2.3 Properties of Functions Even, Odd, or Neither (Symmetry) Increasing and Decreasing Intervals Local Maxima and Minima
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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
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