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
