Introduction to Functions and Graphs Algebra

College Algebra with Modeling and
Visualization
Sixth Edition
Chapter 1
Introduction to
Functions and
Graphs
Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

1.4 Types of Functions and Their
Rates of Change
• Identify linear functions
• Interpret slope as a rate of change
• Identify nonlinear functions
• Identify where a function is increasing or decreasing
• Use and interpret average rate of change
• Calculate the difference quotient

Linear Function
A function f represented by f(x) = mx + b, where m and
b are constants, is a linear function.

Recognizing Linear Functions
A car initially located 30
miles north of the Texas
border, traveling north at
60 miles per hour is
represented by the
function f(x) = 60x + 30
and has the graph:

Rate of Change of a Linear
Function (1 of 2)
In a linear function f, each time x increases by one unit,
the value of f(x) always changes by an amount equal
to m.
That is, a linear function has a constant rate of
change.
The constant rate of change m is equal to the slope of
the graph of f.

Constant Function
A function f represented by f(x) = b, where b is a
constant (fixed number), is a constant function.

Rate of Change of a Linear
Function (2 of 2)
In our car example:
Elapsed time (hours) 0 1 2 3 4 5
Distance (miles) 30 90 150 210 270 330
Throughout the table, as x increases by 1 unit, y
increases by 60 units. That is, the rate of change or the
slope is 60.

Slope of Line as a Rate of Change
The slope m of the line passing through the points (x1, y1)
and (x2, y2) is

Positive Slope
If the slope of a line is positive,
the line rises from left to right.
Slope 2 indicates that the line
rises 2 units for every unit
increase in x.

Negative Slope
If the slope of a line is
negative, the line falls from
left to right.

Slope of 0
Slope 0 indicates that the line is horizontal.

Slope is Undefined
When x1 = x2, the line is vertical and the slope is
undefined.

Example: Calculating the slope of a
line given two points (1 of 2)
Find the slope of the line passing through the points
(−2, 3) and (1, −2). Plot these points together with the
line. Explain what the slope indicates about the line.
Solution

Example: Calculating the slope of a
line given two points (2 of 2)

Zero of a Function
•
Let ƒ be any function. Then any number c for which
ƒ(c) = 0 is called a zero of the function ƒ.

Four Representations of a Linear
Function f

Nonlinear Functions
If a function is not linear, then it is called a nonlinear
function.
The following are characteristics of a nonlinear
function:
• Graph is not a (straight) line.
• Does not have a constant rate of change.
• Cannot be written as ƒ(x) = mx + b.
• Can have any number of zeros.

Graphs of Nonlinear Functions (1 of 2)
There are many nonlinear functions.

Graphs of Nonlinear Functions (2 of 2)
Here are two other common nonlinear functions:

Rock Music’s Share of All U.S. Sales
(Percentage)

Increasing and Decreasing
Functions (1 of 2)

Increasing and Decreasing
Functions (2 of 2)
Suppose that a function f is defined over an interval I on
the number line. If x1 and x2 are in I,
a. f increases on I if, whenever x1 < x2, f(x1) < f(x2);
b. f decreases on I if, whenever x1 < x2, f(x1) > f(x2).

Example: Determining where a
function is increasing or decreasing

Average Rate of Change (1 of 2)
Graphs of nonlinear functions are not straight lines, so
we speak of average rate of change.
The line L is referred to as the
secant line, and the slope of L
represents the average rate
of change of f from x1 to x2.
Different values of x1 and x2
usually yield a different secant
line and a different average
rate of change.

Average Rate of Change (2 of 2)
Let (x1, y1) and (x2, y2) be distinct points on the graph of
a function f. The average rate of change of f from x1
to x2 is
That is, the average rate of change from x1 to x2 equals
the slope of the line passing through (x1, y1) and (x2, y2).

Example: Finding an average rate of
change (1 of 2)
Let f(x) = 2x². Find the average rate of change from
x = 1 to x = 3.
Solution
Calculate f(1) and f(3)
f(1) = 2(1)² = 2
f(3) = 2(3)² = 18
The average rate of change equals the slope of the
line passing through the points (1, 2) and (3, 18).

Example: Finding an average rate of
change (2 of 2)
(1, 2) and (3, 18)
The average rate of change from x = 1 to x = 3 is 8.

Difference Quotient (1 of 2)

Difference Quotient (2 of 2)
The difference quotient of a function f is an
expression of the form

Example: Finding a difference
quotient (1 of 2)
Let f(x) = 3x − 2.
a. Find f(x + h)
b. Find the difference quotient of f and simplify the result.
Solution
a. To find f(x + h), substitute (x + h) for x in the expression
3x – 2.

Example: Finding a difference
quotient (2 of 2)
b. The difference quotient can be calculated as follows:

Introduction to Functions and Graphs Algebra

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