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Modeling, Functions, and Graphs
Katherine Yoshiwara
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Front Matter
Colophon
Preface
1
Functions and Their Graphs
1.1
Linear Models
Tables, Graphs and Equations
Choosing Scales for the Axes
Linear Equations
Intercepts
Intercept Method for Graphing Lines
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.1
1.2
Functions
Definition of Function
Functions Defined by Tables
Functions Defined by Graphs
Functions Defined by Equations
Function Notation
Using Function Notation
Operations with Function Notation
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.2
1.3
Graphs of Functions
Reading Function Values from a Graph
Constructing the Graph of a Function
The Vertical Line Test
Graphical Solution of Equations and Inequalities
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.3
1.4
Slope and Rate of Change
Using Ratios for Comparison
Measuring Steepness
Definition of Slope
Notation for Slope
Lines Have Constant Slope
Meaning of Slope
A Formula for Slope
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.4
1.5
Linear Functions
Slope-Intercept Form
Slope-Intercept Method of Graphing
Finding a Linear Equation from a Graph
Point-Slope Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.5
1.6
Linear Regression
Fitting a Line through Two Points
Scatterplots
Linear Regression
Linear Interpolation and Extrapolation
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 1.6
1.7
Chapter Summary and Review
Key Concepts
Chapter 1 Review Problems
1.8
Projects for Chapter 1
2
Modeling with Functions
2.1
Nonlinear Models
Solving Nonlinear Equations
Solving Formulas
More Extraction of Roots
Compound Interest and Inflation
Other Nonlinear Equations
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.1
2.2
Some Basic Functions
Absolute Value
Examples of Models
Graphs of Eight Basic Functions
Some Properties of Absolute Value
Functions Defined Piecewise
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.2
2.3
Transformations of Graphs
Vertical Translations
Horizontal Translations
Scale Factors
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.3
2.4
Functions as Mathematical Models
The Shape of the Graph
Using the Basic Functions as Models
Modeling with Piecewise Functions
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.4
2.5
The Absolute Value Function
Introduction
Absolute Value Equations
Absolute Value Inequalities
Using the Absolute Value in Modeling
Measurement Error
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.5
2.6
Domain and Range
Definitions of Domain and Range
Finding Domain and Range from a Graph
Finding the Domain from a Formula
Restricting the Domain
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 2.6
2.7
Chapter Summary and Review
Key Concepts
Chapter 2 Review Problems
2.8
Projects for Chapter 2: Periodic Functions
3
Power Functions
3.1
Variation
Direct Variation
The Scaling Property of Direct Variation
Finding a Formula for Direct Variation
Direct Variation with a Power of
\(x\)
Scaling
Inverse Variation
Finding a Formula for Inverse Variation
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 3.1
3.2
Integer Exponents
Negative Exponents
Power Functions
Working with Negative Exponents
Laws of Exponents
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 3.2
3.3
Roots and Radicals
\(n\)
th Roots
Exponential Notation for Radicals
Irrational Numbers
Working with Fractional Exponents
Using Fractional Exponents to Solve Equations
Power Functions
Solving Radical Equations
A Note on Roots of Negative Numbers
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 3.3
3.4
Rational Exponents
Powers of the Form
\(a^{m/n}\)
Power Functions
More about Scaling
Radical Notation
Operations with Rational Exponents
Solving Equations
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 3.4
3.5
Joint Variation
Functions of Two or More Variables
Tables of Values
Joint Variation
Graphs
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 3.5
3.6
Chapter Summary and Review
Key Concepts
Chapter 3 Review Problems
3.7
Projects for Chapter 3
4
Exponential Functions
4.1
Exponential Growth and Decay
Exponential Growth
Growth Factors
Percent Increase
Exponential Decay
Decay Factors
Comparing Linear Growth and Exponential Growth
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.1
4.2
Exponential Functions
Introduction
Graphs of Exponential Functions
Transformations of Exponential Functions
Comparing Exponential and Power Functions
Exponential Equations
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.2
4.3
Logarithms
Introduction
Using the Conversion Equations
Approximating Logarithms
Base 10 Logarithms
Solving Exponential Equations
Application to Exponential Models
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.3
4.4
Properties of Logarithms
Introduction
Using the Properties of Logarithms
Solving Exponential Equations
Applications
Compound Interest
Solving Formulas
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.4
4.5
Exponential Models
Fitting an Exponential Function through Two Points
Doubling Time
Half-Life
Annuities and Amortization
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.5
4.6
Chapter Summary and Review
Key Concepts
Chapter 4 Review Problems
4.7
Projects for Chapter 4
5
Logarithmic Functions
5.1
Inverse Functions
Introduction
Finding a Formula for the Inverse Function
Inverse Function Notation
Graph of the Inverse Function
When Is the Inverse a Function?
Mathematical Properties of the Inverse Function
Symmetry
Domain and Range
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.1
5.2
Logarithmic Functions
Inverse of the Exponential Function
Graphs of Logarithmic Functions
Evaluating Logarithmic Functions
Logarithmic Equations
Using the Properties of Logarithms
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.2
Investigation
5.3
The Natural Base
The Natural Exponential Function
The Natural Logarithmic Function
Properties of the Natural Logarithm
Solving Equations
Exponential Growth and Decay
Continuous Compounding
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.3
5.4
Logarithmic Scales
Introduction
Using Log Scales
Equal Increments on a Log Scale
Acidity and the pH Scale
Decibels
The Richter Scale
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.4
5.5
Chapter Summary and Review
Key Concepts
Chapter 5 Review Problems
5.6
Projects for Chapter 5
6
Quadratic Functions
6.1
Factors and
\(x\)
-Intercepts
Zero-Factor Principle
Solving Quadratic Equations by Factoring
Applications
Solutions of Quadratic Equations
Equations Quadratic in Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.1
6.2
Solving Quadratic Equations
Squares of Binomials
Solving Quadratic Equations by Completing the Square
The General Case
Quadratic Formula
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.2
6.3
Graphing Parabolas
Introduction
The Graph of
\(y = ax^2\)
The Graph of
\(y= x^2 + c\)
The Graph of
\(y = ax^2 + bx\)
Finding the Vertex
The Graph of
\(y = ax^2 + bx + c\)
Number of
\(x\)
-Intercepts
Sketching a Parabola
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.3
6.4
Problem Solving
Maximum or Minimum Values
The Vertex Form for a Parabola
Graphing with the Vertex Form
Systems Involving Quadratic Equations
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.4
6.5
Quadratic Inequalities
Solving Inequalities Graphically
Solving Quadratic Inequalities Algebraically
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.5
6.6
Curve Fitting
Introduction
Finding a Quadratic Function through Three Points
Finding an Equation in Vertex Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.6
6.7
Chapter Summary and Review
Key Concepts
Chapter 6 Review Problems
6.8
Projects for Chapter 6
7
Polynomial and Rational Functions
7.1
Polynomial Functions
Introduction
Products of Polynomials
Special Products
Factoring Cubics
Modeling with Polynomials
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.1
7.2
Graphing Polynomial Functions
Classifying Polynomials by Degree
Cubic Polynomials
Quartic Polynomials
\(x\)
-Intercepts and the Factor Theorem
Zeros of Multiplicity Two or Three
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.2
7.3
Complex Numbers
Introduction
Imaginary Numbers
Complex Numbers
Arithmetic of Complex Numbers
Products of Complex Numbers
Quotients of Complex Numbers
Zeros of Polynomials
Graphing Complex Numbers
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.3
7.4
Graphing Rational Functions
Introduction
Domain of a Rational Function
Vertical Asymptotes
Horizontal Asymptotes
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.4
7.5
Equations That Include Algebraic Fractions
Solving Equations with Fractions Algebraically
Extraneous Solutions
Formulas
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.5
7.6
Chapter Summary and Review
Key Concepts
Chapter 7 Review Problems
7.7
Projects for Chapter 7
8
Linear Systems
8.1
Systems of Linear Equations in Two Variables
Solving Systems by Graphing
Solving Systems Algebraically
Inconsistent and Dependent Systems
Applications
An Application from Economics
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.1
8.2
Systems of Linear Equations in Three Variables
\(3\times 3\)
Linear Systems
Back-Substitution
Gaussian Reduction
Inconsistent and Dependent Systems
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.2
8.3
Solving Linear Systems Using Matrices
Coefficient Matrix and Augmented Matrix of a System
Elementary Row Operations
Matrix Reduction
Reducing a
\(3\times 3\)
Matrix
Solving Larger Systems
Reduced Row Echelon Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.3
8.4
Linear Inequalities
Graphs of Inequalities in Two Variables
Linear Inequalities
Using a Test Point
Systems of Inequalities
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.4
8.5
Linear Programming
The Objective Function and Constraints
Feasible Solutions
The Optimum Solutions
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.5
8.6
Chapter Summary and Review
Key Concepts
Chapter 8 Review Problems
8.7
Projects for Chapter 8
9
Sequences and Series
9.1
Sequences
Definitions and Notation
Applications of Sequences
Recursively Defined Sequences
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.1
9.2
Arithmetic and Geometric Sequences
Arithmetic Sequences
The General Term of an Arithmetic Sequence
Geometric Sequences
The General Term of a Geometric Sequence
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.2
9.3
Series
Introduction
Arithmetic Series
Geometric Series
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.3
9.4
Infinite Geometric Series
Summation Notation
Infinite Series
Repeating Decimals
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.4
9.5
The Binomial Expansion
Powers of Other Binomials
The Binomial Coefficient
Using Pascal’s Triangle
Factorial Notation
The Binomial Coefficient in Factorial Notation
The Binomial Theorem
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.5
9.6
Chapter Summary and Review
Key Concepts
Chapter 9 Review Problems
9.7
Projects for Chapter 9
Appendices
A
Algebra Skills Refresher
A.1
Numbers and Operations
Order of Operations
Parentheses and Fraction Bars
Radicals
Scientific Notation
Section Summary
Vocabulary
SKILLS
Exercises A.1
A.2
Linear Equations and Inequalities
Solving Linear Equations
Formulas
Linear Inequalities
Interval Notation
Section Summary
Vocabulary
SKILLS
Exercises A.2
A.3
Algebraic Expressions and Problem Solving
Problem Solving
Supply and Demand
Percent Problems
Weighted Averages
Section Summary
Vocabulary
SKILLS
Exercises A.3
A.4
Graphs and Equations
Reading a Graph
Graphs of Equations
Section Summary
Vocabulary
SKILLS
Exercises A.4
A.5
Linear Systems in Two Variables
Solving Systems by Substitution
Solving Systems by Elimination
Section Summary
Vocabulary
SKILLS
Exercises A.5
A.6
Laws of Exponents
Product of Powers
Quotients of Powers
Power of a Power
Power of a Product
Power of a Quotient
Section Summary
Vocabulary
SKILLS
Exercises A.6
A.7
Polynomials and Factoring
Polynomials
Products of Polynomials
Products of Binomials
Factoring
Common Factors
Opposite of a Binomial
Polynomial Division
Section Summary
Vocabulary
SKILLS
Exercises A.7
A.8
Factoring Quadratic Trinomials
Special Products and Factors
Section Summary
Vocabulary
SKILLS
Exercises A.8
A.9
Working with Algebraic Fractions
Reducing Fractions
Products of Fractions
Quotients of Fractions
Sums and Differences of Like Fractions
Lowest Common Denominator
Building Fractions
Sums and Differences of Unlike Fractions
Complex Fractions
Negative Exponents
Section Summary
Vocabulary
SKILLS
Exercises A.9
A.10
Working with Radicals
Properties of Radicals
Simplifying Radicals
Sums and Differences of Radicals
Products of Radicals
Rationalizing the Denominator
Simplifying
\(\sqrt[n]{x^n} \)
Extraneous Solutions to Radical Equations
Equations with More than One Radical
Section Summary
Vocabulary
SKILLS
Exercises A.10
A.11
Facts from Geometry
Right Triangles and the Pythagorean Theorem
Isosceles and Equilateral Triangles
The Triangle Inequality
Similar Triangles
Volume and Surface Area
The Distance Formula
The Midpoint Formula
Circles
Section Summary
Vocabulary
SKILLS
Exercises A.11
A.12
Properties of Lines
Horizontal and Vertical Lines
Parallel and Perpendicular Lines
Applications to Geometry
Section Summary
Vocabulary
SKILLS
Reading Questions
Exercises A.12
A.13
The Real Number System
Subsets of the Real Numbers
Rational Numbers
Irrational Numbers
Properties of the Real Numbers
Order Properties of the Real Numbers
Section Summary
Vocabulary
SKILLS
Exercises A.13
B
Using a Graphing Calculator
B.1
Getting Started
On and Off
Numbers and Operations
Clear and Delete
B.2
Entering Expressions
Parentheses
Exponents and Powers
Roots
Absolute Value
Scientific Notation
B.3
Editing Expressions
Overwriting
Recalling an Entry
Inserting a Character
Recalling an Answer
B.4
Graphing an Equation
Standard Window
Tracing
Multiple Graphs
Setting the Window
Intersect Feature
Other Windows
B.5
Making a Table
Using the Auto Option
Using the Ask Option
B.6
Regression
Making a Scatterplot
Finding a Regression Equation
Graphing the Regression Equation
B.7
Function Notation and Transformation of Graphs
Function Notation
Transformation of Graphs
Translations
Vertical Scalings and Reflections
C
Using a GeoGebra Calculator App
C.1
Getting Started
On and Off
Numbers and Operations
Delete and Undo
C.2
Entering Expressions
Parentheses
Fractions
Exponents and Powers
Square Roots
Other Roots
Absolute Value
Scientific Notation
Editing an Entry
C.3
Graphing an Equation
A Basic Graph with intercepts
Translate and Zoom
Graphing a Function, Making a Table, and Zooming One Axis
C.4
More graphing
Finding a Suitable Graphing Window
Multiple Graphs and the Intersect Feature
C.5
Regression
Making a Scatterplot and Finding the Regression Line
C.6
Troubleshooting the GeoGebra App
D
Glossary
E
Properties of Numbers
F
Geometry formulas
G
Answers to Selected Exercises
H
GNU Free Documentation License
Bibliography
Index
Colophon
Colophon
Colophon
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