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Average Rate of Change
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Warm upFind the domain and range of the following graphs.
-10 -8 -6 -4 -2 2 4 6
-10
-8
-6
-4
-2
2
4
6
8
10
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
Coordinate AlgebraUNIT QUESTION: How can we use real-world situations to construct and compare linear and exponential models and solve problems?Standards: MCC9-12.A.REI.10, 11, F.IF.1-7, 9, F.BF.1-3, F.LE.1-3, 5
Today’s Question:How do I find the average rate of change of a function?Standard: MCC9-12.F.IF.6
Average Rate of Change
Rate of Change of a Linear Relationship
The rate of change of a linear relationship is the slope of the line.
rise
run
Rate of Change =
riserun
Rates of change are seen everywhere.
The steepness of the roof of a house is referred to as the pitch of the roof by home builders.
Give one reason why some homes have roofs which have a greater pitch.
There is less snow buildup in the wintertime.
Engineers refer to the rate of change of a road as the grade.
They often represent the rate of change as a percentage.
A grade of 8% would mean for every rise of 8 units there is a run of 100 units.
8100
= 8%
Rate of change = 8
100
The steepness of wheelchair ramps is of great importance for safety.
Rate of change of wheelchair ramp =
112
If the rise is 1.5 m, what is the run?
Answer: 18 m because
11 2
112
1518
.
3 m
5 m
Determine the rate of change (pitch) of the roof.
35change of rate
23
3
3
32
change of rate
33
change of rate
1
Determine the rate of change of each staircase.
Determine the rate of change.Which points will you use to determine rise and run?
= $5/hr
runrise
change of rate
hr 420 $
20
4
Earnings
Number of Hours WorkedWhat does this rate of change represent?
The hourly wage
Rate of Change• Ratio describing how one
quantity changes as another quantity changes
• Slope can be used to describe it
Rate of Change• Positive – increases over time
• Negative – decreases over time
Rate of Change• Linear functions have a
constant rate of change, meaning values increase or decrease at the SAME rate over a period of time
Rate of Change• Horizontal lines have 0
rate of change
• Vertical lines have undefined rate of change
Average Rate of Change
2 1
2
2
1
1
2 1
( ) ( )y ym becomesx x
f x f xx x
Ex 1 Find the Average Rate of Change
f(x) = 2x2 – 3 from [2, 4].
2 1
2 1
( ) ( ) (4) (2)4 2
f x f x f fx x
29 54 2
124
22
Ex 2 Find the Average Rate of Change
f(x) = 3x – 2 from [2, 5].
2 1
2 1
( ) ( ) (5) (2)5 2
f x f x f fx x
241 73
234 787
f(x) = -4x + 10 from [-1, 3].
m = -4
Ex 3 Find the Average Rate of Change
A. Find the rate of change from day 1 to 2.
m = 11
Ex 4 Find the Average Rate of Change
Days (x) Amount of Bacteria f(x)
1 192 303 484 765 1216 192
B. Find the rate of change from day 2 to 5.
91 30.33
In 2008, about 66 million U.S. households had both landline phones & cell phones. Find the rate of change from 2008 – 2011.
m = -5
Ex 5 Find the Average Rate of Change
Year (x) Households in Millions f(x)
2008 662009 612010 562011 51
51 662011 2008
What does this mean?It decreased 5 million households per year from 2008 – 11.
