1.3
Linear Functions, Slope, and
Applications





Determine the slope of a line given two points on
the line.
Solve applied problems involving slope, or average
rate of change.
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.
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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 y  1 x  2
5
Identity Function
y = 1•x + 0 or y = x
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Examples
Constant Function
y = 0•x + b or y = -2
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Not a Function
Vertical line: x = 4
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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
Graph the function 2 x  3 y  3 and determine its slope.
Solution: Calculate two ordered pairs, plot the points,
graph the function, and determine its slope.
x  3:
2(3)  3 y  3
3 y  3  6  3
y  1;
x  9:
(3, 1)
2(9)  3 y  3
3 y  3  18  15
y  5; (9, 5)
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(3,1) (9,5)
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
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
‒2
‒2
‒2
y
3
0
‒4
m

y2  y1
x2  x1
30
3

2  (2) 0
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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Addison Wesley
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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
y2  y1
m
x2  x1
52
0
5 5
–3 5 2

0
2
2


0
52
1
3  0 3
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
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/falls 4 ft for every
horizontal distance of 100 ft.
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Example
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
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
The percent of American adolescents ages 12 to 19 who
are obese increased from about 6.5% in 1985 to 18% in
2008. The graph below illustrates this trend. Find the
average rate of change in the percent of adolescents
who are obese from 1985 to 2008.
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Example
The coordinates of the two points on the graph are
(1985, 6.5%) and (2008, 18%).
Change in y
Slope  Average rate of change 
Change in x
18%  6.5% 11.5%


 0.5%
2008  1985
23
The average rate of change over the 23-yr period was
an increase of 0.5% per year.
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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;
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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Addison Wesley
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Example
There is no proven way to predict a child’s adult
height, but there is a linear function that can be used to
estimate the adult height of a child, given the sum of
the child’s parents heights. The adult height M, in
inches of a male child whose parents’ total height is x,
in inches, can be estimated with the function
M  x   0.5x  2.5.
The adult height F, in inches, of a female child whose
parents’ total height is x, in inches, can be estimated
with the function F x  0.5x  2.5.
 
Estimate the height of a female child whose parents’
total height is 135 in. What is the domain of this
function?
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Example
Solution: We substitute into the function:
F  x   0.5x  2.5.
F 135  0.5 135  2.5
 65
Thus we can estimate the adult height of the
female child as 65 in., or 5 ft 5 in.
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