Post on 19-Dec-2015
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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 y-
intercept. 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
Identity Function
y = 1•x + 0 or y = x1 25
y x
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Examples
Constant Function
y = 0•x + b or y = -2
Not a Function
Vertical line: x = 4
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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.
y = 2
x = 2
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Slope
The slope m of a line containing the points (x1, y1) and (x2, y2) is given by
m rise
run
the change in y
the change in x
y2 y1
x2 x1
y1 y2
x1 x2
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ExampleGraph the function and determine its slope.
2( ) 3 3
3 3 6 3
1; (3,
3
1)
y
y
y
2 3 3x y
Solution: Calculate two ordered pairs, plot the points, graph the function, and determine its slope.
2( ) 3 3
3 3 18
9
15
5; (9, 5)
y
y
y
3:x
9 :x
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m y2 y1
x2 x1
5 1
9 3
4
6
2
3
(3, 1) (9, 5)
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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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Addison Wesley
ExampleGraph each linear equation and determine its slope.
a. x = –2
Choose any number for y ; x must be –2.
Vertical line 2 units to the left of the y-axis. Slope is not defined. Not the graph of a function.
x y
‒2 3
‒2 0
‒2 ‒4
2 1
2 1
3 0 3
( ) 02 2
y ym
x x
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Example (continued)
Graph each linear equation and determine its slope.
b.
Horizontal line 5/2 units above the x-axis. Slope 0. The graph is that of a constant function.
y 5
2Choose any number for x ; y must be
5
2.
x 0 –3 1
5 25 25 2
y 2 1
2 1
5 52 2
00
3
0
3
y ym
x x
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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
The grade, or slope, of the ramp is 8.3%.
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?
m 1
12
m 0.083 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%).
Slope Average rate of change Change in y
Change in x
2008 1985
18% 6. 11.55 %%0.5%
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 they-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
Thus, the slope is and the y-intercept is1
20,
7
6
.
6y 3x 7
1
6( 6y)
1
6( 3x 7)
y 1
2x
7
6
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Addison Wesley
Example
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.
y 2
3x 4
m rise
run
change in y
change in x
2
3
move 2 units down
move 3 units right
Graph
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ExampleThere 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
0.5 2.5. M x x
0.5 2.5. F x x
The adult height F, in inches, of a female child whose parents’ total height is x, in inches, can be estimated with the function
Estimate the height of a female child whose parents’ total height is 135 in. What is the domain of this function?