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Real Numbers: Algebra Essentials

Rational Functions

Introduction to Real Numbers: Algebra Essentials

Introduction to Rational Functions

Classifying a Real Number

Use arrow notation

Performing Calculations Using the Order of Operations

Solve applied problems involving rational functions

Using Properties of Real Numbers

Find the domains of rational functions

Evaluating Algebraic Expressions

Identify vertical and horizontal asymptotes

Formulas

Identify horizontal asymptotes

Simplifying Algebraic Expressions

Graph rational functions

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Exponents and Scientific Notation

Inverses and Radical Functions

Introduction to Exponents and Scientific Notation

Introduction to Inverses and Radical Functions

Using the Product Rule of Exponents

Find the inverse of a polynomial function

Using the Quotient Rule of Exponents

Restrict the domain to find the inverse of a polynomial function

Using the Power Rule of Exponents

Key Concepts & Glossary

Using the Zero Exponent Rule of Exponents

Section Exercises

Using the Negative Rule of Exponents

Solutions

Finding the Power of a Product and a Quotient

Modeling Using Variation

Simplifying Exponential Expressions

Introduction to Modeling Using Variation

Using Scientific Notation

Solve direct variation problems

Key Concepts & Glossary

Solve inverse variation problems

Section Exercises

Solve problems involving joint variation

Solutions

Key Concepts & Glossary

Radicals and Rational Expressions

Section Exercises

Introduction to Radicals and Rational Expressions

Solutions

Evaluating Square Roots

Exponential Functions

Simplifying Square Roots

Introduction to Exponential Functions

Adding and Subtracting Square Roots

Evaluate exponential functions

Rationalizing Denominators

Find the equation of an exponential function

Understanding nth Roots

Use compound interest formulas

Using Rational Exponents

Evaluate exponential functions with base e

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Polynomials

Graphs of Exponential Functions

Introduction to Polynomials

Introduction to Graphs of Exponential Functions

Identifying the Degree and Leading Coefficient of Polynomials

Graph exponential functions

Adding and Subtracting Polynomials

Graph exponential functions using transformations

Multiplying Polynomials

Key Concepts

Special Products of Polynomials

Section Exercises

Performing Operations with Polynomials of Several Variables

Solutions

Key Concepts & Glossary

Logarithmic Functions

Section Exercises

Introduction to Logarithmic Functions

Solutions

Convert from logarithmic to exponential form

Factoring Polynomials

Convert from exponential to logarithmic form

Introduction to Factoring Polynomials

Evaluate logarithms

Factoring the Greatest Common Factor of a Polynomial

Use common logarithms

Factoring a Trinomial with Leading Coefficient 1

Use natural logarithms

Factoring by Grouping

Key Concepts & Glossary

Special Products

Section Exercises

Factoring Expressions with Fractional or Negative Exponents

Solutions

Key Concepts & Glossary

Graphs of Logarithmic Functions

Section Exercises

Introduction to Graphs of Logarithmic Functions

Solutions

Identify the domain of a logarithmic function

Rational Expressions

Graph logarithmic functions

Introduction to Rational Expressions

Graphing Transformations of Logarithmic Functions

Simplifying Rational Expressions

Key Concepts

Multiplying and Dividing Rational Expressions

Section Exercises

Adding and Subtracting Rational Expressions

Solutions

Simplifying Complex Rational Expressions

Logarithmic Properties

Key Concepts & Glossary

Introduction to Logarithmic Properties

Section Exercises

Use the product rule for logarithms

Solutions

Use the quotient and power rules for logarithms

The Rectangular Coordinate Systems and Graphs

Expand logarithmic expressions

Introduction to The Rectangular Coordinate Systems and Graphs

Condense logarithmic expressions

Plotting Ordered Pairs in the Cartesian Coordinate System

Use the change-of-base formula for logarithms

Graphing Equations by Plotting Points

Key Terms & Glossary

Graphing Equations with a Graphing Utility

Section Exercises

Finding x-intercepts and y-intercepts

Solutions

Using the Distance Formula

Exponential and Logarithmic Equations

Using the Midpoint Formula

Introduction to Exponential and Logarithmic Equations

Key Concepts & Glossary

Use like bases to solve exponential equations

Section Exercises

Use logarithms to solve exponential equations

Solutions

Use the definition of a logarithm to solve logarithmic equations

Linear Equations in One Variable

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

Introduction to Linear Equations in One Variable

Solve applied problems involving exponential and logarithmic equations

Solving Linear Equations in One Variable

Key Concepts & Glossary

Solving a Rational Equation

Section Exercises

Finding a Linear Equation

Solutions

Determining Whether Graphs of Lines are Parallel or Perpendicular

Fitting Exponential Models to Data

Key Concepts & Glossary

Introduction to Fitting Exponential Models to Data

Section Exercises

Build an exponential model from data

Solutions

Build a logarithmic model from data

Models and Applications

Build a logistic model from data

Introduction: Models and Applicaitons

Key Concepts

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

Section Exercises

Using a Formula to Solve a Real-World Application

Solutions

Key Concepts & Glossary

Exponential and Logarithmic Models

Section Exercises

Introduction to Exponential and Logarithmic Models

Solutions to Selected Exercises

Model exponential growth and decay

Complex Numbers

Use Newton’s Law of Cooling

Introduction to Complex Numbers

Use logistic-growth models

Express square roots of negative numbers as multiples of i

Choose an appropriate model for data

Plot complex numbers on the complex plane

Key Concepts & Glossary

Add and subtract complex numbers

Section Exercises

Multiply and divide complex numbers

Solutions

Key Concepts & Glossary

Systems of Linear Equations: Two Variables

Section Exercises

Introduction to Systems of Linear Equations: Two Variables

Solutions

Solving Systems of Equations by Graphing

Quadratic Equations

Solving Systems of Equations by Substitution

Introduction: Quadratic Equations

Solving Systems of Equations in Two Variables by the Addition Method

Solving Quadratic Equations by Factoring

Identifying and Expressing Solutions to Systems of Equations

Using the Square Root Property

Using Systems of Equations to Investigate Profits

Completing the Square

Key Concepts & Glossary

Using the Quadratic Formula

Section Exercises

Key Concepts & Glossary

Solutions

Solutions

Systems of Linear Equations: Three Variables

Section Exercises

Introduction to Systems of Linear Equations: Three Variables

Other Types of Equations

Solving Systems of Three Equations in Three Variables

Introduction: Other Types of Equations

Inconsistent and Dependent Systems in Three Variables

Solving Equations Involving Rational Exponents

Key Concepts & Glossary

Solving Equations Using Factoring

Section Exercises

Solving Radical Equations

Solutions

Solving an Absolute Value Equation

Systems of Nonlinear Equations and Inequalities: Two Variables

Solving Other Types of Equations

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

Linear Inequalities and Absolute Value Inequalities

Key Concepts & Glossary

Introduction: Linear Inequalities and Absolute Value Inequalities

Section Exercises

Using Interval Notation

Solutions

Using the Properties of Inequalities

Partial Fractions

Understanding Compound Inequalities

Introduction to Partial Fractions

Solving Absolute Value Inequalities

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

Key Concepts & Glossary

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

Section Exercises

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

Solutions

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

Functions and Function Notation

Key Concepts & Glossary

Introduction to Functions and Function Notation

Section Exercises

Determine whether a relation represents a function

Solutions

Find the input and output values of a function

Matrices and Matrix Operations

Determine whether a function is one-to-one

Introduction: Matrices and Matrix Operations

Use the vertical line test to identify functions

Finding the Sum and Difference of Two Matrices

Graph the functions listed in the library of functions

Finding Scalar Multiples of a Matrix

Key Concepts & Glossary

Finding the Product of Two Matrices

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Domain and Range

Solutions

Introduction to Domain and Range

Solving Systems with Gaussian Elimination

Find the domain of a function defined by an equation

Introduction to Solving Systems with Gaussian Elimination

Use notations to specify domain and range

The Augmented Matrix of a System of Equations

Find domain and range from graphs

Performing Row Operations on a Matrix

Find domains and ranges of the toolkit functions

Solving a System of Linear Equations Using Matrices

Graph piecewise-defined functions

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Solving Systems with Inverses

Rates of Change and Behavior of Graphs

Introduction to Solving Systems with Inverses

Introduction to Rates of Change and Behaviors of Graphs

Finding the Inverse of a Matrix

Find the average rate of change of a function

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

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

Key Concepts & Glossary

Use a graph to locate the absolute maximum and absolute minimum

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Solving Systems with Cramer's Rule

Solutions

Introduction to Solving Systems with Cramer's Rule

Composition of Functions

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

Introduction to Composition of Functions

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

Combine functions using algebraic operations

Understanding Properties of Determinants

Create a new function by composition of functions

Key Concepts & Glossary

Evaluate composite functions

Solutions

Find the domain of a composite function

Section Exercises

Decompose a composite function into its component functions

The Ellipse

Key Concepts & Glossary

Introduction to The Ellipse

Section Exercises

Writing Equations of Ellipses in Standard Form

Solutions

Deriving the Equation of an Ellipse Centered at the Origin

Transformation of Functions

Writing Equations of Ellipses Not Centered at the Origin

Introduction to Transformation of Functions

Graphing Ellipses

Graph functions using vertical and horizontal shifts

Solving Applied Problems Involving Ellipses

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

Key Concepts & Glossary

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

Section Exercises

Graph functions using compressions and stretches

Solutions

Combine vertical and horizontal shifts

The Hyperbola

Key Concepts & Glossary

Introduction to the Hyperbola

Section Exercises

Locating the Vertices and Foci of a Hyperbola

Solutions

Deriving the Equation of a Hyperbola Centered at the Origin

Absolute Value Functions

Writing Equations of Hyperbolas in Standard Form

Introduction to Absolute Value Functions

Graphing Hyperbolas

Graph an absolute value function

Solving Applied Problems Involving Hyperbolas

Solve an absolute value equation

Key Concepts & Glossary

Solve an absolute value inequality

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

The Parabola

Solutions

Introduction to The Parabola

Inverse Functions

Graphing Parabolas with Vertices at the Origin

Introduction to Inverse Functions

Writing Equations of Parabolas in Standard Form

Verify inverse functions

Graphing Parabolas with Vertices Not at the Origin

Determine the domain and range of an inverse function

Solving Applied Problems Involving Parabolas

Find or evaluate the inverse of a function

Key Concepts & Glossary

Use the graph of a function to graph its inverse

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Rotation of Axes

Solutions

Introduction to Rotation of Axes

Linear Functions

Identifying Nondegenerate Conics in General Form

Introduction to Linear Functions

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

Represent a linear function

Writing Equations of Rotated Conics in Standard Form

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

Identifying Conics without Rotating Axes

Calculate and interpret slope

Key Concepts & Glossary

Write the point-slope form of an equation

Section Exercises

Write and interpret a linear function

Solutions

Key Concepts & Glossary

Conic Sections in Polar Coordinates

Section Exercises

Introduction to Conic Sections in Polar Coordinates

Solutions

Identifying a Conic in Polar Form

Graphs of Linear Functions

Graphing the Polar Equations of Conics

Introduction to Graphs of Linear Functions

Deﬁning Conics in Terms of a Focus and a Directrix

Graph linear functions

Key Concepts & Glossary

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

Section Exercises

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

Solutions

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

Sequences and Their Notations

Solve a system of linear equations

Introduction to Sequences and Their Notations

Key Concepts & Glossary

Writing the Terms of a Sequence Defined by an Explicit Formula

Section Exercises

Investigating Alternating Sequences

Solutions

Investigating Explicit Formulas

Modeling with Linear Functions

Writing the Terms of a Sequence Defined by a Recursive Formula

Introduction to Modeling With Linear Functions

Key Concepts & Glossary

Identify steps for modeling and solving

Section Exercises

Build linear models

Solutions

Build systems of linear models

Arithmetic Sequences

Section Exercises

Introduction to Arithmetic Sequences

Solutions

Finding Common Differences

Fitting Linear Models to Data

Using Formulas for Arithmetic Sequences

Introduction to Fitting Linear Models to Data

Finding the Number of Terms in a Finite Arithmetic Sequence

Draw and interpret scatter plots

Key Concepts & Glossary

Find the line of best fit

Section Exercises

Distinguish between linear and nonlinear relations

Solutions

Use a linear model to make predictions

Geometric Sequences

Key Concepts & Glossary

Introduction to Geometric Sequences

Section Exercises

Finding Common Ratios

Solutions

Writing Terms of Geometric Sequences

Quadratic Functions

Solving Application Problems with Geometric Sequences

Introduction to Quadratic Functions

Key Concepts & Glossary

Recognize characteristics of parabolas

Section Exercises

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

Solutions

Determine a quadratic function’s minimum or maximum value

Series and Their Notations

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

Introduction to Series and Their Notations

Key Concepts & Glossary

Using Summation Notation

Section Exercises

Using the Formula for Arithmetic Series

Solutions

Using the Formula for Geometric Series

Power Functions and Polynomial Functions

Finding Sums of Infinite Series

Introduction to Power Functions and Polynomial Functions

Solving Annuity Problems

Describe the end behavior of power functions

Key Concepts & Glossary

Identify power functions

Section Exercises

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

Solutions

Identify polynomial functions

Counting Principles

Describe local behavior of polynomial functions

Introduction to Counting Principles

Key Concepts & Glossary

Using the Addition and Multiplication Principles

Section Exercises

Finding the Number of Permutations of n Distinct Objects

Solutions

Find the Number of Combinations Using the Formula

Graphs of Polynomial Functions

Finding the Number of Subsets of a Set

Introduction to Graphs of Polynomial Functions

Finding the Number of Permutations of n Non-Distinct Objects

Recognize characteristics of graphs of polynomial functions

Key Concepts & Glossary

Use factoring to ﬁnd zeros of polynomial functions

Section Exercises

Zeros, End Behavior, and Turning Points

Solutions

Understand the relationship between degree and turning points

Binomial Theorem

Determine end behavior

Introduction to Binomial Theorem

Graph polynomial functions

Identifying Binomial Coefficients

Use the Intermediate Value Theorem

Using the Binomial Theorem

Key Concepts & Glossary

Using the Binomial Theorem to Find a Single Term

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Dividing Polynomials

Solutions

Introduction to Dividing Polynomials

Probability

Use long division to divide polynomials

Introduction to Probability

Use synthetic division to divide polynomials

Constructing Probability Models

Use polynomial division to solve application problems

Computing the Probability of the Union of Two Events

Key Concepts & Glossary

Computing the Probability of Mutually Exclusive Events

Section Exercises

Using the Complement Rule to Compute Probabilities

Solutions

Computing Probability Using Counting Theory

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

Practice Tests

Use the Rational Zero Theorem to find rational zeros

Functions Practice Test

Find zeros of a polynomial function

Solutions: Functions Practice Test

Use the Fundamental Theorem of Algebra

Linear Functions Practice Test

Use the Linear Factorization Theorem to find polynomials with given zeros

Solutions: Linear Functions Practice Test

Use Descartes’ Rule of Signs

Polynomial and Rational Functions Practice Test

Solve real-world applications of polynomial equations

Solutions: Polynomial and Rational Functions Practice Test

Key Concepts & Glossary

Exponential and Logarithmic Functions Practice Test

Section Exercises

Solutions: Exponential and Logarithmic Functions Practice Test

Solutions