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Algebra - Notes
 Graphing and Functions Previous Chapter Next Chapter Polynomial Functions Transformations Previous Section Next Section Rational Functions

Symmetry

In this section we are going to take a look at something that we used back when we where graphing parabolas.  However, we’re going to take a more general view of it this section.  Many graphs have symmetry to them.

Symmetry can be useful in graphing an equation since it says that if we know one portion of the graph then we will also know the remaining (and symmetric) portion of the graph as well.  We used this fact when we were graphing parabolas to get an extra point of some of the graphs.

In this section we want to look at three types of symmetry.

1. A graph is said to be symmetric about the x-axis if whenever
is on the graph then so is
.  Here is a sketch of a graph that is symmetric about the x-axis.

1. A graph is said to be symmetric about the y-axis if whenever
is on the graph then so is
.  Here is a sketch of a graph that is symmetric about the y-axis.

1. A graph is said to be symmetric about the origin if whenever
is on the graph then so is
.  Here is a sketch of a graph that is symmetric about the origin.

Note that most graphs don’t have any kind of symmetry.  Also, it is possible for a graph to have more than one kind of symmetry.  For example the graph of a circle centered at the origin exhibits all three symmetries.

Tests for Symmetry

 We’ve some fairly simply tests for each of the different types of symmetry.   A graph will have symmetry about the x-axis if we get an equivalent equation when all the y’s are replaced with y. A graph will have symmetry about the y-axis if we get an equivalent equation when all the x’s are replaced with x. A graph will have symmetry about the origin if we get an equivalent equation when all the y’s are replaced with y and all the x’s are replaced with x.

We will define just what we mean by an “equivalent equation” when we reach an example of that.  For the majority of the examples that we’re liable to run across this will mean that it is exactly the same equation.

Let’s test a few equations for symmetry.  Note that we aren’t going to graph these since most of them would actually be fairly difficult to graph.  The point of this example is only to use the tests to determine the symmetry of each equation.