Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1.3 Linear Functions, Slope,...

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.

3 - 2Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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.


Examples

Linear Function

y = mx + b

Identity Function

y = 1•x + 0 or y = x1 25

y x


Examples

Constant Function

y = 0•x + b or y = -2

Not a Function

Vertical line: x = 4


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


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


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


m y2 y1

x2 x1

5 1

9 3

4

6

2

3

(3, 1) (9, 5)


Types of Slopes

Positive—line slants up from left to right

Negative—line slants down from left to right


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.


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.

3 - 12Copyright © 2012 Pearson Education, Inc. Publishing as

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


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


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.


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%


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.


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.


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.


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).


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)


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

3 - 22Copyright © 2012 Pearson Education, Inc. Publishing as

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


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?


Example

Thus we can estimate the adult height of the female child as 65 in., or 5 ft 5 in.

0.5135 13 2 55 . F

65

Solution: We substitute into the function:

0.5 2.5. F x x

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1.3 Linear Functions, Slope,...

Documents

Transcript of Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1.3 Linear Functions, Slope,...