zeros: domain: range: relative maximum: relative minimum: intervals of increase: intervals of...

Zeros:

Domain:

Range:

Relative Maximum:

Relative Minimum:

Intervals of Increase:

Intervals of Decrease:

WARM UP

Essential Question: How do you determine the shape and symmetry

of the graph by the polynomial equation?

Even, Odd, or Neither FunctionsNot to be confused with End behaviorTo determine End Behavior, we check to see if the leading degree is even or oddWith Functions, we are determining symmetry (if the entire function is even, odd, or neither)

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 -x and get the same solution, then it’s even.

Also: It is symmetrical over the y-axis.

If you plug in x and -x and get opposite solutions, then it’s odd.

Also: It is symmetrical over the origin

4

2

-2

-4

-5 5

Y – Axis SymmetryFold the y-axis

52 xy

0 -5

1 -4

2 -1

3 4

4 11

-1 -4

-2 -1

-3 4

(x, y) (-x, y)

Even Function

(x, y) (-x, y)

Test for an Even Function

A function y = f(x) is even if , for each x in the domain of f.

f(-x) = f(x)

Symmetry with respect to the y-axis

4

2

-2

-4

-5 5

Symmetry with respect to the origin

6

4

2

-2

-4

(x, y) (-x, -y)

(2, 2) (-2, -2)

xxy 33

(1, -2) (-1, 2)Odd Function

Test for an Odd Function

A function y = f(x) is odd if , for each x in the domain of f.

f(-x) = -f(x)

Symmetry with respect to the

Origin

6

4

2

-2

-4

f x x( ) Even, Odd or Neither?Ex. 1

( )f x x

Graphically Algebraically

4

4 4

4)

( )

4

4

(f

f

f x x x( ) 3Even, Odd or Neither?

Ex. 2

3( )f x x x Graphically Algebraically

3

3

( ) ( ) ( )

( ) ( )

6

(2 2 2 6)

2 2 2f

f

f x x( ) 2 1Even, Odd or Neither?

2( ) 1f x x Graphically Algebraically

2

2

2( ) ( ) 1

( ) ( )

1

1 2

1

11

f

f

Ex. 3

3( ) 1f x x Even, Odd or Neither?

3( ) 1f x x Graphically Algebraically

32 2( ) ( ) 1 9f 32 2) 1 6(f

Ex. 4

4( ) 2 3f x x

Even, Odd or Neither?

3( )f x x x

What do you notice about the graphs of even functions?

Even functions are symmetric about the y-axis

What do you notice about the graphs of odd functions?

Odd functions are symmetric about the origin