SymmetrySection 3.1

Symmetry

Two Types of Symmetry:

Point SymmetryTwo distinct points P and P’ are symmetric with respect to point M, if and only if M is the midpoint of P and P’

Example: P(3, -4) and P’ are symmetric with respect to M(2, 2). What is P’?

A good example of a graph being symmetric to a point is:

This graph is symmetric to the ORIGIN

P’

P

M

By the way, what is the domain and range of this graph?

D: (-∞, ∞)R: (-∞, ∞)

Interval Notation• A parenthesis ( ) shows an open (not included) endpoint• A bracket [ ] shows a closed [included] endpoint

Examples:• Set A with endpoints 1 and 3, neither endpoint included

(1,3)

• Set B with endpoints 6 and 10, not including 10 [6,10)

• Set C with endpoints 20 and 25, including both endpoints [20,25]

• Set D with endpoints 28 and infinity, not including 28 (28, )

Line symmetryTwo distinct points P and P’ are symmetric with respect with respect to line “l” if “l” is the perpendicular bisector of the line PP’

This graph is symmetric with respect to the y-axis or x = 0

What does the this graph look like:

y = x2

By the way, what is the domain and range of this graph?

D: (-∞, ∞)R: (0, ∞)

What is each graph symmetric with respect to? And tell me the domain and range.

x = 2x-axis

x = 2y = -5P(2, -5)Infinite amount of lines

x-axisy-axisorigin

Finish putting in domain and ranges in this PPT

What is each graph symmetric with respect to?

x = 1 x = -1y = 2P(-1, 2)

y = 2origin

Real quick…Circle equations look like:

Ellipse equations look like:

Same coefficients, both x and y squared

Different POSITIVE coefficients, both x and y squared

A function is odd if f( -x) = - f(x) for every number x in the domain.

A function is even if f( -x) = f(x) for every number x in the domain.

So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd.

125 24 xxxf

1251)(2)(5 2424 xxxxxfEVEN

xxxf 32

xxxxxf 33 2)()(2ODD

If a function is not even or odd we just say neither (meaning neither even nor odd)

15 3 xxf

Determine if the following functions are even, odd or neither.

1515 33 xxxf

Not the original and all terms didn’t change signs, so NEITHER.

23 24 xxxf

232)()(3 2424 xxxxxf

Got f(x) back so EVEN.

Challenge:

Is it even, odd, or neither:

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

Even, Odd or Neither ?

The graph below is a portion of a complete graph. Sketch a complete graph for each of the following symmetries.

With respect to:

The x-axis

The y-axis

The line y = x

The line y = -x

(1, 1) existsDoes (1, -1) exist? NO Not the x-axis

Does (-1, 1) exist? YES Symmetric to y-axis