Precalculus

OpenStax

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0.1 About openstax

0.2 About openstax’s resources

0.3 Customization

0.6 About Precalculus

0.7 Coverage and scope

0.8 Development overview

0.9 Accuracy of the content

0.10 Pedagogical foundations and features

0.11 Learning objectives

0.12 Narrative text

0.15 Supporting features

0.16 Section exercises

0.17 Chapter review features

0.18 Ancillaries

0.19 Online homework

0.20 About our team

0.21 Lead author, senior content expert

0.22 Contributing authors

0.23 Faculty reviewers and consultants

1 Basic functions and identities

Introduction to functions

1 Functions and function notation

1.1 Determining whether a relation represents a function

1.2 Using function notation

1.3 Representing functions using tables

1.4 Finding input and output values of a function

1.5 Evaluation of functions in algebraic forms

1.6 Evaluating functions expressed in formulas

1.7 Evaluating a function given in tabular form

1.8 Finding function values from a graph

1.9 Determining whether a function is one-to-one

1.10 Using the vertical line test

1.11 Using the horizontal line test

1.12 Identifying basic toolkit functions

1.13 Key equations

1.14 Key concepts

1.15 Section exercises

1.20 Technology

1.21 Real-world applications

2 Domain and range

2.1 Finding the domain of a function defined by an equation

2.2 Using notations to specify domain and range

2.3 Finding domain and range from graphs

2.4 Finding domains and ranges of the toolkit functions

2.5 Graphing piecewise-defined functions

2.6 Key concepts

2.7 Section exercises

2.12 Technology

2.14 Real-world applications

3 Rates of change and behavior of graphs

4 Composition of functions

4.1 Combining functions using algebraic operations

4.2 Create a function by composition of functions

4.3 Evaluating composite functions

4.4 Evaluating composite functions using tables

4.5 Evaluating composite functions using graphs

4.6 Evaluating composite functions using formulas

4.7 Finding the domain of a composite function

4.8 Decomposing a composite function into its component functions

4.9 Key equation

4.10 Key concepts

4.11 Section exercises

4.16 Extensions

4.17 Real-world applications

5 Transformation of functions

5.1 Graphing functions using vertical and horizontal shifts

5.2 Identifying vertical shifts

5.3 Identifying horizontal shifts

5.4 Combining vertical and horizontal shifts

5.5 Graphing functions using reflections about the axes

5.6 Determining even and odd functions

5.7 Graphing functions using stretches and compressions

5.8 Vertical stretches and compressions

5.9 Horizontal stretches and compressions

5.10 Performing a sequence of transformations

5.11 Key equations

5.12 Key concepts

5.13 Section exercises

6 Absolute value functions

6.1 Understanding absolute value

6.2 Graphing an absolute value function

6.3 Solving an absolute value equation

6.4 Solving an absolute value inequality

6.5 Key concepts

6.6 Section exercise

6.10 Technology

6.11 Extensions

6.12 Real-world applications

7 Inverse functions

7.1 Verifying that two functions are inverse functions

7.2 Finding domain and range of inverse functions

7.3 Finding and evaluating inverse functions

7.4 Inverting tabular functions

7.5 Evaluating the inverse of a function, given a graph of the original

7.6 Finding inverses of functions represented by formulas

7.7 Finding inverse functions and their graphs

7.8 Key concepts

7.9 Section exercises

7.14 Technology

7.15 Real-world applications

7.16 Chapter review exercises

7.17 Functions and Function Notation

7.18 Domain and Range

7.19 Rates of Change and Behavior of Graphs

7.20 Composition of Functions

7.21 Transformation of Functions

7.22 Absolute Value Functions

7.23 Inverse Functions

7.24 Practice test

Linear functions

Introduction to linear functions

1 Linear functions

1.1 Representing linear functions

1.2 Representing a linear function in word form

1.3 Representing a linear function in function notation

1.4 Representing a linear function in tabular form

1.5 Representing a linear function in graphical form

1.6 Determining whether a linear function is increasing, decreasing, or

1.7 Calculating and interpreting slope

1.8 Writing the point-slope form of a linear equation

1.9 Writing the equation of a line using a point and the slope

1.10 Writing the equation of a line using two points

1.11 Writing and interpreting an equation for a linear function

1.12 Modeling real-world problems with linear functions

1.13 Key equations

1.14 Key concepts

1.15 Section exercises

1.20 Technology

1.21 Extensions

1.22 Real-world applications

2 Graphs of linear functions

2.1 Graphing linear functions

2.2 Graphing a function by plotting points

2.3 Graphing a function using y- Intercept and slope

2.4 Graphing a function using transformations

2.5 Vertical stretch or compression

2.6 Vertical shift

2.7 Writing the equation for a function from the graph of a line

2.8 Finding the x -intercept of a line

2.9 Describing horizontal and vertical lines

2.10 Determining whether lines are parallel or perpendicular

2.11 Writing the equation of a line parallel or perpendicular to a given

2.12 Writing equations of parallel lines

2.13 Writing equations of perpendicular lines

2.14 Solving a system of linear equations using a graph

2.15 Key concepts

2.16 Section exercises

2.20 Extensions

2.21 Real-world applications

3 Modeling with linear functions

4 Fitting linear models to data

4.1 Drawing and interpreting scatter plots

4.2 Finding the line of best fit

4.3 Recognizing interpolation or extrapolation

4.4 Finding the line of best fit using a graphing utility

4.5 Distinguishing between linear and non-linear models

4.6 Predicting with a regression line

4.7 Key concepts

4.8 Section exercises

4.13 Technology

4.14 Extensions

4.15 Real-world applications

4.16 Chapter review exercises

4.17 Linear Functions

4.18 Graphs of Linear Functions

4.19 Modeling with Linear Functions

4.20 Fitting Linear Models to Data

4.21 Practice test

Polynomial and rational functions

Introduction to polynomial and rational functions

1 Complex numbers

2 Quadratic functions

2.1 Recognizing characteristics of parabolas

2.2 Understanding how the graphs of parabolas are related to their quadratic

2.3 Finding the domain and range of a quadratic function

2.4 Determining the maximum and minimum values of quadratic functions

2.5 Finding the x - and y -intercepts of a quadratic function

2.6 Rewriting quadratics in standard form

2.7 Key equations

2.8 Key concepts

2.9 Section exercises

2.14 Technology

2.15 Extensions

2.16 Real-world applications

3 Power functions and polynomial functions

3.1 Identifying power functions

3.2 Identifying end behavior of power functions

3.3 Identifying polynomial functions

3.4 Identifying the degree and leading coefficient of a polynomial function

3.5 Identifying end behavior of polynomial functions

3.6 Identifying local behavior of polynomial functions

3.7 Comparing smooth and continuous graphs

3.8 Key equations

3.9 Key concepts

3.10 Section exercises

3.15 Technology

3.16 Extensions

3.17 Real-world applications

4 Graphs of polynomial functions

5 Dividing polynomials

5.1 Using long division to divide polynomials

5.2 Using synthetic division to divide polynomials

5.3 Using polynomial division to solve application problems

5.4 Key equations

5.5 Key concepts

5.6 Section exercises

5.10 Technology

5.11 Extensions

5.12 Real-world applications

6 Zeros of polynomial functions

7 Rational functions

7.1 Using arrow notation

7.2 Local behavior of f ( x ) = 1 x

7.3 End behavior of f ( x ) = 1 x

7.4 Solving applied problems involving rational functions

7.5 Finding the domains of rational functions

7.6 Identifying vertical asymptotes of rational functions

7.7 Vertical asymptotes

7.8 Removable discontinuities

7.9 Identifying horizontal asymptotes of rational functions

7.10 Graphing rational functions

7.11 Writing rational functions

7.12 Key equations

7.13 Key concepts

7.14 Section exercises

7.19 Technology

7.20 Extensions

7.21 Real-world applications

8 Inverses and radical functions

8.1 Finding the inverse of a polynomial function

8.2 Restricting the domain to find the inverse of a polynomial function

8.3 Solving applications of radical functions

8.4 Solving applications of radical functions

8.5 Determining the domain of a radical function composed with other functions

8.6 Finding inverses of rational functions

8.7 Key concepts

8.8 Section exercises

8.12 Technology

8.13 Extensions

8.14 Real-world applications

9 Modeling using variation

9.1 Solving direct variation problems

9.2 Solving inverse variation problems

9.3 Solving problems involving joint variation

9.4 Key equations

9.5 Key concepts

9.6 Section exercises

9.10 Technology

9.11 Extensions

9.12 Real-world applications

9.13 Chapter review exercises

9.14 Complex Numbers

9.15 Quadratic Functions

9.16 Power Functions and Polynomial Functions

9.17 Graphs of Polynomial Functions

9.18 Dividing Polynomials

9.19 Zeros of Polynomial Functions

9.20 Rational Functions

9.21 Inverses and Radical Functions

9.22 Modeling Using Variation

9.23 Chapter test

Exponential and logarithmic functions

Introduction to exponential and logarithmic functions

1 Exponential functions

1.1 Identifying exponential functions

1.2 Defining an exponential function

1.3 Evaluating exponential functions

1.4 Defining exponential growth

1.5 Finding equations of exponential functions

1.6 Applying the compound-interest formula

1.7 Evaluating functions with base e

1.8 Investigating continuous growth

1.9 Key equations

1.10 Key concepts

1.11 Section exercises

1.15 Technology

1.16 Extensions

1.17 Real-world applications

2 Graphs of exponential functions

3 Logarithmic functions

3.1 Converting from logarithmic to exponential form

3.2 Converting from exponential to logarithmic form

3.3 Evaluating logarithms

3.4 Using common logarithms

3.5 Using natural logarithms

3.6 Key equations

3.7 Key concepts

3.8 Section exercises

3.12 Technology

3.13 Extensions

3.14 Real-world applications

4 Graphs of logarithmic functions

4.1 Finding the domain of a logarithmic function

4.2 Graphing logarithmic functions

4.3 Graphing transformations of logarithmic functions

4.4 Graphing a horizontal shift of f ( x ) = log b ( x )

4.5 Graphing a vertical shift of y = log b ( x )

4.6 Graphing stretches and compressions of y = log b ( x )

4.7 Graphing reflections of f ( x ) = log b ( x )

4.8 Summarizing translations of the logarithmic function

4.9 Key equations

4.10 Key concepts

4.11 Section exercises

4.15 Technology

4.16 Extensions

5 Logarithmic properties

6 Exponential and logarithmic equations

6.1 Using like bases to solve exponential equations

6.2 Rewriting equations so all powers have the same base

6.3 Solving exponential equations using logarithms

6.4 Equations containing e

6.5 Extraneous solutions

6.6 Using the definition of a logarithm to solve logarithmic equations

6.7 Using the one-to-one property of logarithms to solve logarithmic equations

6.8 Solving applied problems using exponential and logarithmic equations

6.9 Key equations

6.10 Key concepts

6.11 Section exercises

6.15 Technology

6.16 Extensions

7 Exponential and logarithmic models

7.1 Modeling exponential growth and decay

7.3 Radiocarbon dating

7.4 Calculating doubling time

7.5 Using newton’s law of cooling

7.6 Using logistic growth models

7.7 Choosing an appropriate model for data

7.8 Expressing an exponential model in base e

7.9 Key equations

7.10 Key concepts

7.11 Section exercises

7.14 Technology

7.15 Extensions

7.16 Real-world applications

8 Fitting exponential models to data

8.1 Building an exponential model from data

8.2 Building a logarithmic model from data

8.3 Building a logistic model from data

8.4 Key concepts

8.5 Section exercises

8.10 Extensions

8.11 Chapter review exercises

8.12 Exponential Functions

8.13 Graphs of Exponential Functions

8.14 Logarithmic Functions

8.15 Graphs of Logarithmic Functions

8.16 Logarithmic Properties

8.17 Exponential and Logarithmic Equations

8.18 Exponential and Logarithmic Models

8.19 Fitting Exponential Models to Data

8.20 Practice test

Trigonometric functions

Introduction to trigonometric functions

1.1 Drawing angles in standard position

1.2 Converting between degrees and radians

1.3 Relating arc lengths to radius

1.4 Using radians

1.5 Identifying special angles measured in radians

1.6 Converting between radians and degrees

1.7 Finding coterminal angles

1.8 Finding coterminal angles measured in radians

1.9 Determining the length of an arc

1.10 Finding the area of a sector of a circle

1.11 Use linear and angular speed to describe motion on a circular path

1.12 Key equations

1.13 Key concepts

1.14 Section exercises

1.18 Real-world applications

1.19 Extensions

2 Unit circle: sine and cosine functions

3 The other trigonometric functions

4 Right triangle trigonometry

4.1 Using right triangles to evaluate trigonometric functions

4.2 Understanding right triangle relationships

4.3 Relating angles and their functions

4.4 Finding trigonometric functions of special angles using side lengths

4.5 Using equal cofunction of complements

4.6 Using trigonometric functions

4.7 Using right triangle trigonometry to solve applied problems

4.8 Key equations

4.9 Key concepts

4.10 Section exercises

4.14 Technology

4.15 Extensions

4.16 Real-world applications

4.17 Review exercises

4.19 Unit Circle: Sine and Cosine Functions

4.20 The Other Trigonometric Functions

4.21 Right Triangle Trigonometry

4.22 Practice test

Periodic functions

Introduction to periodic functions

1 Graphs of the sine and cosine functions

2 Graphs of the other trigonometric functions

3 Inverse trigonometric functions

Trigonometric identities and equations

Introduction to trigonometric identities and equations

1 Solving trigonometric equations with identities

2 Sum and difference identities

3 Double-angle, half-angle, and reduction formulas

4 Sum-to-product and product-to-sum formulas

5 Solving trigonometric equations

6 Modeling with trigonometric equations

Further applications of trigonometry

Introduction to further applications of trigonometry

1 Non-right triangles: law of sines

2 Non-right triangles: law of cosines

3 Polar coordinates

4 Polar coordinates: graphs

5 Polar form of complex numbers

6 Parametric equations

6.1 Parameterizing a curve

6.2 Eliminating the parameter

6.3 Eliminating the parameter from polynomial, exponential, and logarithmic

6.4 Eliminating the parameter from trigonometric equations

6.5 Finding cartesian equations from curves defined parametrically

6.6 Finding parametric equations for curves defined by rectangular equations

6.7 Key concepts

6.8 Section exercises

6.11 Technology

6.12 Extensions

7 Parametric equations: graphs

7.1 Graphing parametric equations by plotting points

7.2 Applications of parametric equations

7.3 Key concepts

7.4 Section exercises

8.1 A geometric view of vectors

8.2 Finding magnitude and direction

8.3 Performing vector addition and scalar multiplication

8.4 Multiplying by a scalar

8.5 Finding component form

8.6 Finding the unit vector in the direction of v

8.7 Performing operations with vectors in terms of i And j

8.8 Performing operations on vectors in terms of i And j

8.9 Calculating the component form of a vector: direction

8.10 Finding the dot product of two vectors

8.11 Key concepts

8.12 Section exercises

8.16 Extensions

8.17 Real-world applications

8.18 Chapter review exercises

8.19 Non-right Triangles: Law of Sines

8.20 Non-right Triangles: Law of Cosines

8.21 Polar Coordinates

8.22 Polar Coordinates: Graphs

8.23 Polar Form of Complex Numbers

8.24 Parametric Equations

8.25 Parametric Equations: Graphs

8.27 Practice test

Systems of equations and inequalities

Introduction to systems of equations and inequalities

1 Systems of linear equations: two variables

1.1 Introduction to systems of equations

1.2 Solving systems of equations by graphing

1.3 Solving systems of equations by substitution

1.4 Solving systems of equations in two variables by the addition method

1.5 Identifying inconsistent systems of equations containing two variables

1.6 Expressing the solution of a system of dependent equations containing

1.7 Using systems of equations to investigate profits

1.8 Key concepts

1.9 Section exercises

1.13 Technology

1.14 Extensions

1.15 Real-world applications

2 Systems of linear equations: three variables

3 Systems of nonlinear equations and inequalities: two variables

4 Partial fractions

5 Matrices and matrix operations

5.1 Finding the sum and difference of two matrices

5.2 Describing matrices

5.3 Adding and subtracting matrices

5.4 Finding scalar multiples of a matrix

5.5 Finding the product of two matrices

5.6 Key concepts

5.7 Section exercises

5.10 Technology

5.11 Extensions

6 Solving systems with gaussian elimination

7 Solving systems with inverses

7.1 Finding the inverse of a matrix

7.2 Finding the multiplicative inverse using matrix multiplication

7.3 Finding the multiplicative inverse by augmenting with the identity

7.4 Finding the multiplicative inverse of 2×2 matrices using a formula

7.5 Finding the multiplicative inverse of 3×3 matrices

7.6 Solving a system of linear equations using the inverse of a matrix

7.7 Key equations

7.8 Key concepts

7.9 Section exercises

7.12 Technology

7.13 Extensions

7.14 Real-world applications

8 Solving systems with cramer's rule

8.1 Evaluating the determinant of a 2×2 matrix

8.2 Using cramer’s rule to solve a system of two equations in two variables

8.3 Evaluating the determinant of a 3 × 3 matrix

8.4 Using cramer’s rule to solve a system of three equations in three

8.5 Understanding properties of determinants

8.6 Key concepts

8.7 Section exercises

8.10 Technology

8.11 Real-world applications

8.12 Review exercises

8.13 Systems of Linear Equations: Two Variables

8.14 Systems of Linear Equations: Three Variables

8.15 Systems of Nonlinear Equations and Inequalities: Two Variables

8.16 Partial Fractions

8.17 Matrices and Matrix Operations

8.18 Solving Systems with Gaussian Elimination

8.19 Solving Systems with Inverses

8.20 Solving Systems with Cramer's Rule

8.21 Practice test

Analytic geometry

Introduction to analytic geometry

2 The hyperbola

3.1 Graphing parabolas with vertices at the origin

3.2 Writing equations of parabolas in standard form

3.3 Graphing parabolas with vertices not at the origin

3.4 Solving applied problems involving parabolas

3.5 Key equations

3.6 Key concepts

3.7 Section exercises

3.11 Extensions

3.12 Real-world applications

4 Rotation of axes

5 Conic sections in polar coordinates

Sequences, probability and counting theory

Introduction to sequences, probability and counting theory

1 Sequences and their notations

2 Arithmetic sequences

2.1 Finding common differences

2.2 Writing terms of arithmetic sequences

2.3 Using recursive formulas for arithmetic sequences

2.4 Using explicit formulas for arithmetic sequences

2.5 Finding the number of terms in a finite arithmetic sequence

2.6 Solving application problems with arithmetic sequences

2.7 Key equations

2.8 Key concepts

2.9 Section exercises

2.13 Technology

2.14 Extensions

3 Geometric sequences

4 Series and their notations

5 Counting principles

6 Binomial theorem

6.1 Identifying binomial coefficients

6.2 Using the binomial theorem

6.3 Using the binomial theorem to find a single term

6.4 Key equations

6.5 Key concepts

6.6 Section exercises

6.10 Extensions

7.1 Constructing probability models

7.2 Computing probabilities of equally likely outcomes

7.3 Computing the probability of the union of two events

7.4 Computing the probability of mutually exclusive events

7.5 Using the complement rule to compute probabilities

7.6 Computing probability using counting theory

7.7 Key equations

7.8 Key concepts

7.9 Section exercises

7.12 Extensions

7.13 Real-world applications

7.14 Chapter review exercises

7.15 Sequences and Their Notation

7.16 Arithmetic Sequences

7.17 Geometric Sequences

7.18 Series and Their Notation

7.19 Counting Principles

7.20 Binomial Theorem

7.21 Probability

7.22 Practice test

Introduction to calculus

Introduction to calculus

1 Finding limits: numerical and graphical approaches

1.1 Understanding limit notation

1.2 Understanding left-hand limits and right-hand limits

1.3 Understanding two-sided limits

1.4 Finding a limit using a graph

1.5 Finding a limit using a table

1.6 Key concepts

1.7 Section exercises

1.11 Extensions

2 Finding limits: properties of limits

Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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