Precalculus I (editable text only)

Content Overview

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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