Copyright © 2013 Pearson Education, Inc.

Section 3.4

Slope and Rates of Change

Slope

The rise, or change in y, is y2 y1, and the run, or change in x, is x2 – x1.

Page 190

Example

Use the two points to find the slope of the line. Interpret the slope in terms of rise and run.

Solution (–4, 1)

(0, –2)

2 1

2 1

( )

4 0

4

3

2

3

4

1

y ym

x x

The rise is 3 units and the run is –4 units.

Page 191

43

43

)4(0

12

OR

Example

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)Solution

2 1

2 1

0 (

a.

4 3

7

( )

3)

3

y ym

x x

Page 192

Example

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)Solution

2 1

2 1

3 ( 3)

6

b.

( )

1

3

2 4

2

y ym

x x

Page 192

Example

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)Solution

2 1

2 1

c.

( )

0

)

4

4

0

2 2

4

(

y ym

x x

Page 192

Example

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)Solution

2 1

2 1

2 5

d.

( )

undef

4 4

0ed

3in

y ym

x x

Page 192

Find the slope of the line containing the points (-3, 4) and (-4,- 2)

34

42

)3(4

42

12

12

xx

yym

Finding Slope of a Line, p 249

6 1

6

Find the slope of the line containing the points (4,-2) and (-1,5)

14

52

)1(4

52

12

12

xx

yym

5

7

5

7

5

7

oror

Slope

Positive slope: rises from left to rightNegative slope: falls from left to right

Page 193

Slope

Zero slope: horizontal lineUndefined slope: vertical line

Page 193

Example

Find the slope of each line. a. b.

Solutiona. The graph rises 2 units for each unit of run

m = 2/1 = 2.b. The line is vertical, so the slope is undefined.

Page 193

Example

Sketch a line passing through the point (1, 2) and having slope 3/4.

SolutionStart by plotting (1, 2).The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4.The line passes through the point (1 + 4, 2 + 3) = (5, 5).

Page 193

Slope as a Rate of Change

When lines are used to model physical quantities in applications, their slopes provide important information.

Slope measures the rate of change in a quantity.

Page 195

Example

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept

represent?

Solutiona. The y-intercept is 35, so the boat

is initially 35 miles from the dock.

Page 195similar to Example 7&8

and #87 from homeworkand #91

Example (cont)

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.b. The graph passes through the point (4, 15). Discuss

the meaning of this point.

Solutionb. The point (4, 15) means that

after 4 hours the boat is 15 miles from the dock.

Page 195similar to Example 7&8

and #87 from homeworkand #91

Example (cont)

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.c. Find the slope of the line. Interpret the slope as a rate

of change.

Solution

c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

15 05

4 7m

Page 195similar to Example 7&8

and #87 from homeworkand #91

Example: #88 p 202

Electricity: The graph shows how voltage is related to amperage in an electrical circuit. The slope corresponds to the resistance in ohms. Find the resistance in this electrical circuit.

a. Find the slope of the line passing through the points. Look at the graph on page 202 and identify two points.

(0,0), (10, 20) and (20, 40) are possible

b. Interpret the slope as resistance in this electrical circuit.0.5 ohm

5.2

1

020

010orm

Example: #91 p 202

Median Household Income: In 2000, median family income was about $42,000, and in 2008 it was about $50,000.

a. Find the slope of the line passing through the points (2000,42000) and (2008,50000)

b. Interpret the slope as rate of change.Median family income increased on average by $1000/year over this time period

c. If this trend continues, estimate the median family income in 2014.

$56,000 ($14000 added to $42000 or $6000 added to $50,000

10008

8000

20002008

4200050000

m

Example

When a street vendor sells 40 tacos, his profit is $24, and when he sells 75 tacos, his profit is $66.a. Find the slope of the line passing through the points

(40, 24) and (75, 66)b. Interpret the slope as a rate of change.Solution

b. Profit increases on average, by $1.20 for each additional taco sold.

66 24 42a. 1.2

75 40 35m

Example #86 on page 202

Profit from Tablet Computers: When a company manufactures 500 tablet computers, its profit is $100,000, and when it manufactures 1500 tablet computers, its profit is $400,000.a. Find the slope of the line passing through the points

(500, 100000) and (1500, 400000)

b. Interpret the slope as a rate of change.The average profit is $300/tablets computer.

3001000

300000

5001500

100000400000

m

DONE

Objectives

• Finding Slopes of Lines

• Slope as a Rate of Change