In general terms, a graph in twodimensions is said to be symmetric about a particular line if the portion of the graph on one side of the line is a mirror image of the portion of the graph that is on the other side of the line. For example, the graph below is said to be symmetric about the yaxis (the line x = 0) because the quarter circle to the left of the yaxis is a mirror image of the quarter circle to the right of the yaxis. In fact, if you could fold this page along the yaxis, the two quarter circles would match up perfectly.
There are four types of symmetry with which we will be concerned: (1) symmetry about the yaxis
Why does anyone care about symmetry? One reason is that knowing that a graph is symmetrical about a line reduces the amount of work that one would have to do in order to describe the curve. If you were trying describe where a graph had a peak, a valley, or a discontinuity then you would only have to investigate one half of the graph  the other half of the graph (its mirror image) would just be a duplicate. This can be particularly useful if you are working in threedimensions as is done in multivariate calculus.
There are several levels of understanding about symmetry that we are going to develop in this class: (1) a general understanding of the concept so that you could eyeball a twodimensional graph and form an opinion as to its possible symmetry (about the yaxis, the xaxis, the origin, or y = x)
This reading is intended to help you develop your intuitive understanding of symmetry into the basis for the tests for symmetry that we use on equations.
Graphical Representation of Symmetry Take a look at this graph of five points. The black dot represents the original point and the colored dots demonstrate the four types of symmetry. The black and red points are symmetric with respect to the yaxis.
Symmetry about the yaxis Look again at the black and red points. Notice that the xcoordinates are additive inverses of each other. That is, if b is an xcoordinate of one point, then  b is the xcoordinate of the other point. That is also how we test an equation of a curve to see if the curve is symmetrical about the yaxis. Test for symmetry about the yaxis: Replace x with (x). Simplfy the equation. If the resulting equation is equivalent to the original equation then the graph is symmetrical about the yaxis. Example: Use the test for symmetry about the yaxis to determine if the graph of y  5x^{2} = 4 is symmetric about the yaxis. original equation: y  5x^{2} = 4
Symmetry about the xaxis The test for symmetry about the xaxis is similar to the last test. Look back at the black and blue points. Notice that now it is the ycoordinates that are additive inverses of each other. That is, if c is a ycoordinate of one point, then  c is the ycoordinate of the other point. That is also how we test an equation of a curve to see if the curve is symmetrical about the xaxis. Test for symmetry about the xaxis: Replace y with (y). Simplfy the equation. If the resulting equation is equivalent to the original equation then the graph is symmetrical about the xaxis. Example: Use the test for symmetry about the xaxis to determine if the graph of y  5x^{2} = 4 is symmetric about the xaxis. original equation: y  5x^{2} = 4 Here is a sketch of the curve.The fact that the curve IS symmetric zbout the yaxis and IS NOT symmetric with respect to the yaxis is pretty evident.
Symmetry about the origin The test for symmetry about the origin also bears similarities with the last tests. Look at the black and green points. Both the x & ycoordinates are additive inverses. That is, (b,c) and (b,c) are symmetric about the origin. You can think of symmetry about the origin as a reflection about the yaxis and also the xaxis. The test for symmetry about the origin combines elements from the first two tests. Test for symmetry about the origin: Replace y with (y) AND x with (x). Simplfy the equation. If the resulting equation is equivalent to the original equation then the graph is symmetrical about the origin. Example: Use the test for symmetry about the origin to determine if the graph of xy  5x^{2} = 4 is symmetric about the origin. original equation: xy  5x^{2} = 4 Here is a sketch of the curve. We must first solve for y (in terms of x) in order to use the graphing calculator. This time symmetry is not quite as easy to see from the sketch.
Symmetry about the line y = x For our last symmetry, look back at the black and pink points. In this case the x & y coordinates have been interchanged. That is, (b,c) and (c,b) are symmetric about the line y = x. Much of our later work with this type of symmetry is going to involve functions. In that case we are going to be interested in creating an equation whose graph is symmetric (about y = x) with a given graph. We do so by interchanging the x's and y's Example: Create an equation of a graph that will be symmetric (about y = x) with the graph of y = x^{3} ,for x > or = 0.
Here are the two graphs. Note that they are mirror images about the line y = x. You will see alot more of this symmetry when we get into our discussion about functions and their inverses. © 1999 Jo Steig
