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### 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/
```
Associated Topics:
High School Functions

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