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A. Function concepts

• 1 Domain and range
• 2 Identify functions
• 3 Evaluate functions
• 4 Find values using function graphs
• 5 Complete a table for a function graph
• 6 Identify graphs: word problems
• 7 Add, subtract, multiply, and divide functions
• 8 Composition of functions

B. Inverse functions

• 1 Identify inverse functions
• 2 Find values of inverse functions from tables
• 3 Find values of inverse functions from graphs
• 4 Find inverse functions and relations

C. Function transformations

• 1 Function transformation rules
• 2 Translations of functions
• 3 Reflections of functions
• 4 Dilations of functions
• 5 Transformations of functions
• 6 Describe function transformations

D. Quadratic functions

• 1 Find the maximum or minimum value of a quadratic function
• 2 Characteristics of quadratic functions
• 3 Graph a quadratic function
• 4 Match quadratic functions and graphs
• 5 Solve a quadratic equation using square roots
• 6 Solve a quadratic equation by factoring
• 7 Solve a quadratic equation by completing the square
• 8 Solve a quadratic equation using the quadratic formula
• 9 Solve quadratic equations: word problems
• 10 Using the discriminant

E. Polynomial expressions and equations

• 1 Polynomial vocabulary
• 2 Divide polynomials using long division
• 3 Divide polynomials using synthetic division
• 4 Evaluate polynomials using synthetic division
• 5 Factor sums and differences of cubes
• 6 Solve equations with sums and differences of cubes
• 7 Factor using a quadratic pattern
• 8 Solve equations using a quadratic pattern
• 9 Pascal's triangle
• 10 Pascal's triangle and the Binomial Theorem
• 11 Binomial Theorem I
• 12 Binomial Theorem II

F. Polynomial functions

• 1 Find the roots of factored polynomials
• 2 Write a polynomial from its roots
• 3 Rational root theorem
• 4 Complex conjugate theorem
• 5 Conjugate root theorems
• 6 Descartes' Rule of Signs
• 7 Fundamental Theorem of Algebra
• 8 Match polynomials and graphs using end behavior
• 9 Match polynomials and graphs using zeros
• 10 Match polynomials and graphs
• 11 Domain and range of polynomials
• 12 Even and odd functions

G. Rational functions

• 1 Rational functions: asymptotes and excluded values
• 2 Solve rational equations
• 3 Check whether two rational functions are inverses

H. Exponential functions

• 1 Domain and range of exponential functions
• 2 Match exponential functions and graphs
• 3 Linear and exponential functions over unit intervals
• 4 Identify linear and exponential functions
• 5 Describe linear and exponential growth and decay

I. Logarithms

• 1 Convert between exponential and logarithmic form
• 2 Evaluate logarithms
• 3 Change of base formula
• 4 Product property of logarithms
• 5 Quotient property of logarithms
• 6 Power property of logarithms
• 7 Properties of logarithms: mixed review
• 8 Evaluate logarithms using properties

J. Logarithmic functions

• 1 Domain and range of logarithmic functions
• 2 Graph logarithmic functions

K. Exponential and logarithmic equations

• 1 Solve exponential equations by rewriting the base
• 2 Solve exponential equations using logarithms
• 3 Solve logarithmic equations with one logarithm
• 4 Solve logarithmic equations with multiple logarithms
• 5 Exponential growth and decay: word problems
• 6 Compound interest: word problems

L. Radical expressions and functions

• 1 Simplify radical expressions with variables
• 2 Nth roots
• 3 Domain and range of radical functions
• 4 Graph square root functions
• 5 Solve radical equations

M. Rational exponents

• 1 Evaluate rational exponents
• 2 Convert between rational exponents and radicals
• 3 Operations with rational exponents
• 4 Simplify expressions involving rational exponents
• 5 Solve equations with rational exponents

N. Systems of equations

• 1 Solve a system of equations by graphing
• 2 Solve a system of equations by graphing: word problems
• 3 Classify a system of equations
• 4 Solve a system of equations using substitution
• 5 Solve a system of equations using substitution: word problems
• 6 Solve a system of equations using elimination
• 7 Solve a system of equations using elimination: word problems
• 8 Solve a system of equations using augmented matrices
• 9 Solve a system of equations using augmented matrices: word problems
• 10 Solve a system of equations in three variables using substitution
• 11 Solve a system of equations in three variables using elimination
• 12 Determine the number of solutions to a system of equations in three variables

O. Systems of inequalities

• 1 Solve systems of linear inequalities by graphing
• 2 Solve systems of linear and absolute value inequalities by graphing
• 3 Find the vertices of a solution set
• 4 Linear programming

P. Nonlinear inequalities

• 1 Graph solutions to quadratic inequalities
• 2 Solve quadratic inequalities
• 3 Graph solutions to higher-degree inequalities
• 4 Solve higher-degree inequalities

Q. Matrices

• 1 Matrix vocabulary
• 2 Matrix operation rules
• 3 Add and subtract matrices
• 4 Multiply a matrix by a scalar
• 5 Add and subtract scalar multiples of matrices
• 6 Multiply two matrices
• 7 Simplify matrix expressions
• 8 Properties of matrices
• 9 Solve matrix equations
• 10 Determinant of a matrix
• 11 Is a matrix invertible?
• 12 Inverse of a 2 x 2 matrix
• 13 Inverse of a 3 x 3 matrix
• 14 Identify inverse matrices
• 15 Solve matrix equations using inverses
• 16 Identify transformation matrices
• 17 Transformation matrices: write the vertex matrix
• 18 Transformation matrices: graph the image

R. Trigonometry

• 1 Convert between radians and degrees
• 2 Radians and arc length
• 3 Quadrants
• 4 Coterminal and reference angles
• 5 Find trigonometric ratios using right triangles
• 6 Find trigonometric ratios using the unit circle
• 7 Find trigonometric ratios of special angles
• 8 Find trigonometric ratios using reference angles
• 9 Inverses of trigonometric functions
• 10 Inverses of trigonometric functions using a calculator
• 11 Solve trigonometric equations
• 12 Trigonometric ratios: find a side length
• 13 Trigonometric ratios: find an angle measure
• 14 Solve a right triangle
• 15 Law of Sines
• 16 Law of Cosines
• 17 Solve a triangle
• 18 Area of a triangle: sine formula
• 19 Area of a triangle: Heron's formula

S. Trigonometric functions

• 1 Find properties of sine functions
• 2 Write equations of sine functions from graphs
• 3 Write equations of sine functions using properties
• 4 Graph sine functions
• 5 Graph translations of sine functions
• 6 Find properties of cosine functions
• 7 Write equations of cosine functions from graphs
• 8 Write equations of cosine functions using properties
• 9 Graph cosine functions
• 10 Graph translations of cosine functions
• 11 Find properties of sine and cosine functions
• 12 Write equations of sine and cosine functions from graphs
• 13 Write equations of sine and cosine functions using properties
• 14 Graph sine and cosine functions
• 15 Graph translations of sine and cosine functions

T. Trigonometric identities

• 1 Complementary angle identities
• 2 Symmetry and periodicity of trigonometric functions
• 3 Find trigonometric ratios using a Pythagorean or reciprocal identity
• 4 Find trigonometric ratios using multiple identities
• 5 Trigonometric sum and difference identities
• 6 Solve trigonometric equations using sum and difference identities

U. Conic sections

• 1 Find properties of parabolas
• 2 Write equations of parabolas in vertex form
• 3 Graph parabolas
• 4 Find properties of circles
• 5 Write equations of circles in standard form
• 6 Graph circles
• 7 Find properties of ellipses
• 8 Find the eccentricity of an ellipse
• 9 Write equations of ellipses in standard form
• 10 Find properties of hyperbolas
• 11 Find the eccentricity of a hyperbola
• 12 Write equations of hyperbolas in standard form
• 13 Convert equations of conic sections from general to standard form

V. Complex numbers

• 1 Add and subtract complex numbers
• 2 Complex conjugates
• 3 Multiply and divide complex numbers
• 4 Add, subtract, multiply, and divide complex numbers
• 5 Absolute values of complex numbers
• 6 Powers of i

W. Complex plane

• 1 Introduction to the complex plane
• 2 Graph complex numbers
• 3 Addition in the complex plane
• 4 Subtraction in the complex plane
• 5 Graph complex conjugates
• 6 Absolute value in the complex plane
• 7 Midpoints in the complex plane
• 8 Distance in the complex plane

X. Polar form

• 1 Find the modulus and argument of a complex number
• 2 Convert complex numbers from rectangular to polar form
• 3 Convert complex numbers from polar to rectangular form
• 4 Convert complex numbers between rectangular and polar form
• 5 Match polar equations and graphs

Y. Two-dimensional vectors

• 1 Find the magnitude of a vector
• 2 Find the component form of a vector
• 3 Find the direction angle of a vector
• 4 Find the component form of a vector from its magnitude and direction angle
• 5 Graph a resultant vector using the triangle method
• 6 Graph a resultant vector using the parallelogram method
• 7 Add vectors
• 8 Subtract vectors
• 9 Find the magnitude and direction of a vector sum
• 10 Multiply a vector by a scalar
• 11 Find the magnitude of a vector scalar multiple
• 12 Determine the direction of a vector scalar multiple
• 13 Find a unit vector
• 14 Linear combinations of vectors

Z. Three-dimensional vectors

• 1 Find the magnitude of a three-dimensional vector
• 2 Find the component form of a three-dimensional vector
• 3 Add and subtract three-dimensional vectors
• 4 Scalar multiples of three-dimensional vectors
• 5 Find a three-dimensional unit vector
• 6 Linear combinations of three-dimensional vectors

AA. Sequences

• 1 Find terms of a sequence
• 2 Find terms of a recursive sequence
• 3 Identify a sequence as explicit or recursive
• 4 Find a recursive formula
• 5 Find recursive and explicit formulas
• 6 Convert a recursive formula to an explicit formula
• 7 Convert an explicit formula to a recursive formula
• 8 Convert between explicit and recursive formulas
• 1 Identify arithmetic and geometric series
• 2 Introduction to sigma notation
• 3 Find the sum of an arithmetic series
• 4 Find the sum of a finite geometric series
• 5 Introduction to partial sums
• 6 Partial sums of arithmetic series
• 7 Partial sums of geometric series
• 8 Partial sums: mixed review
• 9 Convergent and divergent geometric series
• 10 Find the value of an infinite geometric series
• 11 Write a repeating decimal as a fraction

CC. Probability

• 1 Calculate probabilities of events
• 2 Combinations and permutations
• 3 Find probabilities using combinations and permutations
• 4 Find probabilities using two-way frequency tables
• 5 Identify independent events
• 6 Find conditional probabilities
• 7 Independence and conditional probability
• 8 Find conditional probabilities using two-way frequency tables
• 9 Find probabilities using the addition rule

DD. Probability distributions

• 1 Identify probability distributions
• 2 Identify discrete and continuous random variables
• 3 Write a discrete probability distribution
• 4 Graph a discrete probability distribution
• 5 Expected values of random variables
• 6 Variance of random variables
• 7 Standard deviation of random variables
• 8 Write the probability distribution for a game of chance
• 9 Expected values for a game of chance
• 10 Choose the better bet

EE. Binomial and normal distributions

• 1 Find probabilities using the binomial distribution
• 2 Mean, variance, and standard deviation of binomial distributions
• 3 Find probabilities using the normal distribution I
• 4 Find probabilities using the normal distribution II
• 5 Find z-values
• 6 Find values of normal variables
• 7 Distributions of sample means
• 8 The Central Limit Theorem
• 9 Use normal distributions to approximate binomial distributions

FF. Single-variable statistics

• 1 Identify biased samples
• 2 Variance and standard deviation
• 3 Identify an outlier
• 4 Identify an outlier and describe the effect of removing it
• 5 Find confidence intervals for population means
• 6 Find confidence intervals for population proportions
• 7 Interpret confidence intervals for population means
• 8 Experiment design
• 9 Analyze the results of an experiment using simulations

GG. Bivariate statistics

• 1 Outliers in scatter plots
• 2 Match correlation coefficients to scatter plots
• 3 Calculate correlation coefficients
• 4 Find the equation of a regression line
• 5 Interpret regression lines
• 6 Analyze a regression line of a data set
• 7 Analyze a regression line using statistics of a data set
• • New! Exponential regression
• 1 Find limits using graphs
• 2 Find one-sided limits using graphs
• 3 Determine if a limit exists
• 4 Find limits at vertical asymptotes using graphs
• 5 Determine end behavior using graphs
• 6 Find limits using addition, subtraction, and multiplication laws
• 7 Find limits using the division law
• 8 Find limits using power and root laws
• 9 Find limits using limit laws
• 10 Find limits of polynomials and rational functions
• 11 Find limits involving factorization and rationalization

II. Continuity

• 1 Identify graphs of continuous functions
• 2 Determine continuity using graphs
• 3 Determine one-sided continuity using graphs
• 4 Find and analyze points of discontinuity using graphs
• 5 Determine continuity on an interval using graphs

JJ. Introduction to derivatives

• 1 Average rate of change I
• 2 Average rate of change II
• 3 Find instantaneous rates of change
• 4 Velocity as a rate of change
• 5 Find values of derivatives using limits
• 6 Find the slope of a tangent line using limits
• 7 Find equations of tangent lines using limits

Precalculus Examples

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• Points, Lines, and Line Segments
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• Radical Expressions and Equations
• Rational Expressions and Equations
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• Factoring Polynomials
• Conic Sections
• Exponential and Logarithmic Functions
• Trigonometry
• Analytic Trigonometry
• Inequalities
• Linear Equations
• Systems of Equations
• Quadratic Equations
• Sequences and Series
• Analytic Geometry in Rectangular Coordinates
• Analytic Geometry in Polar Coordinates
• Limits and an Introduction to Calculus
• Number Sets
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PreCalculus

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The topics covered are: Lines and Planes, Functions and Transformation of Graphs, Polynomials, Rational Functions, Limits of a Function, Complex Numbers, Exponential Functions, Logarithmic Functions, Conic Sections, Matrices, Sequences and Series, Probability and Combinatorics, Advanced Trigonometry, Vectors and Parametric Equations, and Polar Coordinates.

Parallel Lines, Skew Lines & Planes Proving Parallel Lines

Difference Quotient of Function I Difference Quotient of Function II Domain of a Function Greatest Integer Function I Greatest Integer Function II Piece-Wise Functions I Piece-Wise Functions II Piece-Wise Functions III Even & Odd Functions I Even & Odd Functions II Composite Functions I Composite Functions II Composite Functions III One-to-one Functions Inverse of a Function I Inverse of a Function II Finding the Inverse of a Function or Showing One Does not Exist

Parent Functions & their Graphs Transformation of Linear Functions Transformation of Quadratic Functions Horizontal & Vertical Graph Transformations Horizontal & Vertical Graph Stretches & Compressions Reflective Transformations Examples of Graph Transformations

Power Functions Polynomial Functions Graphing Polynomial Functions I Graphing Polynomial Functions II Families of Polynomials Graphing Polynomials with Repeated Factors Finding the Equation of a Polynomial Function Finding Maximum & Minimum Values of Polynomial Functions Find the Zeros of a Polynomial Function I Find the Zeros of a Polynomial Function II Conjugate Zeros Theorem Reciprocal of a Function

Introduction to Rational Functions Limits of Rational Functions Graphing Rational Functions I Graphing Rational Functions II Horizontal Asymptotes I Horizontal Asymptotes II Vertical Asymptotes Vertical & Horizontal Asymptotes Oblique or Slant Asymptotes Find all Asymptotes of a Rational Function Graphing Rational Functions with Holes Rational Function Problems

Introduction to Limits of Functions Limits of Rational Functions Examples to Learn about Limits Calculating Limits using Different Techniques I Calculating Limits using Different Techniques II Limit Laws to Evaluate a Limit One Sided Limits Squeeze Theorem for Limits The Limit definition of Continuity

Introduction to Complex Numbers I Introduction to Complex Numbers II Adding & Subtracting Complex Numbers Multiplying Complex Numbers Dividing Complex Numbers Trigonometric or Polar Form of Complex Numbers Converting Complex Numbers between Trigonometric Form & Rectangular Multiplying & Dividing Complex Numbers in Trigonometric or Polar Form DeMoivre’s Theorem & Euler Formula Roots of a Complex Number Complex Quadratic Equations I Complex Quadratic Equations II Simplify Complex Rational Expressions

Introduction to Exponential Functions Graphing Exponential Functions I Graphing Exponential Functions II Applying Exponential Functions I Applying Exponential Functions II Solving Exponential Equations with the Same Base Solving Exponential Equations with Different Bases I Solving Exponential Equations with Different Bases II Solving Exponential Equations with Different Bases III

Introduction to Logarithmic Functions Exponential & Logarithmic Functions Logarithm Review Product Rule Logarithm Review Product & Quotient Rules Logarithm Review Power Rule Logarithm Review Change of Base Rule Properties of Logarithms (or Rules of Logarithms) Proof of Logarithm Properties Expanding Logarithmic Expressions Simplifying (or Condensing) Logarithmic Expressions Simplifying & Expanding Logarithmic Expressions Solving Logarithmic Equations I Solving Logarithmic Equations II Graphs of Logarithmic Functions

Introduction to Conic Sections Conic Sections - Circles I Conic Sections - Circles II Conic Sections - Ellipses I Conic Sections - Ellipses II Conic Sections - Parabolas Conic Sections - Hyperbolas I Conic Sections - Hyperbolas II Conic Sections - Hyperbolas III Conic Sections Summary Identify & Graph Conic Sections

Introduction to Matrices Matrix Addition & Subtraction I Matrix Addition & Subtraction II Matrix Scalar Multiplication Matrix Multiplication I Matrix Multiplication II Matrix Multiplication III Identity Matrix Determinant of a 2x2 Matrix Determinant of a 3x3 Matrix I Determinant of a 3x3 Matrix II Simplifying Determinant Inverse of 2x2 Matrix Inverse of 3x3 Matrix Singular Matrix - A Matrix with no Inverse Solving a 2x2 System of Equations Using a Matrix Inverse I Solving a 2x2 System of Equations Using a Matrix Inverse II Solving a 3x3 System of Equations Using a Matrix Inverse Using Gauss-Jordan to Solve a System of Three Linear Equations Row Reducing a Matrix to solve a System of Equations Solving a System of Equations using Matrix Row Transformations Cramer’s Rule Using Determinant to find the Area of a Parallelogram Using Determinant to find the Area of a Triangle & a Polygon Cross Product of Vectors & Determinants - area of triangles and parallelograms

Introduction to Sequences Arithmetic Sequences I Arithmetic Sequences II Arithmetic Sequences Finding the nth Term Geometric Sequences I Geometric Sequences II Geometric Sequences Finding the nth Term Recursion Sequences

Series and Summation Arithmetic Series I Arithmetic Series II Geometric Series I Geometric Series II Infinite Geometric Series

Mathematical Induction Examples of Mathematical Induction I Examples of Mathematical Induction II

Introduction to Probability I Introduction to Probability II Fundamental Counting Principles Permutations I Permutations II Permutations III Combinations I Combinations II Permutations & Combinations Probability using Permutations & Combinations Probability of Multiple Events Probability of Independent Events Probability of Dependent Events Probability of Complementary Events Conditional Probability I Conditional Probability II Conditional Probability III Pascal’s Triangle & the Binomial Theorem Binomial Theorem I Binomial Theorem II

Inverse Trigonometric Functions I Inverse Trigonometric Functions II Factoring Trigonometric Equations Trig Identities cofunctions, reciprocal, quotient, ratio, Pythagorean, & even/odd Sum & Difference Identities I Sum & Difference Identities II Sine and Cosine Addition Formulas (Proofs) Double Angle Identities I Double Angle Identities II Power Reducing Identities Half Angle Identities Half Angle & Double Angle Examples

Geometric Representation of Vectors Algebraic Representation of Vectors Components of a Vector Adding Vectors Graphically or Head-to-Tail Method Adding Vectors using Components I Adding Vectors using Components II Vector Addition & Scalar Multiplication Vector Magnitude & Direction The Resultant of Two Forces Components of a Force Solve Navigation Problems using Vectors Unit Vectors I Unit Vectors II Vector Equation of a Line Parametric Equations & Motion Parameterize a Line Segment & a Circle Parametric Equations Dot Product of Vectors Angle between Two Vectors Vectors in Three Dimensions Lines in 3D Coordinate Systems Vectors and Planes I Vectors and Planes II

Introduction to Polar Coordinates Conversion between Polar Coordinates & Rectangular Coordinates Polar Coordinates Distance Formula Equations of Lines in Polar Coordinates Convert between Polar Equations & Rectangular Equations Symmetry of Polar Graphs Graphing Polar Equations Families of Polar Curves: Circles, Cardiods, Limacon, Roses & Conic Sections

IIT JEE Trigonometry Problem IIT JEE Perpendicular Planes IIT JEE Complex Root Probability IIT JEE Position Vectors ITT JEE Matrix Equations IIT JEE Integral Limit IIT JEE Algebraic Manipulation IIT JEE Function Maxima IIT JEE Diameter Slope IIT JEE Hairy Trig & Algebra IIT JEE Complex Numbers IIT JEE Differentiability & Boundedness IIT JEE Integral with Binomial Expansion IIT JEE Symmetric & Skew-Symmetric Matrices IIT JEE Trace & Determinant IIT JEE Divisible Determinants IIT JEE Circle Hyperbola Intersection IIT JEE Circle Hyperbola Common Tangent IIT JEE Trigonometric Constraints IIT JEE Trigonometric Maximum Vector Triple Product Expansion (very optional) IIT JEE Lagrange’s Formula Tangent Line Hyperbola Relationship (very optional) 2010 IIT JEE Paper 1 Problem 50 Hyperbola Eccentricity Normal vector from plane equation Point distance to plane Distance Between Planes Periodic Definite Integral Complex Determinant Example Series Sum Example Trigonometric System Example Simple Differential Equation Example

PreCalculus Calculator with step by step solutions Relations, Functions, Operations on Functions, Points, Lines, & Line Segments, Absolute Value Expressions & Equations, Radical Expressions & Equations, Rational Expressions & Equations, Polynomial & Rational Functions, Factoring Polynomials, Conic Sections, Exponential & Logarithmic Functions, Trigonometry, Analytic Trigonometry, Inequalities, Linear Equations, Systems of Equations, Quadratic Equations Matrices, Sequences & Series, Analytic Geometry in Rectangular Coordinates, Analytic Geometry in Polar Coordinates, Limits & an Introduction to Calculus, Vectors, Number Sets

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Systems of Inequalities in Two Variables

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Geometry of a Parabola

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Geometry of an Ellipse

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Geometry of a Hyperbola

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Translation and Rotation of Axis

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Polar Equations of Conic Sections

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3-D Cartesian Coordinate System

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Limits, Motion, and the Tangent Line

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Limits, Motion, and Areas

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• Connection to Areas
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• Concepts and Informal Definition of a Limit
• Properties of Limits
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• One-Sided Limits
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• Limits Involving Infinity

Graphs of Trigonometric Functions

• Graphing Trigonometric Functions with Domain and Range
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To derive the function $x^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

Apply natural logarithm to both sides of the equality

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

Derive both sides of the equality with respect to $x$

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\ln\left(x\right)$

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 Intermediate steps

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About Precalculus 2e

Precalculus 2e is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course. The content is organized by clearly-defined learning objectives, and includes worked examples that demonstrate problem-solving approaches in an accessible way.

Coverage and Scope

Precalculus 2e contains twelve chapters, roughly divided into three groups.

Chapters 1-4 discuss various types of functions, providing a foundation for the remainder of the course.

• Chapter 1: Functions
• Chapter 2: Linear Functions
• Chapter 3: Polynomial and Rational Functions
• Chapter 4: Exponential and Logarithmic Functions

Chapters 5-8 focus on Trigonometry. In Precalculus 2e , we approach trigonometry by first introducing angles and the unit circle, as opposed to the right triangle approach more commonly used in college algebra and trigonometry courses.

• Chapter 5: Trigonometric Functions
• Chapter 6: Periodic Functions
• Chapter 7: Trigonometric Identities and Equations
• Chapter 8: Further Applications of Trigonometry

Chapters 9-12 present some advanced precalculus topics that build on topics introduced in chapters 1-8. Most precalculus syllabi include some of the topics in these chapters, but few include all. Instructors can select material as needed from this group of chapters, since they are not cumulative.

• Chapter 9: Systems of Equations and Inequalities
• Chapter 10: Analytic Geometry
• Chapter 11: Sequences, Probability and Counting Theory
• Chapter 12: Introduction to Calculus

All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.

Development Overview

Precalculus 2e is the product of a collaborative effort by a group of dedicated authors, editors, and instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision for the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading numerous drafts, and assimilating field reviews into actionable revision plans for our authors and editors.

The first eight chapters are built on the foundation of Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. Chapters 9-12 were written and developed by our expert and highly experienced author team . All twelve chapters follow a new and innovative instructional design, and great care has been taken to maintain a consistent voice from cover to cover. New features have been introduced to flesh out the instruction, all of the graphics have been redone in a more contemporary style, and much of the content has been revised, replaced, or supplemented to bring the text more in line with mainstream approaches to teaching precalculus.

Accuracy of the Content

We understand that precision and accuracy are imperatives in mathematics, and undertook an dedicated accuracy program led by experienced faculty. Examples, art, problems, and solutions were reviewed by dedicated faculty, with a separate team evaluating the answer key and solutions.

The text also benefits from years of usage by thousands of faculty and students. A core aspect of the second edition revision process included consolidating and ensuring consistency with regard to any errata and corrections that have been implemented during the series' extensive usage and incorporation into homework systems.

Changes to the Second Edition

The Precalculus 2e revision focused on mathematical clarity and accuracy as well as inclusivity. Examples, Exercises, and Solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates, driven by faculty and students from several thousand colleges, were considered and unified across the different formats of the text.

OpenStax and our authors are aware of the difficulties posed by shifting problem and exercise numbers when textbooks are revised. In an effort to make the transition to the 2nd edition as seamless as possible, we have minimized any shifting of exercise numbers.

The revision also focused on supporting inclusive and welcoming learning experiences. The introductory narratives, example and problem contexts, and even many of the names used for fictional people in the text were all reviewed using a diversity, equity, and inclusion framework. Several hundred resulting revisions improve the balance and relevance to the students using the text, while maintaining a variety of applications to diverse careers and academic fields. In particular, explanations of scientific and historical aspects of mathematics have been expanded to include more contributors. For example, the authors added additional historical and multicultural context regarding what is widely known as Pascal’s Triangle, and similarly added details regarding the international process of decoding the Enigma machine (including the role of Polish college students). Several chapter-opening narratives and in-chapter references are completely new, and contexts across all chapters were specifically reviewed for equity in gender representation and connotation.

Pedagogical Foundations and Features

Learning objectives.

Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives. The learning objectives are listed explicitly at the beginning of each section and are the focal point of every instructional element.

Narrative Text

Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide transitions between topics and examples. Throughout this book, we rely on a few basic conventions to highlight the most important ideas:

• Key terms are boldfaced, typically when first introduced and/or when formally defined.
• Key concepts and definitions are called out in a blue box for easy reference.

Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective in order to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. All told, there are more than 650 Examples, or an average of about 55 per chapter.

All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the Solution, spelling out the steps along the way. Finally (for select Examples), we conclude with an Analysis reflecting on the broader implications of the Solution just shown.

Precalculus 2e contains more than 2000 figures and illustrations, the vast majority of which are graphs and diagrams. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish between the different functions or features of a graph.

Supporting Features

Several elements, each marked by a distinctive icon, serve to support Examples.

• A How To is a list of steps necessary to solve a certain type of problem. A How To typically precedes an Example that proceeds to demonstrate the steps in action.
• A Try It exercise immediately follows an Example or a set of related Examples, providing the student with an immediate opportunity to solve a similar problem. In the PDF and the Web View version of the text, answers to the Try It exercises are located in the Answer Key.
• A Q&A may appear at any point in the narrative, but most often follows an Example. This feature pre-empts misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation.
• The Media icon appears at the conclusion of each section, just prior to the Section Exercises. This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section.

While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany Precalculus 2e .

Section Exercises

Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. With over 5900 exercises across the 12 chapters, instructors should have plenty from which to choose.

Section Exercises are organized by question type, and generally appear in the following order:

• Verbal questions assess conceptual understanding of key terms and concepts.
• Algebraic problems require students to apply algebraic manipulations demonstrated in the section.
• Graphical problems assess students’ ability to interpret or produce a graph.
• Numeric problems require the student to perform calculations or computations.
• Technology problems encourage exploration through use of a graphing utility, either to visualize or verify algebraic results or to solve problems via an alternative to the methods demonstrated in the section.
• Extensions pose problems more challenging than the Examples demonstrated in the section. They require students to synthesize multiple learning objectives or apply critical thinking to solve complex problems.
• Real-World Applications present realistic problem scenarios from fields such as physics, geology, biology, finance, and the social sciences.

Chapter Review Features

Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.

• Key Terms provides a formal definition for each bold-faced term in the chapter.
• Key Equations presents a compilation of formulas, theorems, and standard-form equations.
• Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant Example(s) in case students need to review.
• Chapter Review Exercises include 40-80 practice problems that recall the most important concepts from each section.
• Practice Test includes 25-50 problems assessing the most important learning objectives from the chapter. Note that the practice test is not organized by section, and may be more heavily weighted toward cumulative objectives as opposed to the foundational objectives covered in the opening sections.

Answers to Questions in the Book

All answers to Try It questions are provided in the Answer Key. Answers to Examples are provided directly below the question. Students can find odd-numbered answers to Review Exercises, Practice Test, and Section Exercises in the Answer Key. Answers to all odd and even-numbered questions are provided only to instructors in the Instructor Answer Guide via the Instructor Resources page.

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About the Authors

Senior contributing author.

Jay Abramson, Arizona State University Jay Abramson has been teaching Precalculus for over 35 years, the last 20 at Arizona State University, where he is a principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions.

Contributing Authors

Valeree Falduto, Palm Beach State College Rachael Gross, Towson University David Lippman, Pierce College Melonie Rasmussen, Pierce College Rick Norwood, East Tennessee State University Nicholas Belloit, Florida State College Jacksonville Jean-Marie Magnier, Springfield Technical Community College Harold Whipple Christina Fernandez

Faculty Reviewers and Consultants

Nina Alketa, Cecil College Kiran Bhutani, Catholic University of America Brandie Biddy, Cecil College Lisa Blank, Lyme Central School Bryan Blount, Kentucky Wesleyan College Jessica Bolz, The Bryn Mawr School Sheri Boyd, Rollins College Sarah Brewer, Alabama School of Math and Science Charles Buckley, St. Gregory's University Michael Cohen, Hofstra University Kenneth Crane, Texarkana College Rachel Cywinski, Alamo Colleges Nathan Czuba Srabasti Dutta, Ashford University Kristyanna Erickson, Notre Dame of Maryland University Nicole Fernandez, Georgetown University / Kent State University David French, Tidewater Community College Douglas Furman, SUNY Ulster Lance Hemlow, Raritan Valley Community College Erinn Izzo, Nicaragua Christian Academy John Jaffe Jerry Jared, Blue Ridge School Stan Kopec, Mount Wachusett Community College Kathy Kovacs Cynthia Landrigan, Erie Community College Sara Lenhart, Christopher Newport University Wendy Lightheart, Lane Community College Joanne Manville, Bunker Hill Community College Karla McCavit, Albion College Cynthia McGinnis, Northwest Florida State College Lana Neal, University of Texas at Austin Rhonda Porter, Albany State University Steven Purtee, Valencia College William Radulovich, Florida State College Jacksonville Alice Ramos, Bethel College Nick Reynolds, Montgomery Community College Amanda Ross, A. A. Ross Consulting and Research, LLC Erica Rutter, Arizona State University Sutandra Sarkar, Georgia State University Willy Schild, Wentworth Institute of Technology Todd Stephen, Cleveland State University Scott Sykes, University of West Georgia Linda Tansil, Southeast Missouri State University John Thomas, College of Lake County Diane Valade, Piedmont Virginia Community College Allen Wolmer, Atlanta Jewish Academy

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• Authors: Jay Abramson
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• Book title: Precalculus 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
• Section URL: https://openstax.org/books/precalculus-2e/pages/preface

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