Precalculus I

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Course Contents at a Glance

an icon of a pair of binoculars The following list shows a summary of the topics covered in this course. To see all of the course pages, visit the Table of Contents.

Module 1: Functions and Function Notation

  • Determine whether a relation represents a function
  • Find the input and output values 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

Module 2: Domain and Range

  • Find the domain of a function defined by an equation
  • Use notations to specify domain and range
  • Find domain and range from graphs
  • Find domains and ranges of the toolkit functions
  • Graph piecewise-defined functions

Module 3: 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 the absolute maximum and absolute minimum

Module 4: Composition 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

Module 5: Transformation of Functions

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

Module 6: Absolute Value Functions

  • Graph an absolute value function
  • Solve an absolute value equation
  • Solve an absolute value inequality

Module 7: Inverse Functions

  • Verify inverse functions
  • Determine the domain and range of an inverse function
  • Find or evaluate the inverse of a function
  • Use the graph of a function to graph its inverse

Module 8: Linear Functions

  • Represent a linear function
  • Determine whether a linear function is increasing, decreasing, or constant
  • Calculate and interpret slope
  • Write the point-slope form of an equation
  • Write and interpret a linear function

Module 9: Graphs of Linear Functions

  • Graph linear functions
  • Write the equation for a linear function from the graph of a line
  • Given the equations of two lines, determine whether their graphs are parallel or perpendicular
  • Write the equation of a line parallel or perpendicular to a given line
  • Solve a system of linear equations

Module 10: Modeling with Linear Functions

  • Identify steps for modeling and solving
  • Build linear models
  • Build systems of linear models

Module 11: Fitting Linear Models to Data

  • Draw and interpret scatter plots
  • Find the line of best fit
  • Distinguish between linear and nonlinear relations
  • Use a linear model to make predictions

Module 12: 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

Module 13: Quadratic Functions

  • Recognize characteristics of parabolas
  • Understand how the graph of a parabola is related to its quadratic function
  • Determine a quadratic function’s minimum or maximum value
  • Solve problems involving a quadratic function’s minimum or maximum value

Module 14: Power Functions and Polynomial Functions

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

Module 15: Graphs of Polynomial Functions

  • Recognize characteristics of graphs of polynomial functions
  • Use factoring to find zeros of polynomial functions
  • Identify zeros and their multiplicities
  • Determine end behavior
  • Understand the relationship between degree and turning points
  • Graph polynomial functions
  • Solving Polynomial Inequalities
  • Use the Intermediate Value Theorem

Module 16: Dividing Polynomials

  • Use long division to divide polynomials
  • Use synthetic division to divide polynomials
  • Use polynomial division to solve application problems

Module 17: Zeros of Polynomial Functions

  • 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 Fundamental Theorem of Algebra
  • Use the Linear Factorization Theorem to find polynomials with given zeros
  • Use Descartes’ Rule of Signs
  • Solve real-world applications of polynomial equations

Module 18: Rational Functions

  • Use arrow notation
  • Solve applied problems involving rational functions
  • Find the domains of rational functions
  • Identify vertical asymptotes
  • Identify horizontal asymptotes
  • Graph rational functions

Module 19: Inverses and Radical Functions

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

Module 20: Modeling Using Variation

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

Module 21: Exponential Functions

  • Evaluate exponential functions
  • Find the equation of an exponential function
  • Use compound interest formulas
  • Evaluate exponential functions with base e

Module 22: Graphs of Exponential Functions

  • Graph exponential functions
  • Graph exponential functions using transformations

Module 23: Logarithmic Functions

  • Convert from logarithmic to exponential form
  • Convert from exponential to logarithmic form
  • Evaluate logarithms
  • Use common logarithms
  • Use natural logarithms

Module 24: Graphs of Logarithmic Functions

  • Identify the domain of a logarithmic function
  • Graph logarithmic functions
  • Graphing Transformations of Logarithmic Functions

Module 25: Logarithmic Properties

  • Use the product rule for logarithms
  • Use the quotient and power rules for logarithms
  • Expand logarithmic expressions
  • Condense logarithmic expressions
  • Use the change-of-base formula for logarithms

Module 26: 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

Module 27: 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

Module 28: Fitting Exponential Models to Data

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

Module 29: Systems of Linear Equations: Two Variables

  • Solving Systems of Equations by Graphing
  • Solving Systems of Equations by Substitution
  • Solving Systems of Equations in Two Variables by the Addition Method
  • Identifying and Expressing Solutions to Systems of Equations
  • Using Systems of Equations to Investigate Profits

Module 30: Systems of Linear Equations: Three Variables

  • Solving Systems of Three Equations in Three Variables
  • Inconsistent and Dependent Systems in Three Variables

Module 31: Matrices and Matrix Operations

  • Finding the Sum and Difference of Two Matrices
  • Finding Scalar Multiples of a Matrix
  • Finding the Product of Two Matrices

Module 32: Solving Systems with Gaussian Elimination

  • The Augmented Matrix of a System of Equations
  • Performing Row Operations on a Matrix
  • Solving a System of Linear Equations Using Matrices

Module 33: Solving Systems with Inverses

  • Finding the Inverse of a Matrix
  • Solving a System of Linear Equations Using the Inverse of a Matrix

Module 34: Sequences and Their Notations

  • Writing the Terms of a Sequence Defined by an Explicit Formula
  • Investigating Alternating Sequences
  • Investigating Explicit Formulas
  • Writing the Terms of a Sequence Defined by a Recursive Formula

Module 35: Arithmetic Sequences

  • Finding Common Differences
  • Using Formulas for Arithmetic Sequences
  • Finding the Number of Terms in a Finite Arithmetic Sequence

Module 36: Geometric Sequences

  • Finding Common Ratios
  • Writing Terms of Geometric Sequences
  • Solving Application Problems with Geometric Sequences

Module 37: Series and Their Notations

  • Using Summation Notation
  • Using the Formula for Arithmetic Series
  • Using the Formula for Geometric Series
  • Finding Sums of Infinite Series
  • Solving Annuity Problems

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