In this lesson we will…

Discuss what symmetry is and the different types that exist.

Learn to determine symmetry in graphs.

Classify functions as even or odd.

Point Symmetry: Symmetry about one point

Figure will spin about the point and land on itself in less than 360º.

Two distinct points P and P are symmetric to M

iff M is the midpoint of the segment PP .

M

P’

P

This is the main point we look at for symmetry.

Let’s build some symmetry!

The graph of the relation S is symmetric with respect to the origin iff:

, implies ,a b S a b S

Easier: iff ( ) ( )f x f x

If the function is odd

(all odd exponents) then the graph will be

symmetric to the origin.

* plain numbers have exp = 0 *

Easier yet!

Two distinct points and ' are

symmetric with respect to a line

iff is the perpendicular bisector of '.

A point is symmetric to itself

with respect to line iff is on .

P P

PP

P

P

l

l

l l

U D

x-axis

y-axis

y = x

y = -x

, implies ,a b S a b S

, implies ,a b S a b S

Easier: iff ( ) ( )f x f x

If the function is even

(all even exponents) then the graph will be

symmetric to the axis.

* plain numbers have exp = 0 *

Easier yet!

y

, implies ,a b S b a S

, implies ,a b S b a S

Example 5:

1. What does it mean for a function to be symmetric to the line y = x?

2. How would one going about proving that symmetry mathematically?

HW 2.2: P 134 #25 – 35 odd