the inverse of a function

Introduction to The Parabola

Use the graph of a function to graph its inverse

Graphing Parabolas with Vertices at the Origin

Key Concepts & Glossary

Writing Equations of Parabolas in Standard Form

Section Exercises

Graphing Parabolas with Vertices Not at the Origin

Solutions

Solving Applied Problems Involving Parabolas

Linear Functions

Key Concepts & Glossary

Introduction to Linear Functions

Section Exercises

Represent a linear function

Solutions

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

Rotation of Axes

Calculate and interpret slope

Introduction to Rotation of Axes

Write the point-slope form of an equation

Identifying Nondegenerate Conics in General Form

Write and interpret a linear function

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

Key Concepts & Glossary

Writing Equations of Rotated Conics in Standard Form

Section Exercises

Identifying Conics without Rotating Axes

Solutions

Key Concepts & Glossary

Graphs of Linear Functions

Section Exercises

Introduction to Graphs of Linear Functions

Solutions

Graph linear functions

Conic Sections in Polar Coordinates

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

Introduction to Conic Sections in Polar Coordinates

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

Identifying a Conic in Polar Form

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

Graphing the Polar Equations of Conics

Solve a system of linear equations

Deﬁning Conics in Terms of a Focus and a Directrix

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Modeling with Linear Functions

Sequences and Their Notations

Introduction to Modeling With Linear Functions

Introduction to Sequences and Their Notations

Identify steps for modeling and solving

Writing the Terms of a Sequence Defined by an Explicit Formula

Build linear models

Investigating Alternating Sequences

Build systems of linear models

Investigating Explicit Formulas

Section Exercises

Writing the Terms of a Sequence Defined by a Recursive Formula

Solutions

Key Concepts & Glossary

Fitting Linear Models to Data

Section Exercises

Introduction to Fitting Linear Models to Data

Solutions

Draw and interpret scatter plots

Arithmetic Sequences

Find the line of best fit

Introduction to Arithmetic Sequences

Distinguish between linear and nonlinear relations

Finding Common Differences

Use a linear model to make predictions

Using Formulas for Arithmetic Sequences

Key Concepts & Glossary

Finding the Number of Terms in a Finite Arithmetic Sequence

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Complex Numbers

Solutions

Introduction to Complex Numbers

Geometric Sequences

Express square roots of negative numbers as multiples of i

Introduction to Geometric Sequences

Plot complex numbers on the complex plane

Finding Common Ratios

Add and subtract complex numbers

Writing Terms of Geometric Sequences

Multiply and divide complex numbers

Solving Application Problems with Geometric Sequences

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Quadratic Functions

Series and Their Notations

Introduction to Quadratic Functions

Introduction to Series and Their Notations

Recognize characteristics of parabolas

Using Summation Notation

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

Using the Formula for Arithmetic Series

Determine a quadratic function’s minimum or maximum value

Using the Formula for Geometric Series

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

Finding Sums of Infinite Series

Key Concepts & Glossary

Solving Annuity Problems

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Power Functions and Polynomial Functions

Solutions

Introduction to Power Functions and Polynomial Functions

Counting Principles

Describe the end behavior of power functions

Introduction to Counting Principles

Identify power functions

Using the Addition and Multiplication Principles

Use the degree and leading coefficient to describe end behavior of polynomial functions

Finding the Number of Permutations of n Distinct Objects

Identify polynomial functions

Find the Number of Combinations Using the Formula

Describe local behavior of polynomial functions

Finding the Number of Subsets of a Set

Key Concepts & Glossary

Finding the Number of Permutations of n Non-Distinct Objects

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Graphs of Polynomial Functions

Solutions

Introduction to Graphs of Polynomial Functions

Binomial Theorem

Recognize characteristics of graphs of polynomial functions

Introduction to Binomial Theorem

Use factoring to ﬁnd zeros of polynomial functions

Identifying Binomial Coefficients

Zeros, End Behavior, and Turning Points

Using the Binomial Theorem

Understand the relationship between degree and turning points

Using the Binomial Theorem to Find a Single Term

Determine end behavior

Key Concepts & Glossary

Graph polynomial functions

Section Exercises

Use the Intermediate Value Theorem

Solutions

Key Concepts & Glossary

Probability

Section Exercises

Introduction to Probability

Solutions

Constructing Probability Models

Dividing Polynomials

Computing the Probability of the Union of Two Events

Introduction to Dividing Polynomials

Computing the Probability of Mutually Exclusive Events

Use long division to divide polynomials

Using the Complement Rule to Compute Probabilities

Use synthetic division to divide polynomials

Computing Probability Using Counting Theory

Use polynomial division to solve application problems

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Practice Tests

Zeros of Polynomial Functions

Functions Practice Test

Introduction to Zeros of Polynomials

Solutions: Functions Practice Test

Evaluate a polynomial using the Remainder Theorem

Linear Functions Practice Test

Use the Factor Theorem to solve a polynomial equation

Solutions: Linear Functions Practice Test

Use the Rational Zero Theorem to find rational zeros

Polynomial and Rational Functions Practice Test

Find zeros of a polynomial function

Solutions: Polynomial and Rational Functions Practice Test

Use the Fundamental Theorem of Algebra

Exponential and Logarithmic Functions Practice Test

Use the Linear Factorization Theorem to find polynomials with given zeros

Solutions: Exponential and Logarithmic Functions Practice Test

Use Descartes’ Rule of Signs