essential questions 1)what is the difference between an odd and even function? 2)how do you perform...

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.


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.


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!


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.....


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?


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.....


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


Let’s go to the Task….


What happens when What happens when we change the we change the

equations of these equations of these parent functions?parent functions?


14)9()( 2 xxf

3)2()( xxf

Left 9 , Down 14

Left 2 , Down 3


-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?


Write the Equation to this Graph

2)3( 2 xy



1)2( 3 xy



11 xy



3 3( ) ( ) 2 or ( ) ( ) 2f x x f x x


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


Compare:3 3 31

( ) ( ) 4 ( )4

f x x and g x x and h x x


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


Nonrigid Transformations

h(x) = c f(x) c >1

0 < c < 1

Vertical stretch

Vertical shrink

Closer to y-axis

Closer to x-axis


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

essential questions 1)what is the difference between an odd and even function? 2)how do you perform...

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graphicallyalgebraically

graph of f x

graph of y

fx fx reflection

graph of fx

odd functions

xscaped study tip