2.3 Properties of FunctionsEven, Odd, or Neither (Symmetry)

Increasing and Decreasing Intervals

Local Maxima and Minima

Even Function

• A function that is symmetric about the y-axis.

• Algebraically – ( ) ( )f x f x

8

6

4

2

-2

-4 -2 2 4

2( )f x x

2( ) ( )f x x

2( )f x x

( ) ( )f x f x

Odd Function

• A function that is symmetric about the origin

(180⁰ rotational symmetry about the origin)

• Algebraically -( ) ( )f x f x 6

4

2

-2

-4

-6

-4 -3 -2 -1 1 2 3

3( )f x x3( ) ( )f x x

3( )f x x

3( ) ( )f x x

3( )f x x

Increasing

• As the x-values increase, the y-values also increase 4

2

-2

-4

-10 -5 5 10

(5, 1)

(3, 2)(-1, 2)

(-5, -3)

( 5, 1)

Decreasing

• As the x-values increase, the y-values decrease4

2

-2

-4

-10 -5 5 10

(5, 1)

(3, 2)(-1, 2)

(-5, -3)

(3, 5)

Constant

• As the x-values increase, the y-value remain the same. 4

2

-2

-4

-10 -5 5 10

(5, 1)

(3, 2)(-1, 2)

(-5, -3)

( 1,3)

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Find the intercepts

( ,0) (0,0) ( ,0)

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

State the Domain and Range

{ | }x x { | 1 1}y y

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Identify the intervals where it is

increasing

,2 2

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Identify the intervals where it is

decreasing

, ,2 2

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Identify the intervals where it is

constant

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

Determine whether it is even, odd, or

neither

Local Maxima and Minima

• Local Maximum – The largest value of y on an open interval of x.

• Local Minima – The smallest value of y on an open interval of x.

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Identify the local maximum

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-4 -2 2 4

-

, 12

, 12

Identify the local minimum

3( ) 6 12 5 { | 2 2}f x x x x x

Identify the local extrema on the given

interval (*using a calculator)

Find Maximum

3( ) 6 12 5 { | 2 2}f x x x x x

Find Maximum

3( ) 6 12 5 { | 2 2}f x x x x x

Find Maximum

3( ) 6 12 5 { | 2 2}f x x x x x

Find Minimum

3( ) 6 12 5 { | 2 2}f x x x x x

Find Minimum3( ) 6 12 5 { | 2 2}f x x x x x

Assignmentp. 88

# 7, 10 - 28 even,

39 - 46, 63, 65, 66