Testing For Symmetry and Even/Odd Functions

Date: 06/08/98 at 20:33:45
From: Joshua Stevens
Subject: Tests for symmetry and even/odd functions

I'm trying to do my summer Calculus review and I cannot remember the 
tests for the following:

   -- symmetry on the x-axis
   -- symmetry on the y-axis
   -- symmetry at the origin
   -- even functions
   -- odd functions

I know they are simple tests, but I just cannot remember them, nor can 
I find them anywhere.

Thanks!

Date: 06/08/98 at 23:12:09
From: Doctor Pat
Subject: Re: Tests for symmetry and even/odd functions

Joshua,

Here are my approaches to these ideas:

For symmetry on the x-axis      

   If you replace y in the equation with -y and it is the same 
   equation (when simplified), then it is symmetric around 
   the x-axis. For every point (x,y) there must also be a point 
   (x,-y). Fold the graph along the x-axis and the top and bottom 
   match.

For symmetry on the y-axis  

   If you replace x in the equation with -x and it is the same 
   equation (when simplified), then it is symmetric about the y-axis. 
   For every point (x,y) there must be a point (-x,y). Fold along the 
   y-axis and the left and right sides match.  

For symmetry at the origin   

   If you replace both x and y with their negatives and it is the same 
   equation (simplified), then it is symmetric about the origin. For 
   every point (x,y) there is a point (-x,-y). Rotate the graph 180 
   degrees and the picture is the same.

For even functions 

   The process is same as symmetry in the y-axis.

For odd functions     

 The process is the same as symmetry at the origin.  

These names are easy to remember if you graph y= x^a. If a is odd, the 
graph is symmetric in the origin, if a is even the graph is symmetric 
in the y-axis.  

Hope that helps.
 
-Doctor Pat,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/