Math 2
Lesson 7-2: Rate of Change
Name ________________________________
Date ______________________________
Learning Goals:
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Explore the meaning of average rate of change over different intervals on nonlinear functions (symbolically,
graphically, and a table)
I can define interval, rate of change, and average rate of change.
I can explain the connection between average rate of change and the slope formula (change in y over change
in x).
I can calculate the average rate of change of a function, represented either by function notation, a graph, or a
table, over a specific input interval.
I can compare the rates of change of two or more functions when they are represented with function notation,
with a graph, or with a table.
I can interpret the meaning of the average rate of change (using units) as it relates to a real-world problem.
I. Introduction:
In 2012, Cedar Point’s attendance of 3,200,000 visitors was ranked 15th out of the top 20 amusement and
theme parks in North America. (FYI: The Magic Kingdom at Disneyworld in Florida topped out at
17,536,000!)
Many of you contributed to the Cedar Point total. Most likely the route you took to Cedar Point was the one
illustrated below.
The total distance from Mayfield High School to Cedar Point is 82 miles. According to Google Maps, it
should take you 1 hour and 25 minutes to drive to Cedar Point from here.
 If it takes 1 hour and 25 minutes to get to Cedar Point, what is the average speed of the trip?
Some hints:
d=r*t
1 hour and 25 minutes ≠ 1.25 hours.
 If you take this route, you would be on 271, 480, Ohio Turnpike (all highways) for most of the
drive. Does your computation mean you are driving approximately 58 mph on each of these
highways? Explain.
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Page 2
II. Consider the following scenario (fictitious, of course!):
A group of seniors decided to go to Cedar Point the Monday after prom. They drove the same route as
indicated by the map on the front page. Even though they were breaking school rules by cutting, the
seniors did not want to break the speed limit while they were driving to Cedar Point.
The posted speed limits are as follows:
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For route 250: 45 mph
For 271 & 480: 60 mph
For the Ohio Turnpike: 70 mph
Below is a graph comparing distance versus time driving. Answer the questions that follow.
1. During what time interval does it appear the seniors were driving on the Ohio Turnpike? How can
you tell from the graph?
2. During what time interval does it appear the seniors were driving on 271 & 480? How can you tell
from the graph?
3. During what time interval does it appear the seniors were driving on route 250? How can you tell
from the graph?
4. During what time interval does it appear the seniors stopped to eat breakfast? How can you tell from
the graph?
5. What is their average speed for the whole trip?
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III. Average Rate of Change
Study each of the graphs below. Which one do you think shows the greatest average rate of change over the
interval 1  x  6 ?
Write a sentence or two explaining what you think about the question above. Then stop for a class
discussion.
Now, let’s try it again. Which of the two graphs below has a greater average rate of change over the interval
8  x  2 ?
Notes:
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Page 4
Example: A projectile follows along a path given by the formula h(t )  960t  16t .
A.
Complete the table of values & then graph the path of the object.
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B.
Use the formula from above to calculate ARoC over the following intervals
(use the correct units):
1.) 10  t  20
2.) 20  t  30
3.) 30  t  40
4.) 25  t  35
C.
What do your computations in part B tell you about the projectile?
D.
Use a ruler to draw the secant lines going through the pairs of points. Do the direction of the
lines confirm the signs of your computations in part B?
Homework:
1. When it comes to American media habits, the growth of blogs, emails, and music downloads has
exceeded all expectations. (data from Newsweek magazine, summer of 2010)
 Calculate ARoC for each media category. Label each quantity with the correct units.
 Then indicate if the category is increasing or decreasing.
2. In 1998, Ralph purchased a house for $144,000. In 2009, the house was worth $245,000. Find the
annual average rate of change in dollars per year in the value of the house. Round your answer to the
nearest dollar.
3. The graph below represents Mr. Mackar’s marathon.
a.
What is the rate of change for interval A?
b. Explain what you think happened during
interval E.
c. If the rate of change for interval A had remained constant throughout the whole marathon, how long
would it have taken Mr. Mackar to finish the marathon? (There are 26.2 miles in a marathon.)
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4. Consider the following function:
a. Find the average rate of change
from x  7 to x  2 .
f ( x)  | x  4 |
b. Find the average rate of change
from x  3 to x  11.
c. What does your answer from part a mean about the graph? What about part b’s answer?
5. The graph below contains data on beverage consumption.
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Which beverage type has the greater average rate of change from 1980 to 2004? Use
computations to justify your response.
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In what year might bottled water consumption overtake carbonated soft drink consumption?
Use computations to justify your response.
6. Use the below to calculate the average rate of change for the following beverages from 1945 to 2005.
Show your work and label your answers with the appropriate units.
a. Coffee
Ordered Pairs: _________ and _________
b. Carbonated Soft drinks
Ordered Pairs: _________ and _________
c. Tea
Ordered Pairs: _________ and _________
d. ON THE OTHER SIDE OF THIS PAPER, Write a summary of the trends in beverage
consumption from 1945 to 2005. Use data to support your statements.
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