A knowledge of symmetry can increase your efficiency when working with graphs.
This section discusses symmetry about the $x$axis, symmetry about the $y$axis, and origin symmetry.
DEFINITION
$x$axis symmetry
Here are some observations about graphs with $x$axis symmetry:

DEFINITION
$y$axis symmetry
Here are some observations about graphs with $y$axis symmetry:

DEFINITION
origin symmetry
Here are some observations about graphs with origin symmetry:

The following table illustrates the logic and procedure for testing an equation in two variables for symmetry about the $x$axis, $y$axis, and origin.
EXAMPLE: TEST $x^2y^3  x^4y = y$ FOR VARIOUS SYMMETRIES  
THIS FIRST STEP IS THE SAME WHEN TESTING FOR ANY TYPE OF SYMMETRY: Suppose $\,(a,b)\,$ is on the graph. Thus, substitution of ‘$\,a\,$’ for ‘$\,x\,$’ and ‘$\,b\,$’ for ‘$\,y\,$’ makes the equation true. In other words, when every $\,x\,$value is replaced by $\,a\,$, and every $\,y\,$value is replaced by $\,b\,$, the resulting equation is true. Thus, the following equation is assumed to be true: $$ a^2b^3  a^4b = b\qquad\qquad\qquad (\dagger) $$ This equation is given the name $\ \dagger\ $ (read as ‘dagger’) to make it easy to refer to in subsequent steps.  
PROCEDURE  EXAMPLE 
TEST FOR SYMMETRY ABOUT THE $x$AXIS:

$$\begin{gather}
a^2(b)^3  a^4(b)\ \overset{?}{=}\ b \cr\cr\cr
a^2b^3 + a^4b \ \overset{?}{=}\ b\cr\cr
a^2b^3  a^4b = b\cr
(\text{after multiplying both sides by $1$})\cr
\text{(TRUE, by comparison with $\dagger$)}
\end{gather}
$$
since whenever $\,(a,b)\,$ lies on the graph, so does $\,(a,b)\,$. 
TEST FOR SYMMETRY ABOUT THE $y$AXIS:

$$\begin{gather}
(a)^2b^3  (a)^4b\ \overset{?}{=}\ b \cr\cr\cr
a^2b^3  a^4b = b\cr
\text{(TRUE, by comparison with $\dagger$)}
\end{gather}
$$
since whenever $\,(a,b)\,$ lies on the graph, so does $\,(a,b)\,$. 
TEST FOR ORIGIN SYMMETRY:

$$\begin{gather}
(a)^2(b)^3  (a)^4(b)\ \overset{?}{=}\ b \cr\cr\cr
a^2b^3 + a^4b \ \overset{?}{=}\ b\cr\cr\cr
a^2b^3  a^4b = b\cr
(\text{after multiplying both sides by $1$})\cr
\text{(TRUE, by comparison with $\dagger$)}
\end{gather}
$$
since whenever $\,(a,b)\,$ lies on the graph, so does $\,(a,b)\,$. 
By the way, the equation $x^2y^3  x^4y = y$ is NOT easy to graph. However, WolframAlpha is up to the challenge! Type in x^2y^3  x^4y = y (cutandpaste, if you want) and you'll see the following graph. Look at the beautiful symmetries! 
Or, use the WolframAlpha widget below! Click ‘Submit’ to see the picture of all points $\,(x,y)\,$ for which the equation is true. (Notice how the ‘^’ key is used to input powers.) Play with other equations as you do the exercises below. Have fun! (Note: the exercise equations are randomlygenerated for the purpose of giving you lots of practice testing equations for symmetry. Be prepared for some ‘strange’ results from WolframAlpha on some of the equations! Some equations may have only a single point that makes them true. Some equations may not have any real number solutions!) 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
