3-3 Rate of Change and Slope
Find the rate of change represented in each table or graph.
To find the rate of change, use the coordinates (3, 6) and (0, 2).
So, the rate of change is
.
To find the rate of change, use the coordinates (3, 6) and (5, 2).
So, the rate of change is 4.
CCSS SENSE-MAKING
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3-3 Rate of Change and Slope
CCSS SENSE-MAKING Refer to the graph below.
a. Find the rate of change of prices from 2006 to 2008. Explain the meaning of the rate of change.
b. Without calculating, find a two year period that had a greater rate of change than 2006 2008. Explain.
c. Between which years would you guess the new stadium was built? Explain your reasoning.
a. To find the rate of change, use the coordinates (2008, 28.73) and (2006, 26.66).
So, the rate of change is 1.035, which means there was an average increase in ticket price of $1.035 per year.
b. Sample answer: The two year period that had a greater rate of change than 2006 2008 was 1998 2000. There
was a steeper segment, which means a greater rate of change.
c. Sample answer: I would guess that the new stadium was built between 1998 and 2000, because the ticket prices
show a sharp increase.
Determine whether each function is linear. Write yes or no. Explain.
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3-3 Rate of Change and Slope
Determine whether each function is linear. Write yes or no. Explain.
x
y
Slope
7
5
-
4
4
1
3
2
2
5
1
The rate of change,
Rate of
Change
-
, is constant. So, the function is linear.
x
y
Slope
8
12
8
5
-
16
3
20
0
24
2
Rate of
Change
-
The rate of change is not constant. So, the function is not linear.
Find the slope of the line that passes through each pair of points.
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3-3 Rate of Change and Slope
Find the slope of the line that passes through each pair of points.
(5, 3), (6, 9)
So, the slope is 6.
( 4, 3), ( 2, 1)
So, the slope is 1.
(6, 2), (8, 3)
So, the slope is
.
(1, 10), ( 8, 3)
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3-3 Rate of Change and Slope
(1, 10), ( 8, 3)
So, the slope is
.
( 3, 7), ( 3, 4)
Dividing by 0 is undefined. So, the slope is undefined.
(5, 2), ( 6, 2)
So, the slope is 0.
Find the value of r so the line that passes through each pair of points has the given slope.
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3-3 Rate of Change and Slope
Find the value of r so the line that passes through each pair of points has the given slope.
( 4, r), ( 8, 3), m = 5
So, the line goes through ( 4, 17).
(5, 2), ( 7, r), m =
So, the line goes through ( 7, 8).
FindManual
the rate
of change
represented
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in each table or graph.
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3-3 Rate of Change and Slope
Find the rate of change represented in each table or graph.
To find the rate of change, use the coordinates (10, 3) and (5, 2).
So, the rate of change is
.
To find the rate of change, use the coordinates (1, 15) and (2, 9).
So, the rate of change is 6.
Find the rate of change represented in each table or graph.
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3-3 Rate of Change and Slope
Find the rate of change represented in each table or graph.
To find the rate of change, use the coordinates ( 3, 7) and (3, 1).
So, the rate of change is
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.
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3-3 Rate of Change and Slope
To find the rate of change, use the coordinates ( 4, 6) and (4, 2).
So, the rate of change is
.
SPORTS
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3-3 Rate of Change and Slope
SPORTS What was the annual rate of change from 2004 to 2008 for women participating in college lacrosse?
Explain the meaning of the rate of change.
To find the rate of change, use the coordinates (2004, 5545) and (2008, 6830).
So, the rate of change is 321.25. This rate of change means there was an average increase of 321.25 women per
year competing in triathlons.
RETAIL
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3-3 Rate of Change and Slope
RETAIL The average retail price in the spring of 2009 for a used car is shown in the table below.
a. Write a linear function to model the price of the car with respect to age.
b. Interpret the meaning of the slope of the line.
c. Assuming a constant rate of change, predict the average retail price for a 7 year old car.
a. To find a linear function, find the yslope, or rate of change.
Find the y-intercept.
So, a linear function to model the price of the car with respect to age is p = 1221t + 19,820.
b. The slope of $1221 represents how much the car value depreciates by each year.
c.
So, the average retail price for a 7-year-old car is $11,273.
Determine whether each function is linear. Write yes or no. Explain.
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3-3 Rate of Change and Slope
Determine whether each function is linear. Write yes or no. Explain.
x
y
Rate of
Change
-
Slope
4
1
-
2
1
1
0
3
1
2
5
1
4
7
1
The rate of change, 1, is constant. So, the function is linear.
x
y
Slope
7
11
-
5
14
3
17
1
20
0
23
Rate of
Change
-
3
The rate of change is not constant. So, the function is not linear.
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3-3 Rate of Change and Slope
Rate of
Change
-
x
y
Slope
0.2
0.7
-
0
0.4
0.25
0.2
0.1
1
0.4
0.3
2
0.6
0.6
1.5
The rate of change is not constant. So, the function is not linear.
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3-3 Rate of Change and Slope
x
y
Slope
Rate of
Change
-
-
1
2
The rate of change,
, is constant. So, the function is linear.
Find the slope of the line that passes through each pair of points.
(4, 3), ( 1, 6)
So, the slope is
.
(8, 2), (1, 1)
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3-3 Rate of Change and Slope
(8, 2), (1, 1)
So, the slope is
.
(2, 2), ( 2, 2)
So, the slope is 1.
(6, 10), (6, 14)
Dividing by 0 is undefined. So, the slope is undefined.
(5, 4), (9, 4)
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3-3 Rate of Change and Slope
(5, 4), (9, 4)
So, the slope is 0.
(11, 7), ( 6, 2)
So, the slope is
.
( 3, 5), (3, 6)
So, the slope is
.
( 3, 2), (7, 2)
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3-3 Rate of Change and Slope
( 3, 2), (7, 2)
So, the slope is 0.
(8, 10), ( 4, 6)
So, the slope is
.
( 8, 6), ( 8, 4)
Dividing by 0 is undefined. So, the slope is undefined.
( 12, 15), (18, 13)
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3-3 Rate of Change and Slope
( 12, 15), (18, 13)
So, the slope is
.
( 8, 15), ( 2, 5)
So, the slope is
.
Find the value of r so the line that passes through each pair of points has the given slope.
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3-3 Rate of Change and Slope
Find the value of r so the line that passes through each pair of points has the given slope.
(12, 10), ( 2, r), m = 4
So, the line goes through ( 2, 66).
(r, 5), (3, 13), m = 8
So, the line goes through
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.
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3-3 Rate of Change and Slope
(3, 5), ( 3, r), m =
So, the line goes through ( 3,
).
( 2, 8), (r, 4), m =
So, the line goes through (6, 4).
CCSS TOOLS Use a ruler to estimate the slope of each object.
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3-3 Rate of Change and Slope
CCSS TOOLS Use a ruler to estimate the slope of each object.
Refer to photo on Page 178.
Sample answer: about 0.5
Use
Refer to photo on Page 178.
Sample answer: about 1
Use
Because it slopes up from right to left, the slope is negative.
DRIVING When driving up a certain hill, you rise 15 feet for every 1000 feet you drive forward. What is the slope
of the road?
The rise is 15 feet and the run is 1000 feet. Because you are driving up a hill, the slope is positive.
So, the slope of the road is
.
Find the slope of the line that passes through each pair of points.
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3-3 Rate of Change and Slope
Find the slope of the line that passes through each pair of points.
So, the slope is
.
Dividing by 0 is undefined. So, the slope is undefined.
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3-3 Rate of Change and Slope
So, the slope is
.
MULTIPLE REPRESENTATIONS In this problem, you will investigate why the slope of a
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3-3 Rate of Change and Slope
MULTIPLE REPRESENTATIONS In this problem, you will investigate why the slope of a
line through any two points on that line is constant.
a. VISUAL Sketch a line
that contains points A, B, A , and B on a coordinate plane.
ABC and A B C with right angles C and C . Describe
b. GEOMETRIC
and
and
and
How are triangles ABC and A B C related? What does that imply for the slope between any two
distinct points on line ?
a.
. Place four points on the line and label them from left to right A, B, A , and B .
Draw a line straight down from B and straight across from A to form a right angle and label the vertex of the angle
C. Repeat the same process from B and A to form a right angle at a point C .
and
and
c
ABC
A B C are similar because line
is a transversal of parallel segments
, so ABC
A B C since corresponding angles of parallel lines are congruent. C
C
because all right angles are congruent. Therefore, by AA Similarity, ABC
A B C . The ratios formed by
corresponding sides of similar triangles are equal. The slope of a line is defined as
through points A and B, is equal to
. So
, the slope of line
, the slope of line through points A and B .
BASKETBALL The table shown below shows the average points per game (PPG) Michael Redd has scored in
each of his first 9 seasons with the NBA s Milwaukee Bucks.
Season
PPG
1
2.2
2
11.4
3
15.1
4
21.7
5
23.0
6
25.4
7
26.7
8
22.7
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3-3 Rate of Change and Slope
8
22.7
9
21.2
a. Make a graph of the data. Connect each pair of adjacent points with a line.
b. Use the graph to determine in which period Michael Redd s PPG increased the fastest. Explain your reasoning.
c. Discuss the difference in the rate of change from season 1 through season 4, from season 4 through season 7,
from season 7 through season 9.
a. Graph the season on the x-axis and the points per game on the y-axis. Then, plot the ordered pairs from the table
and connect the points with straight lines.
b. To find the period that Michael Redd s PPG increased the fastest, look for the steepest slope or find the slope
and locate the one with the highest rate of change. .
x
y
Slope
Rate of
Change
1
2.2
-
-
2
11.4
9.2
3
15.1
3.7
4
21.7
6.6
5
23.0
1.3
6
25.4
2.4
7
26.7
1.3
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3-3 Rate of Change and Slope
8
22.7
4
9
21.2
1.5
From Season 1 to Season 2, the line is the steepest. So, that is the period Michael Redd s PPG increased the fastest.
c.
x
y
Slope
Rate of
Change
1
2.2
-
-
4
21.7
6.5
7
26.7
1.7
9
21.2
2.8
The rate of change was much more dramatic or steeper in the first four years, it leveled off the next three seasons,
and
was negative and steeper the last two seasons.
REASONING
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3-3 Rate of Change and Slope
REASONING Why does the Slope Formula not work for vertical lines? Explain.
Let (x 1, y 1) = ( 2, 2) and (x 2, y 2) = (4,
Division by 0 is undefined. Thus, the slope is undefined. This occurs when the difference in the x values for vertical
OPEN ENDED
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3-3 Rate of Change and Slope
OPEN ENDED Use what you know about rate of change to describe the function represented by the table.
To find the rate of change, use the coordinates (4, 9.0) and (6, 13.5).
So, the rate of change is
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inches of growth per week.
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3-3 Rate of Change and Slope
CHALLENGE Find the value of d so the line that passes through (a, b) and (c, d) has a slope of
Set the slope formula equal to
.
d using the coordinates (a, b) and (c, d).
So, the value of d so the line that passes through (a, b) and (c, d) has a slope of
.
WRITING IN MATH Explain how the rate of change and slope are related and how to find the slope of a line.
Students answers may vary. Sample answer:
The slope and rate of change are ratios. Rate of change is a ratio that describes how much one quantity changes
with respect to a change in another quantity. The slope of a line is the ratio of the change in the y coordinates to the
change in the x coordinates.
CCSS ARGUMENTS
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a
a
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3-3 Rate of Change and Slope
CCSS ARGUMENTS Kyle and Luna are finding the value of a so the line that passes through (10, a) and ( 2, 8)
has a slope of
. Is either of them correct? Explain.
Luna is correct because she found the change in y divided by the change in x. Kyle is incorrect because he found the
change in x divided by the change in y.
.
The cost of prints from an online photo processor is given by C(p ) = 29.99 + 0.13p . $29.99 is the cost of the
membership, and p is the number of 4 inch by 6 inch prints. What does the slope represent?
A cost per print
B cost of the membership
C cost of the membership and 1 print
D number of prints
The slope is 0.13. It represents the cost per print, or choice A. Choice B is the cost of membership or $29.99. Choice
C is the cost of the membership and one print or $29.99 + $0.13 = $30.12. Choice D is the number of prints or p . So,
the correct choice is A.
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Danita
bought
a computer
for
$1200 and its value depreciated linearly. After 2 years, the value was $250. WhatPage
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3-3 Rate of Change and Slope
Danita bought a computer for $1200 and its value depreciated linearly. After 2 years, the value was $250. What was
the amount of yearly depreciation?
F $950
G $475
H $250
J $225
Since the deprecation is linear, the computer value dropped the same amount every year. Over 2 years it dropped
from $1200 to $250, or $950. To find the amount for one year, divide $950 by 2, which is $475. So, the correct choice
is G.
SHORT RESPONSE The graph represents how much the Wright Brothers National Monument charges visitors.
How much does the park charge each visitor?
The graph shows a constant rate of change, so the cost per visitor is constant. Look at the cost for one visitor. Find 1
visitor on the x-axis. Move your finger up to the corresponding point, then left to the y-axis. There, you see a 4. So,
the cost for 1 visitor is $4.
PROBABILITY
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3-3 Rate of Change and Slope
PROBABILITY At a gymnastics camp, 1 gymnast is chosen at random from each team. The Flipstars Gymnastics
Team consists of 5 eleven year olds, 7 twelve year olds, 10 thirteen year olds, and 8 fourteen year olds. What is
the probability that the age of the gymnast chosen is an odd number?
A
B
C
D
The probability of choosing a girl with an age that is an odd number is
. So the correct choice is C.
Solve each equation by graphing.
3x + 18 = 0
The related function is y = 3x + 18.
The graph intersects the x-axis at 6. So, the solution is 6.
8x
32 = 0
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3-3 Rate of Change and Slope
8x
32 = 0
The related function is y = 8x
32.
The graph intersects the x-axis at 4. So, the solution is 4.
0 = 12x
48
The related function is y = 12x
48.
The graph intersects the x-axis at 4. So, the solution is 4.
Find the x- and y-intercepts of the graph of each linear function.
The x-coordinate of the point at which the graph of an equation crosses the x-axis is the x-intercept. The ycoordinate of the point at which the graph of an equation crosses the y-axis is the y-intercept. The graph crosses the
x-axis at ( 2, 0) and the y-axis at (0, 2). So the x-intercept is 2 and the y-intercept is 2.
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3-3 Rate of Change and Slope
The x-intercept is the point where the line crosses the x-axis. The y-value at that point is 0. So, the x-intercept is 1.
The y-intercept is the point where the line crosses the y-axis. The x-value at that point is 0. So, the y-intercept is 2.
HOMECOMING Dance tickets are $9 for one person and $15 for two people. If a group of seven students wishes
to go to the dance, write and solve an equation that would represent the least expensive price p of their tickets.
Let p be the price, x be the number of couples tickets and y be the number of single tickets. If seven students are
going, then they can form three couples with one student left over.
So, the least expensive price p of their tickets is $54.
Find each quotient.
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3-3 Rate of Change and Slope
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