Download average rate of change

Introduction
The average running speed of a human being is 5–8
mph. Imagine you set out on an 8-mile run that takes
you no longer than 1 hour. You run often, so you run
consistently and at the same speed for the entire hour.
The rate of change of your position for your runs is
always 8 mph and can be modeled linearly, because the
rate of change is constant.
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2.2.3: Identifying the Average Rate of Change
Introduction, continued
However, a friend of yours, averaging the same distance
in an hour, hasn’t built up the endurance needed to run 8
miles consistently. Sometimes your friend runs 9 mph,
sometimes he stops to rest, sometimes he walks, and
then he resumes running at 8 mph. He might do this
several times in the hour, sometimes running faster than
you, sometimes slower, and sometimes not running at
all. The rate of change of your friend's speed is not
constant and cannot be modeled linearly.
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2.2.3: Identifying the Average Rate of Change
Introduction, continued
The average rate of change for your friend’s speed, the
ratio of the change in the output of a function to the
change in the input for specific intervals, is inconsistent.
In this lesson, you will practice calculating the average
rate of change of quadratic functions over various
intervals.
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2.2.3: Identifying the Average Rate of Change
Key Concepts
• The average rate of change of a function is the rate of
change between any two points of a function; it is a
measure of how a quantity changes over some
interval.
• The average can be found by calculating the ratio of
the difference of output values to the difference of the
f (b) - f (a)
corresponding input values,
, from x = a to
b-a
x = b. This formula is often referred to as the average
rate of change formula.
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2.2.3: Identifying the Average Rate of Change
Key Concepts, continued
• Recall that the slope of a linear function is found
y 2 - y1 Dy rise
using the formula
=
=
.
x2 - x1 Dx run
• Although the formula for calculating the average rate
of change looks quite different from the formula used
to find the slope of a linear function, they are actually
quite similar.
• Both formulas are used to find the rate of change
between two specific points.
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2.2.3: Identifying the Average Rate of Change
Key Concepts, continued
• The rate of change of a linear function is always
constant, whereas the average rate of change of a
quadratic function is not constant.
• Choosing different x-values and their corresponding
y-values will result in different rates of change.
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2.2.3: Identifying the Average Rate of Change
Common Errors/Misconceptions
• assuming the rate of change is the same for every
interval of a quadratic function
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2.2.3: Identifying the Average Rate of Change
Guided Practice
Example 1
Calculate the average rate of change for the function
f(x) = x2 + 6x + 9 between x = 1 and x = 3.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
1. Evaluate the function for x = 3.
f(x) = x2 + 6x + 9
Original function
f(3) = (3)2 + 6(3) + 9
Substitute 3 for x.
f(3) = 36
Simplify.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
2. Evaluate the function for x = 1.
f(x) = x2 + 6x + 9
Original function
f(1) = (1)2 + 6(1) + 9
Substitute 1 for x.
f(1) = 16
Simplify.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
3. Use the average rate of change formula to
determine the average rate of change
between x = 1 and x = 3.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
f (b) - f (a) Average rate of
Average rate of change =
change formula
b-a
Average rate of change =
Average rate of change =
f (3) - f (1)
3 -1
36 - 16
2
Average rate of change = 10
Substitute 1 for a
and 3 for b.
Substitute the values
for f(3) and f(1).
Simplify.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
The average rate of change of f(x) = x2 + 6x + 9
between x = 1 and x = 3 is 10.
✔
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 1, continued
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2.2.3: Identifying the Average Rate of Change
Guided Practice
Example 2
Use the graph of the
function at right to
calculate the average rate
of change between x = –3
and x = –2.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
1. Use the graph to identify f (–2).
According to the graph, f (–2) = –1.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
2. Use the graph to identify f (–3).
According to the graph, f (–3) = 2.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
3. Use the average rate of change formula to
determine the average rate of change
between x = –3 and x = –2.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
Average rate of change =
Average rate of change =
Average rate of change =
f (b) - f (a)
b-a
f (-2) - f (-3)
(-2) - (-3)
-1- 2
1
Average rate of change = - 3
Average rate of
change formula
Substitute –3 for
a and –2 for b.
Substitute the values
for f (–3) and f (–2)
found from the
graph.
Simplify.
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
The average rate of the change of the function
between x = –3 and x = –2 is –3.
✔
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2.2.3: Identifying the Average Rate of Change
Guided Practice: Example 2, continued
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2.2.3: Identifying the Average Rate of Change