Course Learning Outcomes

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.