1. 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 = 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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5. 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
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Example
Graph the function and determine its slope.
Solution: Calculate two ordered pairs, plot the points, graph the function, and determine its slope.
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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 y ‒2 3 ‒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. b.
Choose any number for x ; y must be
x 0 –3 1 y
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?
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 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%).
The average rate of change over the 23-yr period was an increase of 0.5% per year.
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19. 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:
Thus, the slope is and the y-intercept is
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Example
Graph
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.
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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
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?
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Example
Solution: We substitute into the function:
Thus we can estimate the adult height of the female child as 65 in., or 5 ft 5 in.
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