essential questions 1)what is the difference between an odd and even function? 2)how do you perform...
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Essential Questions
1)What is the difference between an odd and even function?
2)How do you perform transformations on polynomial functions?
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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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Even and Odd Functions (algebraically)
A function is even if f(-x) = f(x)
A function is odd if f(-x) = -f(x)
If you plug in -x and get the original function, then it’s even.
If you plug in -x and get the opposite function, then it’s odd.
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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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f x x( ) Even, Odd or Neither?Ex. 1
f x x( ) Graphically Algebraically
(1) 1 1f ( 1) 1 1f
They are the same, so it is.....
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f x x x( ) 3Even, Odd or Neither?
Ex. 2
3( )f x x x Graphically Algebraically
3(2) (2) (2)f 6
63( 2) ( 2) ( 2)f
They are opposite, so…
What happens if we plug in 1?
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f x x( ) 2 1Even, Odd or Neither?
2( ) 1f x x Graphically Algebraically
2(1) (1) 1f
22( 1) ( 1) 1f
Ex. 3
2They are the same, so.....
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3( ) 1f x x Even, Odd or Neither?
3( ) 1f x x Graphically Algebraically
3(1) (1) 1f
0
2
3( 1) ( 1) 1f
They are not = or opposite, so...
Ex. 4
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What happens when What happens when we change the we change the
equations of these equations of these parent functions?parent functions?
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14)9()( 2 xxf
3)2()( xxf
Left 9 , Down 14
Left 2 , Down 3
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-f(x)
f(-x)
Reflection in the x-axis
Reflection in the y-axis
What did the negative on the outside do?
What do you think the negative on the inside will do?
Study tip: If the sign is on the outside it has “x”-scaped
Study tip: If the sign is on the inside, say “y” am I in here?
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Write the Equation to this Graph
2)3( 2 xy
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Write the Equation to this Graph
1)2( 3 xy
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Write the Equation to this Graph
11 xy
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Write the Equation to this Graph
3 3( ) ( ) 2 or ( ) ( ) 2f x x f x x
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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.
x
y
y = x4
y = – x4f (x) = – (x + 2)4
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Compare:3 3 31
( ) ( ) 4 ( )4
f x x and g x x and h x x
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Compare…
What does the “a” do?
2 2( ) to ( ) 3f x x f x x
Compare… 2 21( ) to ( )
2f x x f x x
• What does the “a” do?
Vertical stretch
Vertical shrink
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Nonrigid Transformations
h(x) = c f(x) c >1
0 < c < 1
Vertical stretch
Vertical shrink
Closer to y-axis
Closer to x-axis
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Polynomial functions of the form f (x) = x n, n 1 are called
power functions.
If n is even, their graphs resemble the graph of
f (x) = x2.
If n is odd, their graphs resemble the graph of
f (x) = x3.
x
y
x
y
f (x) = x2
f (x) = x5f (x) = x4
f (x) = x3
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