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CHAPTER 1:
Graphs, Functions,
and Models
1.1
1.2
1.3
1.4
1.5
1.6
Introduction to Graphing
Functions and Graphs
Linear Functions, Slope, and Applications
Equations of Lines and Modeling
Linear Equations, Functions, Zeros and Applications
Solving Linear Inequalities
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1.3
Linear Functions, Slope, and
Applications





Determine the slope of a line given two points on
the line.
Solve applied problems involving slope.
Find the slope and the y-intercept of a line given the
equation y = mx + b, or f (x) = mx + b.
Graph a linear equation using the slope and the yintercept.
Solve applied problems involving linear functions.
Copyright © 2009 Pearson Education, Inc.
Linear Functions
A function f is a linear function if it can be written as
f (x) = mx + b, where m and b are constants.
If m = 0, the function is a constant function f (x) = b.
If m = 1 and b = 0, the function is the identity
function f (x) = x.
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Examples
Linear Function
y = mx + b
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Identity Function
y = 1•x + 0 or y = x
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Examples
Constant Function
y = 0•x + b or y = b
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Not a Function
Vertical line: x = a
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Horizontal and Vertical Lines
Horizontal lines are
given by equations of the
type y = b or f(x) = b.
They are functions.
Vertical lines are given
by equations of the type
x = a. They are not
functions.
x=2
y=2
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Slope
The slope m of a line containing the points (x1, y1)
and (x2, y2) is given by
rise
m
run
the change in y

the change in x
y2  y1 y1  y2


x2  x1 x1  x2
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Example
2
Graph the function f (x)   x  1 and determine its
3
slope.
Solution: Calculate two ordered pairs, plot the points,
graph the function, and determine its slope.
2
f (3)   (3)  1  2  1  1
3
2
f (9)    9  1  6  1  5
3
y2  y1
m
x2  x1
5  1 4
2



93
6
3
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Types of Slopes
Positive—line slants up
from left to right
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Negative—line slants down
from left to right
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Horizontal Lines
If a line is horizontal, the change in y for any two
points is 0 and the change in x is nonzero. Thus a
horizontal line has slope 0.
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Vertical Lines
If a line is vertical, the change in y for any two points
is nonzero and the change in x is 0. Thus the slope is
not defined because we cannot divide by 0.
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Example
Graph each linear equation and determine its slope.
a. x = –2
Choose any number for y ; x must be –2.
x y
-2 3
-2 0
-2 -4
Vertical line 2 units to the left
of the y-axis. Slope is not
defined. Not the graph of a
function.
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Example (continued)
Graph each linear equation and determine its slope.
5
b. y 
2
5
Choose any number for x ; y must be .
2
x
y
52
0
–3 5 2
52
1
Horizontal line 5/2 units
above the x-axis. Slope 0.
The graph is that of a
constant function.
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Applications of Slope (optional)
The grade of a road is a number expressed as a percent
that tells how steep a road is on a hill or mountain. A
4% grade means the road rises 4 ft for every horizontal
distance of 100 ft.
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Example (optional)
Construction laws regarding access ramps for the
disabled state that every vertical rise of 1 ft requires a
horizontal run of 12 ft. What is the grade, or slope, of
such a ramp?
1
m
12
m  0.083  8.3%
The grade, or slope, of the ramp is 8.3%.
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Average Rate of Change (optional)
Slope can also be considered as an average rate of
change. To find the average rate of change between
any two data points on a graph, we determine the
slope of the line that passes through the two points.
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Example (optional)
The percent of travel bookings online has increased
from 6% in 1999 to 55% in 2007. The graph below
illustrates this trend. Find the average rate of change in
the percent of travel bookings made online from 1999
to 2007.
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Example (optional)
The coordinates of the two points on the graph are
(1999, 6%) and (2007, 55%).
Change in y
Slope  Average rate of change 
Change in x
55  6
49
1


6
2007  1999 8
8
The average rate of change over the 8-yr period was
1
an increase of 6 % per year.
8
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Slope-Intercept Equation
The linear function f given by f (x) = mx + b is written
in slope-intercept form. The graph of an equation in
this form is a straight line parallel to f (x) = mx.
The constant m
is called the
slope, and the
y-intercept is
(0, b).
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Example
Find the slope and y-intercept of the line with
equation y = –0.25x – 3.8.
Solution: y = –0.25x – 3.8
Slope = –0.25;
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y-intercept = (0, –3.8)
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Example
Find the slope and y-intercept of the line with equation
3x – 6y  7 = 0.
Solution: We solve for y:
3x  6y  7  0
6y  3x  7
1
1
 (6y)   (3x  7)
6
6
1
7
y x
2
6
1
7

Thus, the slope is
and the y-intercept is  0,  .

2
6
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Example
2
Graph y   x  4
3
Solution: The equation is
in slope-intercept form,
y = mx + b.
The y-intercept is (0, 4).
Plot this point, then use
the slope to locate a
second point.
rise change in y 2  move 2 units down
m


run change in x
3  move 3 units right
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Example (optional)
An anthropologist can use linear functions to estimate
the height of a male or a female, given the length of the
humerus, the bone from the elbow to the shoulder. The
height, in centimeters, of an adult male with a humerus
of length x, in centimeters, is given by the function
M x   2.89x  70.64
The height, in centimeters, of an adult female with a
humerus of length x is given by the function .
F x   2.75x  71.48
A 26-cm humerus was uncovered in a ruins.
Assuming it was from a female, how tall was she?
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Example (optional)
Solution: We substitute into the function:
f 26   2.75 26   71.48
 142.98
Thus, the female was 142.98 cm tall.
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