Math 110
In Class Exercises, Lecture 8
Page 1 / 12
Symmetry ; Average Rate Of Change
X
Symmetry: Graphically
Goal: Identify equations that are symmetric, and their type of symmetry
1) Does the graph to the left illustrate
an equation with any symmetry? If it
10
does, what symmetry does it illustrate?
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2 3 4 5 6 7 8 9 10
X
Y
10
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2) Does the graph to the left illustrate
an equation with any symmetry? If it
does, what symmetry does it illustrate?
2 3 4 5 6 7 8 9 10
Y
Math 110
©, ® 2003, Michael Panitz
Page 1 / 12
X
Math 110
In Class Exercises, Lecture 8
10
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
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3) Does the graph to the left illustrate
an equation with any symmetry? If it
does, what symmetry does it illustrate?
2 3 4 5 6 7 8 9 10
X
Y
10
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
4) Does the graph to the left illustrate
an equation with any symmetry? If it
does, what symmetry does it illustrate?
2 3 4 5 6 7 8 9 10
Y
Math 110
©, ® 2003, Michael Panitz
Page 2 / 12
Math 110
In Class Exercises, Lecture 8
Page 3 / 12
Symmetry: With Tables
Goal: Identify equations that are symmetric, and their type of symmetry – from a table!
Previously, you’ve seen something like this:
5) Given the points to the left, does the
equation that generated those points appear to
(0, 0), (2, 4), (6, 8), (10, 12), (-2, -4), (-6, -8), (-10,have any particular symmetry?
12)
We can also write the exact same information out
like, in a table of points:
X
Y
0 2
0 4
6
8
10
12
-2
-4
-6
-8
-10
-12
If it does, what symmetry does it illustrate?
How do you know?
6) Given the points to the left, does the
equation that generated those points appear to
have any particular symmetry?
Table Of Points:
X
Y
0
-1
1
1
2
7
3
17
-1
1
-2
7
-3
17
If it does, what symmetry does it illustrate?
How do you know?
Math 110
©, ® 2003, Michael Panitz
Page 3 / 12
Math 110
In Class Exercises, Lecture 8
7) Given the points to the left, does the
equation that generated those points appear to
have any particular symmetry?
Table Of Points:
X
Y
0 2
0 4
6
8
Page 4 / 12
10
12
-2
-3
-6
-7
-10
-11
If it does, what symmetry does it illustrate?
How do you know?
8) Given the points to the left, does the
equation that generated those points appear to
have any particular symmetry?
Table Of Points:
X
Y
10
30
2
3
1
1
-1
0
2
-3
1
-1
10
-30
If it does, what symmetry does it illustrate?
How do you know?
Math 110
©, ® 2003, Michael Panitz
Page 4 / 12
Math 110
In Class Exercises, Lecture 8
Page 5 / 12
Symmetry: With Formulas
Goal: Identify equations that are symmetric, and their type of symmetry – from a table!
9) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  x2  2
Show your work to test for the following types of symmetry:
x-axis
y-axis
origin
10) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  x 6  3x 4  x 2  2
Show your work to test for the following types of symmetry:
x-axis
y-axis
Math 110
©, ® 2003, Michael Panitz
origin
Page 5 / 12
Math 110
In Class Exercises, Lecture 8
Page 6 / 12
11) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  3x 8  3x 6  17 x 4  x 2  2
Show your work to test for the following types of symmetry:
x-axis
y-axis
origin
12) If an equation is a polynomial (it only has terms like 3x2, or 2x3 – nothing funky like absolute
value, etc), how can you quickly tell if it’s symmetric about the y-axis?
Why are functions that are symmetric about the y-axis called even functions?
Math 110
©, ® 2003, Michael Panitz
Page 6 / 12
Math 110
In Class Exercises, Lecture 8
Page 7 / 12
13) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  x3  x
Show your work to test for the following types of symmetry:
x-axis
y-axis
origin
14) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  2x 5  x 3  7 x
Show your work to test for the following types of symmetry:
x-axis
y-axis
Math 110
©, ® 2003, Michael Panitz
origin
Page 7 / 12
Math 110
In Class Exercises, Lecture 8
Page 8 / 12
15) Is the following formula symmetric in any way? If so, how is it symmetric? How do you know?
y  3x 9  3x 7  17 x 7  x 5  x 3
Show your work to test for the following types of symmetry:
x-axis
y-axis
origin
16) If an equation is a polynomial (it only has terms like 3x2, or 2x3 – nothing funky like absolute
value, etc), how can you quickly tell if it’s symmetric about the origin?
Why are functions that are symmetric about the origin called odd functions?
Math 110
©, ® 2003, Michael Panitz
Page 8 / 12
Math 110
In Class Exercises, Lecture 8
Page 9 / 12
X
Average Rate Of Change
Goal: Identify the rate of change of a function, given it's picture ; intuitively understand ARoC
17) What is the average rate of
change between x = -8 and x = -4?
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
Draw the secant line between these
two points on the graph to the left
2 3 4 5 6 7 8 9 10
What is the average rate of change
between x = 2 and x = 3?
Y
X
Draw the secant line between these
two points on the graph to the left
18) What is the average rate of
change between x = -8 and x = -4?
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
Draw the secant line between these
two points on the graph to the left
2 3 4 5 6 7 8 9 10
What is the average rate of change
between x = 4 and x = 6?
Y
Draw the secant line between these
two points on the graph to the left
Math 110
©, ® 2003, Michael Panitz
Page 9 / 12
Math 110
In Class Exercises, Lecture 8
Page 10 / 12
19) Let's say you were looking at the graph of
1 2
y
x (pictured left), and wanted to find a
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formula for the average rate of change between
x=-2, and any other point on the graph.
Using the formula given in class, find a formula for
the average rate of change from x = -2 to any
other point.
20) Let's say you were looking at the graph of
1 3
y
x  2 x 2 (pictured left), and wanted to find a
60
formula for the average rate of change between
x=-2, and any other point on the graph.
Using the formula given in class, find a formula for
the average rate of change from x = -2 to any
other point.
Math 110
©, ® 2003, Michael Panitz
Page 10 / 12
Math 110
In Class Exercises, Lecture 8
Page 11 / 12
21) Examine the picture to the left. For these
questions, you may assume that the interval to be
considered is pictured in it's entirely
Where is the function ascending?
Where is the function descending?
Where is the function constant?
How many local maxima are there? What are
they?
How many local minima are there? What are
they?
22) Examine the picture to the left. For these
questions, you may assume that the interval to be
considered is pictured in it's entirely
Where is the function ascending?
Where is the function descending?
Where is the function constant?
How many local maxima are there? What are
they?
Math 110
How many local minima are there? What are
they?
©, ® 2003, Michael Panitz
Page 11 / 12
X
Math 110
In Class Exercises, Lecture 8
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
Page 12 / 12
23) Examine the picture to the left.
For these questions, you may
assume that the interval to be
considered is pictured in it's entirely
Where is the function ascending?
2 3 4 5 6 7 8 9 10
Where is the function
descending?
Where is the function constant?
Y
How many local maxima are there? What are they?
Math 110
©, ® 2003, Michael Panitz
How many local minima are
there? What are they?
Page 12 / 12