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Intermediate Algebra:
Functions and Graphs
Katherine Yoshiwara
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Front Matter
Colophon
Preface
1
Linear Models
1.1
Linear Models
1.1.1
Tables, Graphs, and Equations
1.1.2
Equations for Linear Models
1.1.3
Problem Set 1.1
1.2
Graphs and Equations
1.2.1
Equations and Solutions
1.2.2
Linear Inequalities
1.2.3
Equations in Two Variables
1.2.4
What is the Graph of an Equation?
1.2.5
Graphical Solution of Equations
1.2.6
Graphical Solution of Inequalities
1.2.7
Using a Graphing Utility
1.2.8
Finding Coordinates with a Graphing Calculator
1.2.9
Problem Set 1.2
1.3
Intercepts
1.3.1
Intercepts of a Graph
1.3.2
Meaning of the Intercepts
1.3.3
General Form for a Linear Equation
1.3.4
Intercept Method of Graphing
1.3.5
Two Forms for Linear Equations
1.3.6
Problem Set 1.3
1.4
Slope
1.4.1
Glacier Melt
1.4.2
Rate of Change
1.4.3
Review of Slope
1.4.4
Interpreting Slope as a Rate
1.4.5
Lines Have Constant Slope
1.4.6
Problem Set 1.4
1.5
Equations of Lines
1.5.1
Polar Ice
1.5.2
Slope-Intercept Form
1.5.3
Coordinate Formula for Slope
1.5.4
Point-Slope Formula
1.5.5
Finding a Linear Model
1.5.6
Summary
1.5.7
Problem Set 1.5
1.6
Chapter Summary and Review
1.6.1
Glossary
1.6.2
Key Concepts
1.6.3
Chapter 1 Review Problems
2
Applications of Linear Models
2.1
Linear Regression
2.1.1
Shrinking Rain Forest
2.1.2
Line of Best Fit
2.1.3
Interpolation and Extrapolation
2.1.4
Scatterplots
2.1.5
Problem Set 2.1
2.2
Linear Systems
2.2.1
Systems of Equations
2.2.2
Solving Systems by Graphing
2.2.3
The Intersect Feature
2.2.4
Inconsistent and Dependent Systems
2.2.5
Applications
2.2.6
Problem Set 2.2
2.3
Algebraic Solution of Systems
2.3.1
Substitution Method
2.3.2
Elimination Method
2.3.3
Inconsistent and Dependent Systems
2.3.4
Problem Set 2.3
2.4
Gaussian Reduction
2.4.1
\(3\times 3\)
Linear Systems
2.4.2
Back-Substitution
2.4.3
Gaussian Reduction
2.4.4
Inconsistent and Dependent Systems
2.4.5
Applications
2.4.6
Problem Set 2.4
2.5
Linear Inequalities in Two Variables
2.5.1
Graphs of Inequalities in Two Variables
2.5.2
Linear Inequalities
2.5.3
Using a Test Point
2.5.4
Systems of Inequalities
2.5.5
Problem Set 2.5
2.6
Chapter Summary and Review
2.6.1
Glossary
2.6.2
Key Concepts
2.6.3
Chapter 2 Review Problems
3
Quadratic Models
3.1
Extraction of Roots
3.1.1
Introduction
3.1.2
Graphs of Quadratic Equations
3.1.3
Solving Quadratic Equations
3.1.4
Geometric Formulas
3.1.5
Solving Formulas
3.1.6
More Extraction of Roots
3.1.7
An Application: Compound Interest
3.1.8
Problem Set 3.1
3.2
Intercepts, Solutions, and Factors
3.2.1
Zero-Factor Principle
3.2.2
X-Intercepts of a Parabola
3.2.3
Solving Quadratic Equations by Factoring
3.2.4
An Application
3.2.5
More About Solutions of Quadratic Equations
3.2.6
Problem Set 3.2
3.3
Graphing Parabolas
3.3.1
Introduction
3.3.2
The Graph of
\(y = ax^2\)
3.3.3
The Graph of
\(y= x^2 + c\)
3.3.4
The Graph of
\(y = ax^2 + bx\)
3.3.5
A Formula for the Vertex
3.3.6
Problem Set 3.3
3.4
Completing the Square
3.4.1
Squares of Binomials
3.4.2
Solving Quadratic Equations by Completing the Square
3.4.3
The General Case
3.4.4
Problem Set 3.4
3.5
Chapter 3 Summary and Review
3.5.1
Glossary
3.5.2
Key Concepts
3.5.3
Chapter 3 Review Problems
4
Applications of Quadratic Models
4.1
Quadratic Formula
4.1.1
A New Formula
4.1.2
Applications
4.1.3
Complex Numbers
4.1.4
Number of
\(x\)
-Intercepts
4.1.5
Solving Formulas
4.1.6
Problem Set 4.1
4.2
The Vertex
4.2.1
Finding the Vertex
4.2.2
Maximum or Minimum Values
4.2.3
The Vertex Form for a Parabola
4.2.4
Using the Vertex Form
4.2.5
Problem Set 4.2
4.3
Curve Fitting
4.3.1
Finding a Quadratic Equation through Three Points
4.3.2
Applications
4.3.3
Using a Calculator for Quadratic Regression
4.3.4
Choosing an Appropriate Model
4.3.5
Problem Set 4.3
4.4
Quadratic Inequalities
4.4.1
Solving Inequalities Graphically
4.4.2
Using the
\(x\)
-Intercepts
4.4.3
Interval Notation
4.4.4
Solving Quadratic Inequalities Algebraically
4.4.5
Applications
4.4.6
Problem Set 4.4
4.5
Chapter 4 Summary and Review
4.5.1
Glossary
4.5.2
Key Concepts
4.5.3
Chapter 4 Review Problems
5
Functions and Their Graphs
5.1
Functions
5.1.1
Definitions and Notation
5.1.2
Functions Defined by Tables
5.1.3
Functions Defined by Graphs
5.1.4
Functions Defined by Equations
5.1.5
Function Notation
5.1.6
Using Function Notation
5.1.7
Problem Set 5.1
5.2
Graphs of Functions
5.2.1
Reading Function Values from a Graph
5.2.2
Constructing the Graph of a Function
5.2.3
The Vertical Line Test
5.2.4
Graphical Solution of Equations and Inequalities
5.2.5
More about Notation
5.2.6
Problem Set 5.2
5.3
Some Basic Graphs
5.3.1
Cube Roots
5.3.2
Absolute Value
5.3.3
Eight Basic Graphs
5.3.4
Problem Set 5.3
5.4
Direct Variation
5.4.1
Direct Variation
5.4.2
The Scaling Property of Direct Variation
5.4.3
Finding a Formula for Direct Variation
5.4.4
Direct Variation with a Power of
\(x\)
5.4.5
Scaling
5.4.6
Problem Set 5.4
5.5
Inverse Variation
5.5.1
Inverse Variation
5.5.2
Finding a Formula for Inverse Variation
5.5.3
Inverse Variation with a Power
5.5.4
Problem Set 5.5
5.6
Functions as Models
5.6.1
The Shape of the Graph
5.6.2
Using the Basic Functions as Models
5.6.3
The Absolute Value and Distance
5.6.4
Absolute Value Equations
5.6.5
Absolute Value Inequalities
5.6.6
Problem Set 5.6
5.7
Chapter 5 Summary and Review
5.7.1
Glossary
5.7.2
Key Concepts
5.7.3
Chapter 5 Review Problems
6
Powers and Roots
6.1
Integer Exponents
6.1.1
Negative Exponents
6.1.2
Power Functions
6.1.3
Working with Negative Exponents
6.1.4
Review of Scientific Notation
6.1.5
Problem Set 6.1
6.2
Roots and Radicals
6.2.1
\(n^\text{th}\)
Roots
6.2.2
Exponential Notation for Radicals
6.2.3
Irrational Numbers
6.2.4
Working with Fractional Exponents
6.2.5
Using Fractional Exponents to Solve Equations
6.2.6
Power Functions
6.2.7
Solving Radical Equations
6.2.8
Roots of Negative Numbers
6.2.9
Problem Set 6.2
6.3
Rational Exponents
6.3.1
Powers of the Form
\(a^{m/n}\)
6.3.2
Power Functions
6.3.3
Radical Notation
6.3.4
Operations with Rational Exponents
6.3.5
Solving Equations
6.3.6
Problem Set 6.3
6.4
Working with Radicals
6.4.1
Properties of Radicals
6.4.2
Simplifying Radicals
6.4.3
Sums and Differences of Radicals
6.4.4
Products and Quotients of Radicals
6.4.5
Rationalizing the Denominator
6.4.6
Problem Set 6.4
6.5
Radical Equations
6.5.1
Solving a Radical Equation
6.5.2
Extraneous Solutions
6.5.3
Solving Formulas
6.5.4
Equations with More than One Radical
6.5.5
Simplifying
\(\sqrt[n]{a^n}\)
6.5.6
Problem Set 6.5
6.6
Chapter 6 Summary and Review
6.6.1
Glossary
6.6.2
Key Concepts
6.6.3
Chapter 6 Review Problems
7
Exponential Functions
7.1
Exponential Growth and Decay
7.1.1
Introduction
7.1.2
Growth Factors
7.1.3
Comparing Linear Growth and Exponential Growth
7.1.4
Exponential Decay
7.1.5
Percent Increase
7.1.6
Percent Decrease
7.1.7
Problem Set 7.1
7.2
Exponential Functions
7.2.1
Introduction
7.2.2
Graphs of Exponential Functions
7.2.3
Comparing Exponential and Power Functions
7.2.4
Exponential Equations
7.2.5
Applications
7.2.6
Graphical Solution of Exponential Equations
7.2.7
Problem Set 7.2
7.3
Logarithms
7.3.1
A Logarithm is an Exponent
7.3.2
Logs and Exponential Equations
7.3.3
Approximating Logarithms
7.3.4
Base 10 Logarithms
7.3.5
Solving Exponential Equations
7.3.6
Application to Exponential Models
7.3.7
Problem Set 7.3
7.4
Properties of Logarithms
7.4.1
Logarithms are Exponents
7.4.2
Using the Properties of Logarithms
7.4.3
Solving Exponential Equations
7.4.4
Solving Formulas
7.4.5
Problem Set 7.4
7.5
Exponential Models
7.5.1
Fitting an Exponential Function through Two Points
7.5.2
Doubling Time
7.5.3
Half-Life
7.5.4
Problem Set 7.5
7.6
Chapter 7 Summary and Review
7.6.1
Glossary
7.6.2
Key Concepts
7.6.3
Chapter 7 Review Problems
8
Polynomial and Rational Functions
8.1
Polynomial Functions
8.1.1
Introduction
8.1.2
Products of Polynomials
8.1.3
Special Products
8.1.4
Factoring Cubics
8.1.5
Modeling with Polynomials
8.1.6
Problem Set 8.1
8.2
Algebraic Fractions
8.2.1
Introduction
8.2.2
Reducing Fractions
8.2.3
Opposite of a Binomial
8.2.4
Rational Functions
8.2.5
Problem Set 8.2
8.3
Operations on Algebraic Fractions
8.3.1
Products of Fractions
8.3.2
Quotients of Fractions
8.3.3
Adding and Subtracting Like Fractions
8.3.4
Unlike Fractions
8.3.5
Finding the Lowest Common Denominator
8.3.6
Adding and Subtracting Unlike Fractions
8.3.7
Applications
8.3.8
Problem Set 8.3
8.4
More Operations on Fractions
8.4.1
Complex Fractions
8.4.2
Negative Exponents
8.4.3
Applications
8.4.4
Polynomial Division
8.4.5
Problem Set 8.4
8.5
Equations with Fractions
8.5.1
Solving Algebraically
8.5.2
Proportions
8.5.3
Extraneous Solutions
8.5.4
Solving Graphically
8.5.5
Applications
8.5.6
Formulas
8.5.7
Problem Set 8.5
8.6
Chapter 8 Summary and Review
8.6.1
Glossary
8.6.2
Key Concepts
8.6.3
Chapter 8 Review Problems
9
Equations and Graphs
9.1
Properties of Lines
9.1.1
Horizontal and Vertical Lines
9.1.2
Parallel and Perpendicular Lines
9.1.3
Applications to Geometry
9.1.4
Problem Set 9.1
9.2
The Distance and Midpoint Formulas
9.2.1
Distance in a Coordinate Plane
9.2.2
The Distance Formula
9.2.3
Finding the Midpoint
9.2.4
The Midpoint Formula
9.2.5
Circles
9.2.6
General Form for Circles
9.2.7
Problem Set 9.2
9.2.8
Investigation
9.3
Conic Sections: Ellipses
9.3.1
Circles and Ellipses
9.3.2
Translated Ellipses
9.3.3
Writing in Standard Form
9.3.4
Finding the Equation of an Ellipse
9.3.5
Problem Set 9.3
9.4
Conic Sections: Hyperbolas
9.4.1
Asymptotes of Hyperbolas
9.4.2
The Central Conics
9.4.3
Translated Hyperbolas
9.4.4
Writing in Standard Form
9.4.5
General Quadratic Equation in Two Variables
9.4.6
Problem Set 9.4
9.5
Nonlinear Systems
9.5.1
Systems Involving Quadratic Equations
9.5.2
Systems Involving Conic Sections
9.5.3
Solving Systems by Elimination
9.5.4
Problem Set 9.5
9.6
Chapter 9 Summary and Review
9.6.1
Glossary
9.6.2
Key Concepts
9.6.3
Chapter 9 Review Problems
10
Logarithmic Functions
10.1
Logarithmic Functions
10.1.1
Inverse of a Function
10.1.2
Inverse of an Exponential Function
10.1.3
Graphs of Logarithmic Functions
10.1.4
Using Logarithmic Functions
10.1.5
Solving Logarithmic Equations
10.1.6
Problem Set 10.1
10.2
Logarithmic Scales
10.2.1
Making a Log Scale
10.2.2
Labeling a Log Scale
10.2.3
Acidity and the pH Scale
10.2.4
Decibels
10.2.5
The Richter Scale
10.2.6
Problem Set 10.2
10.3
The Natural Base
10.3.1
The Natural Exponential Function
10.3.2
The Natural Logarithmic Function
10.3.3
Solving Equations
10.3.4
Exponential Growth and Decay
10.3.5
Continuous Compounding
10.3.6
Homework 10.3
10.4
Chapter 10 Summary and Review
10.4.1
Glossary
10.4.2
Key Concepts
10.4.3
Chapter 10 Review Problems
Appendices
A
Using a GeoGebra Calculator App
A.1
Getting Started
A.1.1
On and Off
A.1.2
Numbers and Operations
A.1.3
Delete and Undo
A.2
Entering Expressions
A.2.1
Parentheses
A.2.2
Fractions
A.2.3
Exponents and Powers
A.2.4
Square Roots
A.2.5
Other Roots
A.2.6
Absolute Value
A.2.7
Scientific Notation
A.2.8
Editing an Entry
A.3
Graphing an Equation
A.3.1
A Basic Graph with intercepts
A.3.2
Translate and Zoom
A.3.3
Graphing a Function, Making a Table, and Zooming One Axis
A.4
More graphing
A.4.1
Finding a Suitable Graphing Window
A.4.2
Multiple Graphs and the Intersect Feature
A.5
Regression
A.5.1
Making a Scatterplot and Finding the Regression Line
A.6
Troubleshooting the GeoGebra App
B
Answers to Selected Exercises
Index
Colophon
Colophon
Colophon
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