0-2: Smart Graphing

Objectives:•Identify symmetrical graphs•Identify odd/even functions•Sketch the graphs of functions using translations, reflections & dilations

© 2002 Roy L. Gover (roygover@att.net) Modified by Mike Efram 2004

DefinitionPoint Symmetry: Two points, P & P’ are symmetric with respect to a point M if M is the midpoint of

'PP

P P’M

...For a graph to have point symmetry with respect to a point M, M must be the midpoint of every set of points P & P’ on the graph. Examples...

Example

2 2 2x y r Point SymmetryConsider:

3( )f x x

M

Example

Point Symmetry:

M

A graph that is symmetrical with the point (0,0) is symmetric with respect to the origin.

Definition

Definition

A function f(x) is symmetric with respect to the origin if and only if

f(-x)=-f(x)

Examplef(x)=x3 is symmetric with the origin because -30

-20

-10

0

10

20

30

1 2 3 4 5 6 7

f(-x)=-f(x). ie f(-2)=-8 & f(2)=8,therefore f(-2)=-f(2)

Try This

Is f(x)=x2

symmetric with respect to the origin?

No

Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly.

Important Idea

Examples of Line Symmetry

Symmetry with respect to x=0 ( y-axis ) exists if and only if:f(x)=f(-x)

Example: f(x)=x2-3

Definition

Symmetry is useful in graphing functions. If you graph part of the function and understand the symmetry, the rest of the graph can be sketched.

Important Idea

DefinitionEven Functions are functions symmetric with the y axis. They have exponents that are all even.

Definition

Odd functions are functions symmetric with the origin. They have exponents that are all odd.

Try ThisAre the following functions even, odd or neither:4 2 6y x x

3( )f x x x 5 3( ) 1g x x x

Even

Odd

Neither

SummaryOdd functions:

f(-x) = -f(x)

Symmetry with origin (0, 0)

SummaryEven functions:

f(x) = f(-x)

Symmetry with y-axis

DefinitionReflections: the mirror image of a graph.

Example

f(x)=x2 f(x)=-x2

Try This

Without using a graphing calculator, graph f(x)=-x3 using its parent graph as a starting point.

Solution

3y x 3y x

Definition

Translation: the sliding of a graph vertically or horizontally without changing its size or shape.

Examples

f(x)=x2-3

f(x)=(x+3)2

f(x)=x2+3

f(x)=(x-3)2

VerticalTranslations

HorizontalTranslations

Try ThisWrite the equation of this graph based on its parent graph.Hint: a vertical & horizontal translation is required.f x x( ) ( ) 3 32

Try ThisWrite the equation of this graph based on its parent graph.Hint: a reflection & horizontal translation is required. f x x( ) ( ) 2 2

Try ThisWithout using your calculator, sketch the graph of:

p x x( ) 2 2

DefinitionDilation: changing a graph’s size. Making it either smaller or larger. Examples:

f x x( ) f x x( ) 1

4f x x( ) 4

Example

The graph of f(x) is pictured at the right. Sketch a graph of:a) f(x+3)

b) f(x+3)-2

c) -f(x-3)-2

d) 2f(x+2)+3