section 3.4
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Section 3.4. Slope and Rates of Change. Page 190. Slope. The rise , or change in y, is y 2 y 1 , and the run , or change in x, is x 2 – x 1. Example. Page 191. Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution. ( –4 , 1 ). - PowerPoint PPT PresentationTRANSCRIPT
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