Module 1: Algebra Essentials

Real Numbers

• Classify a real number
• Perform calculations using order of operations.
• Use the properties of real numbers
• Evaluate and simplify algebraic expressions.

Exponents and Scientific Notation

• Use the rules of exponents to simplify exponential expressions
• Use scientific notation

Radicals and Rational Exponents

• Evaluate and simplify square roots
• Rationalize a denominator that contains a square root
• Rewrite a radical expression using rational exponents

Module 2: Polynomial and Rational Expressions

Polynomial Basics

• Identify the degree, leading coefficient, and leading term of a polynomial expression
• Perform algebraic operations on polynomial expressions

Factoring Polynomials

• Identify the greatest common factor of a polynomial expression
• Factor a wide variety of polynomials including those with fractional or negative exponents

Rational Expressions

• Simplify and perform algebraic operations on rational expressions

Module 3: The Rectangular Coordinate System and Equations of Lines

Points and Lines in the Plane

• Plot ordered pairs, and graph equations by plotting points
• Use a graphing utility to graph equations
• Find the x and y intercepts of a graphed equation
• Use the distance and midpoint formulas

Equations of Lines

• Write equations of lines in slope-intercept, point-slope, and standard forms
• Identify the equations and graphs of horizontal and vertical lines
• Determine whether two lines are parallel, perpendicular, or neither
• Write equations of lines that are parallel or perpendicular to another line

Models and Applications of Linear Equations

• Develop a problem solving method
• Write an equation to model an application
• Solve distance, rate and time problems
• Solve perimeter, area, and volume problems

Module 4: Equations and Inequalities

Equation-Solving Techniques

• Solve equations involving rational exponents.
• Solve equations using factoring.
• Solve radical equations.
• Solve absolute value equations

Models and Applications

• Set up a linear equation to solve a real-world application.
• Use a formula to solve a real-world application.

Quadratic Equations

• Solve quadratic equations by factoring.
• Solve quadratic equations by the square root property.
• Solve quadratic equations by completing the square.
• Solve quadratic equations by using the quadratic formula.

Linear Inequalities and Absolute Value Inequalities

• Use interval notation.
• Use properties of inequalities.
• Solve inequalities in one variable algebraically.
• Solve absolute value inequalities

Module 5: Function Basics

Characteristics of Functions and Their Graphs

• Determine whether a relation represents a function.
• Find the value of a function.
• Determine whether a function is one-to-one.
• Use the vertical line test to identify functions.
• Graph the functions listed in the library of functions

Domain and Range of Functions

• Find the domain of a function defined by an equation
• Write Domain and Range Using Standard Notations
• Find Domain and Range from a Graph
• Define Domain and Range of Toolkit Functions
• Graph Piecewise-Defined Functions

Rates of Change and Behavior of Graphs

• Find the average rate of change of a function.
• Use a graph to determine where a function is increasing, decreasing, or constant.
• Use a graph to locate local maxima and local minima.
• Use a graph to locate the absolute maximum and absolute minimum.

Module 6: Algebraic Operations on Functions

Compositions of Functions

• Combine functions using algebraic operations.
• Create a new function by composition of functions.
• Evaluate composite functions.
• Find the domain of a composite function.
• Decompose a composite function into its component functions.

Transformations of Functions

• Graph functions using vertical and horizontal shifts.
• Graph functions using reflections about the [latex]x[/latex] -axis and the [latex]y[/latex] -axis.
• Determine whether a function is even, odd, or neither from its graph.
• Graph functions using compressions and stretches.
• Combine transformations.

Inverse Functions

• Verify inverse functions.
• Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
• Find or evaluate the inverse of a function.
• Use the graph of a one-to-one function to graph its inverse function on the same axes.

Module 7: Linear and Absolute Value Functions

Linear Functions

• Represent a linear function with an equation, words, a table and a graph
• Determine whether a linear function is increasing, decreasing, or constant.
• Write and interpret a linear function.

Graphs of Linear Functions

• Graph linear functions by plotting points, using the slope and y-intercept, and by using transformations
• Write the equation of a linear function given it’s graph, including vertical and horizontal lines, match linear equations with their graphs
• Find the equations of vertical and horizontal lines
• Graph an absolute value function, find it’s intercepts

Modeling With Linear Functions

• Identify steps for modeling and solving.
• Build linear models from verbal descriptions.
• Draw and interpret scatter plots.
• Find the line of best fit using the Desmos calculator.
• Distinguish between linear and nonlinear relations.
• Use a linear model to make predictions.

Module 8: Quadratic Functions

Complex Numbers

• Express square roots of negative numbers as multiples of i
• Plot complex numbers on the complex plane
• Add and subtract complex numbers
• Multiply and divide complex numbers

Graphs of Quadratic Functions

• Recognize characteristics of parabolas
• Understand how the graph of a parabola is related to its quadratic function

Analysis of Quadratic Functions

• Use the quadratic formula and factoring to find both real and complex roots (x-intercepts) of quadratic functions
• Use algebra to find the y-intercepts of a quadratic function
• Solve problems involving the roots and intercepts of a quadratic function
• Use the discriminant to determine the nature (real or complex) and quantity of solutions to quadratic equations
• Determine a quadratic function’s minimum or maximum value
• Solve problems involving a quadratic function’s minimum or maximum value

Module 9: Power and Polynomial Functions

Characteristics of Power and Polynomial Functions

• Identify power functions.
• Identify end behavior of power functions.
• Identify polynomial functions.
• Identify the degree and leading coefficient of polynomial functions.
• Identify local behavior of polynomial functions.

Graphs of Polynomial Functions

• Identify zeros of polynomial functions with even and odd multiplicity
• Use the degree of a polynomial to determine the number of turning points of its graph
• Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the intermediate value theorem
• Write the equation of a polynomial function given it’s graph

Divide Polynomials

• Use long division to divide polynomials.
• Use synthetic division to divide polynomials.

Methods for Finding Zeros of Polynomials

• Evaluate a polynomial using the Remainder Theorem.
• Use the Factor Theorem to solve a polynomial equation.
• Use the Rational Zero Theorem to find rational zeros.
• Find zeros of a polynomial function.
• Use the Linear Factorization Theorem to find polynomials with given zeros.
• Use Descartes’ Rule of Signs.
• Solve real-world applications of polynomial equations

Module 10: Rational and Radical Functions

Rational Functions

• Use arrow notation to describe end behavior of rational functions
• Solve applied problems involving rational functions.
• Find the domains of rational functions.
• Identify vertical and horizontal asymptotes of graphs of rational functions
• Graph rational functions.

Radical Functions

• Find the inverse of a polynomial function.
• Restrict the domain to find the inverse of a polynomial function.

Variation

• Solve direct variation problems.
• Solve inverse variation problems.
• Solve problems involving joint variation.

Module 11: Exponential and Logarithmic Functions

Exponential Functions

• Evaluate an exponential growth function with different bases
• Use a compound interest Formula
• Write an exponential function
• Find an exponential function given a graph
• Use a graphing calculator to find an exponential function
• Find an exponential function that models continuous growth or decay

Graphs of Exponential Functions

• Graph exponential functions, determine whether a graph represents exponential growth or decay
• Graph exponential functions using transformations.

Logarithmic Functions

• Convert from logarithmic to exponential form.
• Convert from exponential to logarithmic form.
• Evaluate common and natural logarithms.

Graphs of Logarithmic Functions

• Identify the domain of a logarithmic function.
• Graph logarithmic functions using transformations, and identify intercepts and the vertical asymptote
• Identify why and how a logarithmic function is an inverse of an exponential function

Module 12: Exponential and Logarithmic Equations and Models

Logarithmic Properties

• Use power, product, and quotient rules to expand and condense logarithms
• Use the change-of-base formula for logarithms.

Exponential and Logarithmic Equations

• Use like bases to solve exponential equations.
• Use logarithms to solve exponential equations.
• Use the definition of a logarithm to solve logarithmic equations.
• Use the one-to-one property of logarithms to solve logarithmic equations.
• Solve applied problems involving exponential and logarithmic equations.

Exponential and Logarithmic Models

• Model exponential growth and decay.
• Use Newton’s Law of Cooling.
• Use logistic-growth models.
• Choose an appropriate model for data.
• Express an exponential model in base e.

Fitting Exponential Models to Data

• Build an exponential model from data.
• Build a logarithmic model from data.
• Build a logistic model from data.

Module 13: Systems of Equations and Inequalities

Systems of Linear Equations: Two Variables

• Solve systems of equations by graphing, substitution, and addition.
• Identify inconsistent systems of equations containing two variables.
• Express the solution of a system of dependent equations containing two variables using standard notations.

Systems of Nonlinear Equations and Inequalities

• Solve a system of nonlinear equations using substitution or elimination.
• Graph a nonlinear inequality.
• Graph a system of nonlinear inequalities.

Systems of Linear Equations: Three Variables

• Solve systems of three equations in three variables.
• Identify inconsistent systems of equations containing three variables.
• Express the solution of a system of dependent equations containing three variables using standard notations.

Partial Fractions: an Application of Systems

• Decompose   [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] ,  where  Q( x )  has only nonrepeated linear factors.
• Decompose  [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] ,  where  Q( x )  has repeated linear factors.
• Decompose  [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] ,  where  Q( x )  has a nonrepeated irreducible quadratic factor.
• Decompose  [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] ,  where  Q( x )  has a repeated irreducible quadratic factor.

Module 14: Solve Systems With Matrices

Matrices and Matrix Operations

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Gaussian Elimination

• Write the augmented matrix of a system of equations.
• Write the system of equations from an augmented matrix.
• Perform row operations on a matrix.
• Solve a system of linear equations using matrices.

Solve Systems with Inverses

• Find the inverse of a matrix.
• Solve a system of linear equations using an inverse matrix.

Module 15: Conic Sections

The Ellipse

• Write equations of ellipses in standard form.
• Graph ellipses centered at the origin.
• Graph ellipses not centered at the origin.
• Solve applied problems involving ellipses.

The Hyperbola

• Locate a hyperbola’s vertices and foci.
• Write equations of hyperbolas in standard form.
• Graph hyperbolas centered at the origin.
• Graph hyperbolas not centered at the origin.
• Solve applied problems involving hyperbolas.

The Parabola

• Graph parabolas with vertices at the origin.
• Write equations of parabolas in standard form.
• Graph parabolas with vertices not at the origin.
• Solve applied problems involving parabolas.

Module 16: Sequences and Series

Sequences and Their Notations

• Write the terms of a sequence defined by an explicit formula.
• Write the terms of a sequence defined by a recursive formula.
• Use factorial notation.

 Arithmetic Sequences

• Find the common difference for an arithmetic sequence.
• Write terms of an arithmetic sequence.
• Use a recursive formula for an arithmetic sequence.
• Use an explicit formula for an arithmetic sequence.

Geometric Sequences

• Find the common ratio for a geometric sequence.
• List the terms of a geometric sequence.
• Use a recursive formula for a geometric sequence.
• Use an explicit formula for a geometric sequence.

Series and Their Notations

• Use summation notation.
• Use the formula for the sum of the first [latex]n[/latex] terms of an arithmetic series.
• Use the formula for the sum of the first [latex]n[/latex] terms of a geometric series.
• Use the formula for the sum of an infinite geometric series.
• Solve annuity problems.

Module 17: Counting Principles

Counting Principles

• Solve counting problems using the Addition Principle and the Multiplication Principle.
• Solve counting problems using permutations and combinations  involving n distinct objects.
• Find the number of subsets of a given set.
• Solve counting problems using permutations involving n non-distinct objects.
• Apply the Binomial Theorem

Probability

• Construct probability models.
• Compute probabilities of equally likely outcomes.
• Compute probabilities of the union of two events.
• Use the complement rule to find probabilities.
• Compute probability using counting theory.