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### 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.

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 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/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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