Lesson 3.7
Average Rate of Change
(exponential functions)
EQ: How do we analyze a real world scenario to
interpret an average rate of change? (F.IF.6)
Vocabulary: Average Rate of Change
1
3.3.3: Recognizing Average Rate of Change
Introduction
In the previous unit, we found the constant rate of
change of linear equations and functions using the slope
y 2 - y1
formula, m=
x2 - x1
The slope of a line is the ratio of the change in
y-values to the change in x-values.
The rate of change can be determined from graphs,
tables, and equations themselves. In this lesson, we will
extend our understanding of the slope of linear functions
to that of intervals of exponential functions.
2
3.3.3 Recognizing Average Rate of Change
Constant Rate
of Change
VS.
Average Rate
of Change
Key Concepts
Finding the rate of change of a non-linear
function, in our case, exponential functions, is
very similar to that of a linear function. You
still use the slope formula to calculate the rate
of change, but you are told by the interval
which two points to use for (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 )
4
3.3.3: Recognizing Average Rate of Change
Key Concepts, continued
• An interval is a continuous portion of a function.
• The rate of change of an interval is the average rate of
change for that period.
• Intervals can be noted using the format [a, b], where a
represents the initial x value of the interval and b
represents the final x value of the interval. Another
way to state the interval is a ≤ x ≤ b.
• For example, the interval [2, 7] means the portion of
the function where x = 2 through x = 7. You would
use the points (2, y) & (7, y) in the slope formula when
calculating the rate of change for the function.
5
3.3.3 Recognizing Average Rate of Change
Key Concepts, continued
Steps to Calculating Average
Rate of Change
1.
2.
3.
4.
Identify the interval to be observed.
Identify (x1, y1) as the starting point of the interval.
Identify (x2, y2) as the ending point of the interval.
Substitute (x1, y1) and (x2, y2) into the slope formula to
𝑦 −𝑦
calculate the rate of change. 𝑚 = 2 1
𝑥2 −𝑥1
5. The result is the average rate of change for the
interval between the two points identified.
6. Interpret your answer in the context of the problem.
6
3.3.3: Recognizing Average Rate of Change
Guided Practice
Example 1
In 2008, about 66 million
Year (x)
U.S. households had both
2008
landline phones and cell
2009
phones. Use the table
2010
to the right to calculate
and interpret the rate of
2011
change for the interval [2008, 2011].
Households in
millions (f(x))
66
61
56
51
7
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 1, continued
1. Determine the interval to be observed.
The interval to be observed is [2008, 2011], or where
2008 ≤ x ≤ 2011.
2. Determine (x1, y1).
The initial x-value is 2008 and the corresponding
y-value is 66; therefore, (x1, y1) is (2008, 66).
3. Determine (x2, y2).
The ending x-value is 2011 and the corresponding
y-value is 51; therefore, (x2, y2) is (2011, 51).
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3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 1, continued
4. Substitute (x1, y1) and (x2, y2) into the
slope formula to calculate the rate of
change.
y 2 - y1
=
x2 - x1
51- 66
2011- 2008
=
-15
3
= –5
Slope formula
Substitute (2008, 66) and (2011, 51)
for (x1, y1) and (x2, y2).
Simplify as needed.
9
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 1, continued
5. Interpret your answer in the context of the
problem.
The average rate of change for the interval [2008,
2011] is 5 million households per year. Meaning,
from 2008 to 2011 there were 5 million households
less per year who no longer had both a landline and
cell phone.
10
3.3.3: Recognizing Average Rate of Change
You Try 1
The table below represents a type of bacteria that
doubles every 36 hours. A Petri dish starts out with
12 of these bacteria.
Calculate and interpret the average rate of change over
the interval [2,5].
3.3.3: Recognizing Average Rate of Change
11
Guided Practice
Example 2
Jasper has invested an
amount of money into a
savings account. The graph
to the right shows the value
of his investment over a
period of time. What is the
rate of change for the
interval [1, 3]?
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3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued
1. Determine the interval to be observed.
The interval to observe is [1, 3], or where 1 ≤ x ≤ 3.
2. Identify the starting point of the interval.
The x-value of the starting point is 1. The corresponding yvalue is approximately 550.
The starting point of the interval is (1, 550).
3. Identify the ending point of the interval.
The x-value for the ending point is 3. The corresponding yvalue is approximately 1,100.
The ending point of the interval is (3, 1100).
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3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued
14
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued
4. Substitute (1, 550) and (3, 1100) into the
slope formula to calculate the rate of change.
y 2 - y1
x2 - x1
=
=
1100 - 550
3 -1
550
Slope formula
Substitute (1, 550) and (3, 1100)
for (x1, y1) and (x2, y2).
Simplify as needed.
2
= 275
3.3.3: Recognizing Average Rate of Change
15
Guided Practice: Example 2, continued
5. Interpret your answer in the context of the
problem.
The average rate of change for this function over the
interval [1, 3] is approximately $275 per year.
Meaning, from year 1 to year 3, Jasper’s average
investment value per year was $275.
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3.3.3: Recognizing Average Rate of Change
Guided Practice
Example 3
Jasper is curious about how
the rate of change differs for
the interval [3, 6]. Calculate
the rate of change using the
graph from Example 2.
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3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 3, continued
1. Determine the interval to be observed.
The interval to observe is [3, 6], or where 3 ≤ x ≤ 6.
2. Identify the starting point of the interval.
The x-value of the starting point is 3. The
corresponding y-value is approximately 1,100.
The starting point of the interval is (3, 1100).
3. Identify the ending point of the interval.
The x-value for the ending point is 6. The
corresponding y-value is approximately 3,100. The
ending point of the interval is (6, 3100).
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3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 3, continued
19
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 3, continued
4. Substitute (3, 1100) and (6, 3100) into the
slope formula to calculate the rate of change.
y 2 - y1
Slope formula
x2 - x1
=
=
3100 -1100
6-3
2000
Substitute (3, 1100) and (6, 3100)
for (x1, y1) and (x2, y2).
Simplify as needed.
3
» 666.67
3.3.3: Recognizing Average Rate of Change
20
Guided Practice: Example 3, continued
5. Interpret your answer in the context of the
problem.
The rate of change for this function over the interval
[3, 6] is approximately $666.67 per year. Meaning,
from year 3 to year 6, Jasper’s average investment
value per year was $666.67.
Notice that the rate of change for the interval [3, 6] is
much steeper than that of the interval [1, 3].
What can you conclude about Jasper’s investment
value over time?
21
3.3.3: Recognizing Average Rate of Change
You Try 2
Each year, volunteers at a three-day music festival
record the number of people who camp on the
festival grounds. The graph below shows the
number of campers for each of the last 20 years.
Calculate and interpret the average rate of change over
the intervals [3,9] and [9,16].
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3.3.3: Recognizing Average Rate of Change
Linear Rate
of Change
• You choose your
two points
• Rate of Change
is Constant
VS.
Both use
the slope
formula:
y 2 - y1
x2 - x1
Non-linear Rate
of Change
• You are told by
the interval
which two points
to use
• Rate of Change
is different at
different
intervals