Symmetry, odd, even functions with Alex Babrak

Lecture: Symmetry & Odd, Even Functions

Hello class!

Unfortunately Professor Little will not be joining us today because she won the $800,00 DC lottery! As a result, she decided to retire early and move to Puerto Rico in hopes of living an even happier life! Do not worry, I will be your new substitute professor! 

Today, we will learn about symmetry and its application with odd and even functions. 

Lets begin with symmetry. 

What is symmetry? The simple definition of symmetry states that it is the quality of being made up of similar parts facing each other or around an axis. Symmetry is important because it can be applied to algebra, geometry and graphing. 

There are many forms of symmetry. Here are a few examples:

- Vertical Symmetry

- Horizontal Symmetry

- Rotational Symmetry 

- Diagonal Symmetry  

TABLE A

Th rectangle in Table A has 2 lines of symmetry (Horizontal and Vertical)

The Triangle has 3 lines of symmetry (Diagonal, Vertical and Rotational Symmetry)
The Square has 4 lines of symmetry (Rotational, Vertical, Horizontal, Diagonal)

Now lets examine a rhombus:

Notice that the rhombus has no lines of symmetry because corners a,b,c, and d are not equal.

Okay, so now you guys understand the various types of symmetry involving geometrical objects. Did you know that symmetry can also be related to functions? Lets move on to Odd and Even Functions!

Odd Functions

Here is the equation for odd functions:

y=f(x)

f(-x)=-f(x)

 and there has to be symmetry about the origins 

Confuses? Lets do an example:

f(x) = x^3 

 f(-x)= (-x)^3 

THIS IS AN ODD FUNCTION because it reflects the formula AND there is symmetry about the origin.

Notice that is is an Even Function because it is symmetric about the origin. 

Even Function

Here is the equation for even functions:

for y=f(x)

  if f(-x)=f(x)

    AND there is symmetry about the Y-axis

Example: 

f(x)=x^4

f(-x)=(-x)^4 THIS IS and even function because it reflects the equations AND there is symmetry about the y axis. 

A Parabula is symmetric about the Y axis!