Lesson #6: Average Rate of Change and
Symmetry of Functions
Lesson #6: Average Rate of
Change and Symmetry of Functions
Objectives:
• Interpret slope as a rate
of change
• Find a function’s
average rate of change
• Symmetry of a function
Think of slope as a ratio or a rate of change
• If x- and y-axes have the same unit of
measure, the slope has no units and is
therefore a ratio.
Example:
4 inches to 8 inches
Or 1 : 2
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Think of slope as a ratio or a rate of change
• If x- and y-axes have the different units of
measure, the slope is a rate or a rate of change.
Example:
100 miles per 2 gallons
This is a rate of change.
Gallons are changing with
respect to miles driven
Ex 1) The population of Colorado was 3,827,000 in 1995
and 4,665,000 in 2005. Over this 10-year period, what
was the average rate of change of the population?
Remember: rate of change is slope!
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
This rate of change is an average rate of change.
Average rate of change is always calculated over
an interval. [1995, 2005]
In Calculus, we look at another type of rate of
change called instantaneous rate of change.
Is the rate of change of a parabola constant?
Examine the intervals [0,1] and [1,2] below.
We can estimate the average rate of
change of a curve on intervals.
Average Rate of Change = Slope
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Ex 1) Use the graph of the function below to calculate
the average rate of change on the given intervals:
(a) 2,0
(b) 1,0
Ex 2) Find the average rate of change of from
0 to 1.
Ex 3) Find the average rate of change of 2 from
2 to 1.
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Types of Symmetry
There are many types of symmetry a relation can have,
however we are going to look at 3 types of symmetry in
this unit.
Symmetric with
respect to the -axis
Symmetric with
respect to the -axis
Symmetric with
respect to the origin
Types of Symmetry
1. If , and , are points on a graph ,
then is symmetric with respect to the -axis.
Test for Symmetry WRT the -axis:
Evaluate .
If , then is
symmetric with respect to
the -axis.
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Ex 1) Determine if 2 3 1 is symmetric
with respect to the -axis.
A function that is symmetric
with respect to the -axis is
called an even function.
Types of Symmetry
2. If , and , are points on a graph ,
then is symmetric with respect to the -axis.
Test for Symmetry WRT the -axis:
Evaluate .
If , then is
symmetric with respect to
the -axis.
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Ex 2) Determine if is symmetric with
respect to the -axis.
A relation that is symmetric with
respect to the -axis is not a function.
Types of Symmetry
3. If , and , are points on a graph ,
then is symmetric with respect to the origin.
Test for Symmetry WRT the origin:
Evaluate .
If , then is symmetric with respect to
the origin.
Algebra II with Trigonometry: Unit 1
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Lesson #6: Average Rate of Change and
Symmetry of Functions
Ex 3) Determine if respect to the origin.
is symmetric with
A function that is symmetric
with respect to the origin is
called an odd function.
Ex 4) Determine whether each function is even, odd, or
neither.
(a) 1 (b) 4 2 Algebra II with Trigonometry: Unit 1
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