Unit 2: Graphing Equations
Lesson 7: Rate of Change
Formula for Finding Slope Given Two Points
Using two points: (x1, y1) (x2, y2)
In real world problems, slope is related to how something changes (usually over time). In this
case, slope is referred to as rate of change. Therefore, in real world problems when you are
asked to find the rate of change, you are actually finding the slope. The same formulas that
you use for slope will also be used for rate of change.
Example 1
1. What is the average rate of change
between hours 4 and 9?
2. What is the average rate of change
between hours 9 and 11? What most likely
occurred during this time?
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Unit 2: Graphing Equations
Example 2
Jerry purchased a house in 1994 for $230,000. In 2008, he sold the house for $378,000. What is
the average rate of change in the value of the house per year?
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Unit 2: Graphing Equations
Lesson 7: Rate of Change Practice
The following graph shows the weekly price of a stock over a ten week period. Use the graph to
answer the following questions:
1a. How much was the stock worth at the very beginning (week 0)? ___________
Write this as an ordered pair.
(# of week, Price of stock)
(0, ____)
b. How much was the stock worth during week 3? ___________
Write this as an ordered pair.
(# of week, Price of stock)
(3, ____)
c. What is the average rate of change during this time frame (week 0 to week 3)?
Write your two ordered pairs: (___ , ____)
Slope =
y2 – y1
x2 - x1
(___ , ____). Find the slope. (Slope and rate of
change are synonymous)
The rate of change from week 0 to week 3 is ________________. This means
that_________________________________________________________________________
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Unit 2: Graphing Equations
2. What is the average rate of change from week 3 to week 5?
Week 3 ordered pair: ___________
Rate of Change = slope =
Week 5 ordered pair _____________
y2 – y 1
x2 - x1
The rate of change from week 3 to week 5 is _____________.
6b. Explain what the rate of change means during this time frame.
3. Find the rate of change for the company’s final downfall, between weeks 7 and 10.
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Unit 2: Graphing Equations
4. As part of a New Year’s Resolution plan, Brenda started a weight loss program on January 1,
2009. On January 1, 2009 she weighed 185 pounds. Six months later she recorded a weight of
140 pounds. Find the average monthly rate of change of her weight in pounds per month.
Ordered Pairs: ___________________
January weight
___________________________
June weight
5. Jonathan was driving from Maryland to Virginia for vacation. He started his trip at 5:00am. By
7:00 am he had travelled 120 miles. By 9:00 am he had travelled 210 miles. He stopped for
breakfast between 9 and 10:00 am. By 12:00 pm he had reached his destination and had travelled
480 miles.
a. Create a graph to illustrate this situation. Don’t forget to label your axis and title your graph.
b. Find the average rate of change in miles per hour between 5:00 am and 9:00 am.
c. Find the average rate of change in miles per hour for the entire trip.
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Unit 2: Graphing Equations
6. Michael bought 573 shares of stock for $730 in 1995. When he sold the stock in 2007, the stock
was valued at $1735. What is the average rate of change of the price of the stock per year?
7. Claudia bought her first brand new car in 2004 for $23,458. She sold that car in 2010 for
$12,010. What is the average rate of change of the price of the car per year?
8. In the first year that Patricia started her career as a teacher, she made $26,800 per year. She is
now in her 17th year of teaching and her annual salary is $62,000 per year. What is the average
rate of change in dollars per year?
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Unit 2: Graphing Equations
1. Josie tracks the balance in her savings account
very carefully. She opened an account and in
January she deposited $480. In February she was
able to deposit $130 dollars more into her account.
In March, she did not work much, so she made no
deposits or withdrawals. In April she deposited
another $440. In May she bought a computer, so
she withdrew $500.
A. Create a graph to represent the balance in Josie’s savings account. (3 points)
B. What is the average rate of change between January and April? (2 points)
C. What is the average rate of change from the time she opened the account to the end of
May?
(2 points)
D. What is the average rate of change between February and March? How do you know?
(2 points)
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Unit 2: Graphing Equations
Lesson 7: Rate of Change – Answer Key
The following graph shows the weekly price of a stock over a ten week period. Use the graph to
answer the following
questions:
1a. How much was the stock worth at the very beginning (week 0)? $12
Write this as an ordered pair.
(# of week, Price of stock)
(0, 12)
b. How much was the stock worth during week 3? $19
Write this as an ordered pair.
(# of week, Price of stock)
(3, 19)
c. What is the average rate of change during this time frame (week 0 to week 3)?
Write your two ordered pairs: (0, 12)
Slope =
y2 – y1 =
x2 - x1
19 - 12
3 -0
=7
(3, 19)
Find the slope. (Slope and rate of
change are synonymous)
3
The rate of change from week 0 to week 3 is 7/3. This means that between weeks 0 and 3, the
average price of the stock rose $2.33 cents per week. (Since we are talking about money, I
converted the fraction 7/3 to a decimal. 7/3 = 2 – 1/3 which is 2.33333…)
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Unit 2: Graphing Equations
2. What is the average rate of change from week 3 to week 5?
Week 3 ordered pair: (3, 19)
Rate of Change = slope =
Week 5 ordered pair (5, 8)
y2 – y1 = 8 – 19
x2 - x1
5–3
= -11
2
The rate of change from week 3 to week 5 is -11/2.
2b. Explain what the rate of change means during this time frame.
During the time period between weeks 3 and 5, the average price of the stock fell $5.50 per week. I
changed 11/2 to a decimal since we were talking about money. 11/2 = 5.50. The price of the stock
fell because the slope is negative.
3. Find the rate of change for the company’s final downfall, between weeks 7 and 10.
Week 7 ordered pair (7, 13)
x1 y1
y2 – y1 = 2 – 13 = -11
x2 - x1 10 - 7
3
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Week 10 ordered pair (10, 2)
x2 y2
The rate of change between weeks 7 and 10 is -11/3
or -3.67. The average price of the stock fell $3.67 per
week between the weeks of 7 and 10.
Unit 2: Graphing Equations
4. As part of a New Year’s Resolution plan, Brenda started a weight loss program on January
1, 2009. On January 1, 2009 she weighed 185 pounds. Six months later she recorded a
weight of 140 pounds. Find the average monthly rate of change of her weight in pounds per
month.
Ordered Pairs:
(1, 185)
January weight
y2 – y1 = 140- 185 = -45 = -9
x2 - x1
6-1
5
(6, 140)
June weight
Brenda’s average rate of change is -9. This means
that she lost an average of 9 pounds per month
between January and June.
5. Jonathan was driving from Maryland to Virginia for vacation. He started his trip at
5:00am. By 7:00 am he had travelled 120 miles. By 9:00 am he had travelled 210 miles. He
stopped for breakfast between 9 and 10:00 am. By 12:00 pm he had reached his destination
and had travelled 480 miles.
a. Create a graph to illustrate this situation. Don’t forget to label your axis and title your
graph.
b. Find the average rate of change in miles per hour between 5:00 am and 9:00 am.
(5, 0) (9, 210)
y2 – y1 = 210-0 = 210 = 52.5
x2 - x1
9-5
4
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The average rate of change between 5 and 9 am is 52.5
miles per hour.
Unit 2: Graphing Equations
c. Find the average rate of change in miles per hour for the entire trip.
(5, 0) (12, 480)
y2 – y1 = 480-0 = 480 = 68.6
x2 - x1
12-5
7
The average rate of change over the entire trip
is 68.6 miles per hour.
6. Michael bought 573 shares of stock for $730 in 1995. When he sold the stock in 2007, the stock
was valued at $1735. What is the average rate of change of the price of the stock per year?
(year, price of stock)
(1995, 730) (2007, 1735)
**Note: The number of shares (573) is
irrelevant information in this problem. It is not
needed to find the rate of change.
y2 – y1 = 1735 – 730 = 1005 = 83.75
x2 - x1
2007-1995
12
The stock rose an average of $83.75 per year.
7. Claudia bought her first brand new car in 2004 for $23,458. She sold that car in 2010 for
$12,010. What is the average rate of change of the price of the car per year?
(year, price of car)
(2004, 23,458) (2010, 12,010)
y2 – y1 = 12010-23458 = -11448 = -1908
x2 - x1
2010-2004
6
The car’s value decreased in value by $1908 a year.
8. In the first year that Patricia started her career as a teacher, she made $26,800 per year. She is
now in her 17th year of teaching and her annual salary is $62,000 per year. What is the average
rate of change in dollars per year?
(year, salary)
(1, 26800) (17, 62000)
y2 – y1 = 62000-26800 = 35200 = 2200
x2 - x1
17-1
16
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Patricia’s salary has an average rate of change
of $2200 per year. Therefore, her salary
increases by about $2200 per year.
Unit 2: Graphing Equations
1. Josie tracks the balance in her savings account
very carefully. She opened an account on
December 31st and in January she deposited $480.
In February she was able to deposit $130 dollars
more into her account. In March, she did not work
much, so she made no deposits or withdrawals. In
April she deposited another $440. In May she
bought a computer, so she withdrew $500.
A. Create a graph to represent the balance in Josie’s savings account. (3 points)
B. What is the average rate of change between January and April? (2 points)
January (1, 480)
April (4, 1050)
y2 – y1 1050 – 480 = 570 = $190
x2 - x1
4-1
3
The rate of change between January and April is $190 per month.
C. What is the average rate of change from the time she opened the account to the end of
May?
(2 points)
Open account (0,0)
May (5,550)
y2 – y1 550-0 = 550 = $110
x2 - x1 5-0
5
The rate of change from the time she opened the account to the end of May is $110 per
month.
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Unit 2: Graphing Equations
D. What is the average rate of change between February and March? How do you know?
(2 points)
Between February and March, the rate of change is 0 because she did not make any
deposits or withdrawals. Her account balance remained the same. The graph shows a
horizontal line which also indicates a slope of 0.
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