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Intermediate Algebra
Functions and Graphs
Katherine Yoshiwara
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1
s
p
Front Matter
Colophon
Preface
1
Linear Models
1.1
Creating a Linear Model
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
Graphical Solution of Equations and Inequalities
1.2.5
Using a Graphing Utility
1.2.6
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
Rate of Change
1.4.2
Review of Slope
1.4.3
Interpreting Slope as a Rate
1.4.4
Lines Have Constant Slope
1.4.5
Problem Set 1.4
1.5
Equations of Lines
1.5.1
Slope-Intercept Form
1.5.2
Coordinate Formula for Slope
1.5.3
Point-Slope Formula
1.5.4
Finding a Linear Model
1.5.5
Summary
1.5.6
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
Inconsistent and Dependent Systems
2.2.4
Applications
2.2.5
Problem Set 2.2
2.3
Algebraic Solution of Systems
2.3.1
Substitution Method
2.3.2
Elimination Method
2.3.3
Problem Solving with Systems
2.3.4
Inconsistent and Dependent Systems
2.3.5
Problem Set 2.3
2.4
Gaussian Reduction
2.4.1
3x3 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
Finding the Solutions
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
Extracting Roots
3.1.5
Geometric Formulas
3.1.6
Solving Formulas
3.1.7
More Extraction of Roots
3.1.8
An Application: Compound Interest
3.1.9
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
=
a
x
2
3.3.3
The Graph of
y
=
x
2
+
c
3.3.4
The Graph of
y
=
a
x
2
+
b
x
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
Proportion and 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
th
n
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
6.3.7
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
a
n
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
Graphical Solution of Exponential Equations
7.2.6
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 and 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
Compound Interest
7.5.5
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
What are Polynomials?
8.1.3
Products of Polynomials
8.1.4
Modeling with Polynomials
8.1.5
Cubic Polynomials
8.1.6
Approximations with Polynomials
8.1.7
Special Products
8.1.8
Factoring Cubics
8.1.9
An Application of Cubic Splines
8.1.10
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
Using Factoring in Multiplication and Division
8.3.4
Adding and Subtracting Like Fractions
8.3.5
Unlike Fractions
8.3.6
Finding the Lowest Common Denominator
8.3.7
Adding and Subtracting Unlike Fractions
8.3.8
Applications
8.3.9
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
Summary
8.5.8
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
The Midpoint Formula
9.2.4
Circles
9.2.5
General Form for Circles
9.2.6
Investigation
9.2.7
Problem Set 9.2
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
Logarithms and Exponents
10.1.2
Inverse of a Function
10.1.3
Inverse of an Exponential Function
10.1.4
Graphs of Logarithmic Functions
10.1.5
Using Logarithmic Functions
10.1.6
Solving Logarithmic Equations
10.1.7
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
Comparing Quantities on a Log Scale
10.2.7
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
Properties of the Natural Logarithm
10.3.4
Solving Equations
10.3.5
Exponential Growth and Decay
10.3.6
Continuous Compounding
10.3.7
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
Algebra Toolkit
A.1
Chapter 1 Linear Models
A.1.1
Creating a Linear Model
A.1.1.1
Write a linear model
A.1.1.2
Plot points and graph a line
A.1.1.3
Solve a linear equation
A.1.1.4
Solve a linear inequality
A.1.2
Graphs and Equations
A.1.2.1
Verify a solution
A.1.2.2
Solve linear equations and inequalities with parentheses
A.1.2.3
Solutions of an equation in two variables
A.1.3
Intercepts
A.1.3.1
Graph a linear equation by the intercept method
A.1.3.2
Interpret the intercepts
A.1.3.3
Solve an equation for one of the variables
A.1.4
Slope
A.1.4.1
Use ratios for comparison
A.1.4.2
Calculate slope from a graph
A.1.4.3
Interpret the slope
A.1.5
Equations of Lines
A.1.5.1
Slope-Intercept Form
A.1.5.2
Calculate slope using a formula
A.1.5.3
Point-Slope Form
A.1.5.4
Graphing a linear equation by the point-slope method
A.2
Chapter 2 Applications of Linear Models
A.2.1
Linear Regression
A.2.1.1
Read a scatterplot
A.2.1.2
Sketch a line of best fit
A.2.1.3
Fit a line through two points
A.2.2
Linear Systems
A.2.2.1
Write equations in two variables
A.2.2.2
Identify the solution of a system
A.2.2.3
Use the formula for profit,
P
=
R
−
C
A.2.3
Algebraic Solution of Systems
A.2.3.1
Use the interest formula,
I
=
P
r
A.2.3.2
Use the percent formula,
P
=
r
W
A.2.3.3
Use the distance formula,
d
=
r
t
A.2.4
Gaussian Reduction
A.2.4.1
Write an equation in standard form
A.2.4.2
Clear fractions from an equation
A.2.4.3
Write a 3x3 linear system
A.2.5
Linear Inequalities in Two Variables
A.2.5.1
Solve a linear inequality
A.2.5.2
Graph a line
A.2.5.3
Solve a 2x2 system
A.3
Chapter 3 Quadratic Models
A.3.1
Extraction of Roots
A.3.1.1
Evaluate quadratic expressions
A.3.1.2
Use square roots
A.3.1.3
Use geometric formulas
A.3.1.4
Use the Pythagorean theorem
A.3.2
Intercepts, Solutions, and Factors
A.3.2.1
Multiply binomials
A.3.2.2
Factor quadratic trinomials
A.3.2.3
Write algebraic expressions
A.3.3
Graphing Parabolas
A.3.3.1
Find the average of two numbers
A.3.3.2
Solve quadratic equations
A.3.3.3
Find the coordinates of points on a parabola
A.3.4
Completing the Square
A.3.4.1
Squares of binomials
A.3.4.2
Multiply fractions
A.3.4.3
Add fractions
A.4
Chapter 4 Applications of Quadratic Models
A.4.1
Quadratic Formula
A.4.1.1
Simplify square roots
A.4.1.2
Use the order of operations
A.4.1.3
Use the quadratic formula
A.4.2
The Vertex
A.4.2.1
Parabolas in vertex form
A.4.2.2
Product of binomials
A.4.2.3
Solve for a parameter
A.4.3
Curve Fitting
A.4.3.1
Points on a graph
A.4.3.2
Solve a 2x2 linear system
A.4.3.3
Special 3x3 linear system
A.4.4
Quadratic Inequalities
A.4.4.1
Solve a linear inequality
A.4.4.2
x
-intercepts of a parabola
A.5
Chapter 5 Functions
A.5.1
Functions
A.5.1.1
New vocabulary
A.5.1.2
Solve non-linear equations
A.5.2
Graphs of Functions
A.5.2.1
Function notation
A.5.2.2
Solve graphically
A.5.3
Some Basic Graphs
A.5.3.1
Evaluate cube roots
A.5.3.2
Evaluate absolute values
A.5.3.3
Use the order of operations
A.5.4
Direct Variation
A.5.4.1
Find the constant of variation
A.5.4.2
Sketch a variation graph
A.5.4.3
Solve a variation equation
A.5.5
Inverse Variation
A.5.5.1
Solve an inverse variation equation
A.5.5.2
Sketch a variation graph
A.5.5.3
Find the constant of variation
A.5.6
Functions as Models
A.5.6.1
Properties of the basic functions
A.5.6.2
Simplify absolute value functions
A.5.6.3
Use interval notation
A.6
Chapter 6 Powers and Roots
A.6.1
Integer Exponents
A.6.1.1
Evaluate powers with negative exponents
A.6.1.2
Use the laws of exponents
A.6.1.3
Use scientific notation
A.6.2
Roots and Radicals
A.6.2.1
Use the definition of root
A.6.2.2
Irrational numbers
A.6.2.3
Using exponents and roots
A.6.3
Rational Exponents
A.6.3.1
Operations on fractions
A.6.3.2
Convert between fractions and decimals
A.6.3.3
Solve equations
A.6.4
Working with Radicals
A.6.4.1
Perfect squares
A.6.4.2
Apply properties of radicals
A.6.4.3
Simplify radicals
A.6.4.4
Rationalize the denominator
A.6.5
Radical Equations
A.6.5.1
Solve radical equations
A.6.5.2
Square binomials containing radicals
A.6.5.3
Use absolute value
A.7
Chapter 7 Exponential Functions
A.7.1
Exponential Growth and Decay
A.7.1.1
Compute percent increase and decrease
A.7.1.2
Use the order of operations
A.7.1.3
Simplify expressions
A.7.2
Exponential Functions
A.7.2.1
Evaluate exponential functions
A.7.2.2
Find an unknown exponent
A.7.2.3
Solve equations graphically
A.7.3
Logarithms
A.7.3.1
Convert between radicals and powers
A.7.3.2
Estimate logs
A.7.3.3
Using logs
A.7.4
Properties of Logarithms
A.7.4.1
Apply the laws of exponents
A.7.4.2
Apply the distributive law
A.7.4.3
Solve power and exponential equations
A.7.5
Exponential Models
A.7.5.1
Interpret function notation
A.7.5.2
Doubling time and half-life
A.7.5.3
Analyze graphs of exponential functions
A.8
Chapter 8 Polynomial and Rational Functions
A.8.1
Polynomial Functions
A.8.1.1
Compute sums and products
A.8.1.2
Use formulas
A.8.1.3
Square binomials
A.8.2
Algebraic Fractions
A.8.2.1
Factor a polynomial
A.8.2.2
Find the opposite of a binomial
A.8.2.3
Asymptotes
A.8.3
Operations on Algebraic Fractions
A.8.3.1
Multiply fractions
A.8.3.2
Divide fractions
A.8.3.3
Use factoring
A.8.3.4
Find an LCD
A.8.3.5
Build fractions
A.8.3.6
Add or subtract fractions
A.8.4
More Operations on Fractions
A.8.4.1
Use negative exponents
A.8.4.2
Improper fractions
A.8.4.3
Check a division
A.8.5
Equations with Fractions
A.8.5.1
Solve quadratic equations
A.8.5.2
Solve equations graphically
A.8.5.3
Choose the correct technique
A.9
Chapter 9 Equations and Graphs
A.9.1
Properties of Lines
A.9.1.1
Find the slope of a line
A.9.1.2
Use the point-slope formula
A.9.1.3
Use properties of geometric figures
A.9.2
The Distance and Midpoint Formulas
A.9.2.1
Use radicals
A.9.2.2
Use the equation for a circle
A.9.2.3
Complete the square
A.9.3
Conic Sections: Ellipses
A.9.3.1
Find points on a graph
A.9.3.2
Complete the square
A.9.4
Conic Sections: Hyperbolas
A.9.4.1
Write a quadratic equation in standard form
A.9.4.2
Find an asymptote
A.9.5
Nonlinear Systems
A.9.5.1
Write a system of equations
A.9.5.2
Use substitution
A.9.5.3
Use elimination
A.10
Chapter 10 Logarithmic Functions
A.10.1
Logarithmic Functions
A.10.1.1
Graph log functions
A.10.1.2
Use function notation
A.10.1.3
Use the properties of logarithms
A.10.2
Logarithmic Scales
A.10.2.1
Plot a log scale
A.10.2.2
Compare quantities
A.10.3
The Natural Base
A.10.3.1
Graphs of
y
=
e
x
and
y
=
ln
(
x
)
A.10.3.2
Using growth and decay laws with base
e
B
Using a GeoGebra Calculator App
B.1
Getting Started
B.1.1
On and Off
B.1.2
Numbers and Operations
B.1.3
Delete and Undo
B.2
Entering Expressions
B.2.1
Parentheses
B.2.2
Fractions
B.2.3
Exponents and Powers
B.2.4
Square Roots
B.2.5
Other Roots
B.2.6
Absolute Value
B.2.7
Scientific Notation
B.2.8
Editing an Entry
B.3
Graphing an Equation
B.3.1
A Basic Graph with intercepts
B.3.2
Translate and Zoom
B.3.3
Graphing a Function, Making a Table, and Zooming One Axis
B.4
More graphing
B.4.1
Finding a Suitable Graphing Window
B.4.2
Multiple Graphs and the Intersect Feature
B.5
Regression
B.5.1
Making a Scatterplot and Finding the Regression Line
B.6
Troubleshooting the GeoGebra App
C
Answers to Selected Exercises
Index
Colophon
Intermediate Algebra
Functions and Graphs
Katherine Yoshiwara
Department of Mathematics
Los Angeles Pierce College
June 7, 2025
Colophon
Preface
