Precalculus I (editable text only)

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Text

Precalculus I is intended as a first semester course for college-level precalculus students. Included, you will find all of the content that might be covered in any first semester precalculus or college algebra course.

The text contains several short modules, with each page typically covering one learning outcome.  This layout may not suit everyone, but this text is fully editable so you can reorganize the pages to fit your needs.

Topics covered include:

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

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

 

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

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

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

 

Absolute Value Functions

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

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

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

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

 

Modeling with Linear Functions

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

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

 

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

 

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

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

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

Dividing Polynomials

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

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

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

Inverses and Radical Functions

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

 

Modeling Using Variation

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

Exponential Functions

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

Graphs of Exponential Functions

  • Graph exponential functions
  • Graph exponential functions using transformations

Logarithmic Functions

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

Graphs of Logarithmic Functions

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

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

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

Fitting Exponential Models to Data

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

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

Systems of Linear Equations: Three Variables

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

Matrices and Matrix Operations

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

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

Solving Systems with Inverses

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

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

Arithmetic Sequences

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

 

Geometric Sequences

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

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

Practice Tests