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
