Symmetry Tests

Date: 01/12/99 at 15:50:16
From: Lindsey Scardino
Subject: Functions

I need an explanation on how to find out whether a function is 
symmetric to the x-axis, the y-axis, or the origin. We have information 
stating how to test it, but I don't understand what it means. The 
information we have is: for the function to be symmetric to the y-axis 
f(x) = f(-x); for the function to be symmetric to the x-axis  
f(x) = -f(x); and for something to be symmetric to the origin then 
f(-x) = -f(x). 

Any information you can provide on this would be greatly appreciated. 
Thanks for your time. 

Lindsey.

Date: 01/12/99 at 17:51:45
From: Doctor Pat
Subject: Re: Functions

Lindsey,

First, a function on x can NOT be symmetric in the x-axis. If it were, 
it would not pass the vertical line test (there would be two outcomes 
for one input). Relations and equations can be symmetric in x however.  
Usually it helps to understand these better if you write them as 
equations using x and y rather than f(x) (just put y in place of f(x) 
in the expression. Then here are some rules:
 
  Symmetric in Y-axis TEST
  ------------------------
  Replace all x values with (-x) and simplify. If the equation is the 
  same as you started with, it is symmetric about the y-axis.
  Example: y = x^2 - 5 
     Replacing x with -x, we have y = (-x)^2 - 5 = x^2 - 5
     Since it is the same, y = x^2 - 5 is symmetric about the y-axis

   Symmetric in X-axis TEST
   ------------------------
   Replace all y values with (-y) and simplify. If the equation is the 
   same then it is symmetric about the x-axis.
   Example: x^3 +y^2 = 4   
      Replacing y with -y, we have x^3 + (-y)^2 = 4, which simplifies 
      to x^3 + y^2 = 4. Since it is the same, x^3 + y^2 = 4 is 
      symmetric about the x-axis

   Symmetric in Origin TEST
   ------------------------
   Replace BOTH x and y by their opposites and simplify. If they are 
   the same then the graph is symmetric about the origin.
   Example: y = x^3
      Replacing both x and y, we get (-y) = (-x)^3, which simplifies 
      to -y = -x^3, and then y = x^3. Since we get the same equation, 
      y = x^3 is symmetric about the origin.

For another example, is 3xy = x^2 y^2 symmetric in x, y, or the origin?

Test y-axis: 3(-x)y = (-x)^2 y^2  =>  -3xy = x^2 y^2 
   Thus, 3xy = x^2 y^2 is not symmetric about the y-axis.

Test x-axis: 3x(-y) = x^2 (-y)^2  =>  -3xy = x^2 y^2 
   Thus, 3xy = x^2 y^2 is not symmetric about the x-axis.

Test origin  3 (-x)(-y)= (-x)^2 (-y)^2  => 3xy = x^2 y^2
   Thus, 3xy = x^2 y^2 is not symmetric about the origin.

But what do these symmetries mean? Well, if a graph is symmetric about 
something, all it means is that you can fold the graph in some way, 
and the different parts of the graph will overlap. A graph that is 
symmetric about the y-axis will overlap if you fold the graph along 
the y-axis. Similarly, a graph that is symmetric about the x-axis will 
overlap if you fold the graph along the x-axis. Finally, a graph that 
is symmetric about the origin will overlap if you fold the graph along 
the x-axis and the y-axis. 

Here's a graph of y = x^3 to help you see the folds. 

First, here is the graph folded along the y-axis:



The graph folded along the x-axis:

 

And finally, the graph folded along both the x-axis and the y-axis:

 

Hope this helps. This is the same test your teacher uses, but worded a 
little differently.  
   
- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/