Learning Outcomes

Module 1: Algebra Essentials

Evaluate and simplify expressions that contain both real numbers and variables

• Classify a real number
• Perform calculations using order of operations.
• Use the properties of real numbers
• Evaluate and simplify algebraic expressions.
• Use the rules of exponents to simplify exponential expressions
• Use scientific notation
• 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

• Identify the degree, leading coefficient, and leading term of a polynomial expression
• Perform algebraic operations on polynomial expressions
• Identify the greatest common factor of a polynomial expression
• Factor a wide variety of polynomials including those with fractional or negative exponents
• Simplify and perform algebraic operations on rational expressions

Module 3: The Rectangular Coordinate System and Equations of Lines

• 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
• 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
• 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

• Solve equations involving rational exponents
• Solve equations using factoring
• Solve radical equations
• Solve absolute value equations
• Set up a linear equation to solve a real-world application
• Use a formula to solve a real-world application
• 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
• Use interval notation
• Use properties of inequalities
• Solve inequalities in one variable algebraically
• Solve absolute value inequalities

Module 5: Function Basics

• 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
• 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
• 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

• 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
• 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
• 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

• 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.

• 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

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

Module 8: Quadratic Functions

• 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

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

• 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

• 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.

• 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

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

• 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.
• Find the inverse of a polynomial function.
• Restrict the domain to find the inverse of a polynomial function.
• Solve direct variation problems.
• Solve inverse variation problems.
• Solve problems involving joint variation.

Module 11: Exponential and Logarithmic 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
• Graph exponential functions, determine whether a graph represents exponential growth or decay
• Graph exponential functions using transformations.
• Convert from logarithmic to exponential form.
• Convert from exponential to logarithmic form.
• Evaluate common and natural logarithms.
• 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

• Use power, product, and quotient rules to expand and condense logarithms
• Use the change-of-base formula for logarithms.
• 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.
• 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.
• Build an exponential model from data.

Module 13: Systems of Equations and Inequalities

• 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.

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

• 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.

• 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

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.
• 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.
• Find the inverse of a matrix.
• Solve a system of linear equations using an inverse matrix.

Module 15: Conic Sections

• 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
• 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
• 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

• 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
• 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
• 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
• 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: Probability and 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
• 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

Licenses & Attributions