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HomeStudy GuidesPrecalculus I

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Course Contents

Module 1: Introduction to Functions

Function and Function NotationSolutions 1: Functions and Function NotationDomain and RangeSolutions 2: Domain and RangeRates of Change and Behavior of GraphsSolutions 3: Rates of Change and Behavior of GraphsComposition of FunctionsSolutions 4: Composition of FunctionsTransformation of FunctionsSolutions 5: Transformation of FunctionsAbsolute Value FunctionsSolutions 6: Absolute Value FunctionsInverse FunctionsSolutions 7: Inverse Functions

Module 2: Linear Functions

Module 3: Polynomial and Rational Functions

Complex NumbersSolutions 12: Complex NumbersQuadratic FunctionsSolutions 13: Quadratic FunctionsPower Functions and Polynomial FunctionsSolutions 14: Power Functions and Polynomial FunctionsGraphs of Polynomial FunctionsSolutions 15: Graphs of Polynomial FunctionsDividing PolynomialsSolutions 16: Dividing PolynomialsZeros of Polynomial FunctionsSolutions 17: Zeros of Polynomial FunctionsRational FunctionsSolutions 18: Rational FunctionsInverses and Radical FunctionsSolutions 19: Inverses and Radical FunctionsModeling Using VariationSolutions 20: Modeling Using Variation

Module 4: Exponential and Logarithmic Functions

Exponential FunctionsSolutions 21: Exponential FunctionsGraphs of Exponential FunctionsSolutions 22: Graphs of Exponential FunctionsLogarithmic FunctionsSolutions 23: Logarithmic FunctionsGraphs of Logarithmic FunctionsSolutions 24: Graphs of Logarithmic FunctionsLogarithmic PropertiesSolutions 25: Logarithmic PropertiesExponential and Logarithmic EquationsSolutions 26: Exponential and Logarithmic EquationsExponential and Logarithmic ModelsSolutions 27: Exponential and Logarithmic ModelsFitting Exponential Models to DataSolutions 28: Fitting Exponential Models to Data

Module 5: Sequences, Probability, and Counting Theory

Module 6: Systems of Linear Equations: Two Variables

Introduction to Systems of Linear Equations: Two VariablesSolving Systems of Equations by GraphingSolving Systems of Equations by SubstitutionSolving Systems of Equations in Two Variables by the Addition MethodIdentifying and Expressing Solutions to Systems of EquationsUsing Systems of Equations to Investigate ProfitsKey Concepts & GlossaryProblem Set 29: Systems of Linear Equations: Two VariablesSolutions 29: Systems of Linear Equations: Two Variables

Module 7: Systems of Linear Equations: Three Variables

Introduction to Systems of Linear Equations: Three VariablesSolving Systems of Three Equations in Three VariablesInconsistent and Dependent Systems in Three VariablesKey Concepts & GlossaryProblem Set 30: Systems of Linear Equations: Three VariablesSolutions 30: Systems of Linear Equations: Three Variables

Module 8: Matrices and Matrix Operations

Module 9: Solving Systems with Gaussian Elimination

Introduction to Solving Systems with Gaussian EliminationThe Augmented Matrix of a System of EquationsPerforming Row Operations on a MatrixSolving a System of Linear Equations Using MatricesKey Concepts & GlossaryProblem Set 32: Solving Systems with Gaussian EliminationSolutions 32: Solving Systems with Gaussian Elimination

Module 10: Solving Systems with Inverses

Practice Tests

Functions Practice TestSolutions: Functions Practice TestLinear Functions Practice TestSolutions: Linear Functions Practice TestPolynomial and Rational Functions Practice TestSolutions: Polynomial and Rational Functions Practice TestExponential and Logarithmic Functions Practice TestSolutions: Exponential and Logarithmic Functions Practice Test

Appendix

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

## Module 2: Domain and Range

**Module 3: **Rates of Change and Behavior of Graphs

**Module 4: **Composition of Functions

**Module 5: **Transformation of Functions

**Module 6: **Absolute Value Functions

**Module 7: **Inverse Functions

**Module 8: **Linear Functions

**Module 9: **Graphs of Linear Functions

**Module 10: **Modeling with Linear Functions

**Module 11: **Fitting Linear Models to Data

**Module 12: **Complex Numbers

**Module 13: **Quadratic Functions

**Module 14: **Power Functions and Polynomial Functions

**Module 15: **Graphs of Polynomial Functions

**Module 16: **Dividing Polynomials

**Module 17: **Zeros of Polynomial Functions

**Module 18: **Rational Functions

**Module 19: **Inverses and Radical Functions

**Module 20: **Modeling Using Variation

**Module 21: **Exponential Functions

**Module 22: **Graphs of Exponential Functions

**Module 23: **Logarithmic Functions

**Module 24: **Graphs of Logarithmic Functions

**Module 25: **Logarithmic Properties

**Module 26: **Exponential and Logarithmic Equations

**Module 27: **Exponential and Logarithmic Models

**Module 28: **Fitting Exponential Models to Data

**Module 29: **Systems of Linear Equations: Two Variables

**Module 30: **Systems of Linear Equations: Three Variables

**Module 31: **Matrices and Matrix Operations

**Module 32: **Solving Systems with Gaussian Elimination

**Module 33: **Solving Systems with Inverses

**Module 34: **Sequences and Their Notations

**Module 35: **Arithmetic Sequences

**Module 36: **Geometric Sequences

**Module 37: **Series and Their Notations

### Licenses and Attributions

- 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

- 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

- 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

- 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 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

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

- 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

- 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

- 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

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

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

- 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
- Determine a quadratic function’s minimum or maximum value
- Solve problems involving a quadratic function’s minimum or maximum value

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

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

- 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

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

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

- Graph exponential functions
- Graph exponential functions using transformations

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

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

- 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

- 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

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

- 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

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

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

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

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

- 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

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

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

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