Each part of graph is described as:

1) Increasing : function values increase from left to right

2) Decreasing: function values decrease

3) Constant function values are constant

1. Definition: Increasing/Decreasing/Constant

Use x-intervals to describe this property!

1) Increasing

2) Decreasing

3) Constant

2. Determine the intervals of Increasing/Decreasing/Constant

3. Definition: Local Maxima and Minima

local maximum:“peaks” or “high points” on graph ( is NOT a max!)local minimum: “valleys” or “low points” on graph

Local maximum at x = ___

Value of local maximum is f(a)

Local maxima are the points:

Local minimum at x =

Local minima at:

Local Minima and MaximaList intervals of increasing/decreasing and all maxima/minima.p. 89 #12.

4 a) Using the Graphing Calculator

Use a graphing utility 1. Approximate all local maxima or minima.

2. Find all intervals where f is increasing and decreasing.

)2,2( 23)( 3 xxxf

To find max/min on calculator1. y1=2. 2nd TRACE (CALC)3. 3:minimum or (4:maximum)4. move left of min or max ENTER5. move right of min or max ENTER6. ENTER

A function changes from increasing to decreasing at a maximumA function changes from decreasing to increasing at a minimum

5. SymmetrySome functions have graphs with symmetry:

Even function

symmetry with respect to y-axis

ODD function

symmetry with respect to origin

no symmetry

5) Even and Odd FunctionsTo show algebraically:All EVEN functions satisfy

f ( x) f (x)

f ( x) f (x)

Ex : f (x) x 2 2

Ex : f (x) 2x 3

3

To show algebraically:All ODD functions satisfy

Even functionsymmetry with respect to y-axis

Odd functionsymmetry with respect to origin

5 b) Examples. Determine if even/oddDetermine whether the following functions are even, odd, or neither. Then state any symmetry (origin/y-axis)

1)

2)

3)

4)x

xxf

22)(

6

4)(

2

5

x

xxf

f (x) x 2 3x 1

xxxf 55)(

5) Even and Odd Functions

Is the same as ?

Evaluate and simplify: Evaluate and simplify: )( xf

Evaluate :Evaluate : )(xfThen is an Even function

)( xf

Is the same as ?)( xf )(xf

Then is an ODD function

)(xf

)(xf

is neither and no symmetry

)(xf

YES NO

YES NO

)(xf