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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 Change1
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
formula, m=
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
Average Rate of Change
VS.
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 () and ()
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
6
3.3.3: Recognizing Average Rate of Change
Steps to Calculating Average Rate of Change
1. Identify the interval to be observed.
2. Identify (x1, y1) as the starting point of the interval.
3. Identify (x2, y2) as the ending point of the interval.
4. Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change.
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.
Guided Practice
Example 1In 2008, about 66 million U.S. households had both landline phones and cell phones. Use the table to the right to calculate and interpret the rate of
change for the interval [2008, 2011].
7
3.3.3: Recognizing Average Rate of Change
Year (x)Households in millions (f(x))
2008 66
2009 61
2010 56
2011 51
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).
8
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.
Slope formula
Substitute (2008, 66) and (2011, 51)
for (x1, y1) and (x2, y2).
Simplify as needed.
= –59
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 1The 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].
11
3.3.3: Recognizing Average Rate of Change
Guided Practice
Example 2Jasper 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]?
12
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued1. 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 y-value 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 y-value is approximately 1,100.
The ending point of the interval is (3, 1100). 13
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.
Slope formula
Substitute (1, 550) and (3, 1100)
for (x1, y1) and (x2, y2).
Simplify as needed.
= 27515
3.3.3: Recognizing Average Rate of Change
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.
16
3.3.3: Recognizing Average Rate of Change
Guided Practice
Example 3Jasper 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.
17
3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 3, continued1. 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).
18
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.
Slope formula
Substitute (3, 1100) and (6, 3100)
for (x1, y1) and (x2, y2).
Simplify as needed.
20
3.3.3: Recognizing Average Rate of Change
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 2Each 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]. 22
3.3.3: Recognizing Average Rate of Change
Linear Rate of Change
Non-linear Rate of Change
VS.
• You choose your two points
• Rate of Change is Constant
• You are told by the interval which two points to use
• Rate of Change is different at different intervals
Both use the slope formula:
![Sample Exercise 14.1 Calculating an Average Rate of … average rate of reaction over this time interval. Plan: The average rate is given by the change in concentration, Δ[A], divided](https://static.documents.pub/doc/80x56/5afeed9a7f8b9a994d8facbb/sample-exercise-141-calculating-an-average-rate-of-average-rate-of-reaction.jpg)
![2 Average Rate of Change of f over [a, b]: Difference Quotient The average rate of change of the function f over the interval [a, b] is Average rate of.](https://static.documents.pub/doc/80x56/56649d6d5503460f94a4cf3b/2-average-rate-of-change-of-f-over-a-b-difference-quotient-the-average.jpg)
