Essential Questions
1)What is the difference
between an odd and even
function?
2)How do you perform
transformations on
polynomial functions?
Even and Odd Functions
(graphically)
If the graph of a function is symmetric with
respect to the y-axis, then it’s even.
If the graph of a function is symmetric with
respect to the origin, then it’s odd.
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2
Even and Odd Functions
(algebraically)
A function is even if f(-x) = f(x)
If you plug in -x and get the original function,
then it’s even.
A function is odd if f(-x) = -f(x)
If you plug in -x and get the opposite function,
then it’s odd.
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3
Let’s simplify it a little…

We are going to plug in a number to
simplify things. We will usually use 1
and -1 to compare, but there is an
exception to the rule….we will see
soon!
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4
Ex. 1
Even, Odd or Neither?
Graphically
f ( x)  x
Algebraically
f ( x)  x
f (1)  1  1
f (1)  1  1
They are the same, so it is.....
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5
Ex. 2
Even, Odd or Neither?
f ( x)  x  x
Graphically
3
What happens
if we plug in 1?
Algebraically
3
f ( x)  x  x
f (2)  (2)  (2) 
6
3
f (2)  (2)  (2)
3
 6
They are
opposite, so…
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6
Ex. 3
Even, Odd or Neither?
f ( x)  x  1
Graphically
2
Algebraically
f ( x)  x  1
2
f (1)  (1)  1
2
2
2
f (1)  (1)  1
2
They are the same, so.....
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7
Ex. 4
Even, Odd or Neither?
f ( x)  x 1
Graphically
3
Algebraically
f ( x)  x 1
3
f (1)  (1) 1
3
0
3
f (1)  (1) 1
 2
They are not = or opposite, so...
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8
Let’s go to the Task….
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9
What happens when
we change the
equations of these
parent functions?
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10
f ( x)  ( x  9) 14
Left 9 , Down 14
f ( x)  ( x  2)  3
Left 2 , Down 3
2
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11
What did the negative on the outside
do?
-f(x)
Reflection in the x-axis
Study tip: If the sign is on the outside it
has “x”-scaped
What do you think the negative on the
inside will do?
f(-x)
Reflection in the y-axis
Study tip: If the sign is on the inside, say
“y” am I in here?
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12
Write the Equation to this Graph
y  ( x  3)  2
2
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13
Write the Equation to this Graph
y  ( x  2) 1
3
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14
Write the Equation to this Graph
y   x 1  1
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15
Write the Equation to this Graph
f ( x)  ( x)  2 or f ( x)  ( x)  2
3
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3
16
Example: Sketch the graph of f (x) = – (x + 2)4 .
This is a shift of the graph of y = – x 4 two units to the left.
This, in turn, is the reflection of the graph of y = x 4 in the x-axis.
y
y = x4
x
f (x) = – (x + 2)4
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y = – x4
17
Compare:
1 3
f ( x)  x and g ( x)  4 x and h( x)  x
4
3
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3
18
Compare…

f ( x)  x to f ( x)  3x
What does the “a”
do?
Compare…
• What does the “a” do?
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2
2
Vertical stretch
1 2
f ( x)  x to f ( x)  x
2
2
Vertical shrink
19
Nonrigid Transformations
h(x) = c f(x)
c >1
Vertical stretch
Closer to y-axis
0<c<1
Vertical shrink
Closer to x-axis
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20
Polynomial functions of the form f (x) = x n, n  1 are called
power functions.
5
f
(x)
=
x
4
y
f (x) = x
y
f (x) = x2
f (x) = x3
x
If n is even, their graphs
resemble the graph of
f (x) = x2.
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x
If n is odd, their graphs
resemble the graph of
f (x) = x3.
21