# A Study of Symmetry Instructor: Dr.Jo Steig

 In general terms, a graph in two-dimensions 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 y-axis (the line x = 0) because the quarter circle to the left of the y-axis is a mirror image of the quarter circle to the right of the y-axis. In fact, if you could fold this page along the y-axis, the two quarter circles would match up perfectly. There are four types of symmetry with which we will be concerned: (1) symmetry about the y-axis (2) symmetry about the x-axis (3) symmetry about the origin (4) symmetry about the line y = x   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 three-dimensions 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 eye-ball a two-dimensional graph and form an opinion as to its possible symmetry (about the y-axis, the x-axis, the origin, or y = x) (2) a spatial perspective so that you could draw a sketch of a graph that would be symmetric to a given graph (3) ability to test the equation of a graph for symmetry before you ever see the graph. This last one is particularly helpful when we move into three-dimensional graphs and symmetry is harder to tell by looking at a shape.   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 y-axis. The black and blue points are symmetric with respect to the x-axis. The black and green points are symmetric with respect to the origin The black and pink points are symmetric with respect to y = x     Symmetry about the y-axis Look again at the black and red points. Notice that the x-coordinates are additive inverses of each other. That is, if b is an x-coordinate of one point, then - b is the x-coordinate of the other point. That is also how we test an equation of a curve to see if the curve is symmetrical about the y-axis. Test for symmetry about the y-axis: Replace x with (-x). Simplfy the equation. If the resulting equation is equivalent to the original equation then the graph is symmetrical about the y-axis. Example: Use the test for symmetry about the y-axis to determine if the graph of y - 5x2 = 4 is symmetric about the y-axis. original equation: y - 5x2 = 4 test: y - 5(-x)2 = 4 simplify: y - 5x2 = 4 Conclusion: Since the resulting equation is equivalent to the original equation then the graph is symmetrical about the y-axis   Symmetry about the x-axis The test for symmetry about the x-axis is similar to the last test. Look back at the black and blue points. Notice that now it is the y-coordinates that are additive inverses of each other. That is, if c is a y-coordinate of one point, then - c is the y-coordinate of the other point. That is also how we test an equation of a curve to see if the curve is symmetrical about the x-axis. Test for symmetry about the x-axis: Replace y with (-y). Simplfy the equation. If the resulting equation is equivalent to the original equation then the graph is symmetrical about the x-axis. Example: Use the test for symmetry about the x-axis to determine if the graph of y - 5x2 = 4 is symmetric about the x-axis. original equation: y - 5x2 = 4 test: (-y) - 5x2 = 4 simplify: - y - 5x2 = 4 Conclusion: Since the resulting equation is NOT equivalent to the original equation then the graph is NOT symmetrical about the x-axis Here is a sketch of the curve.The fact that the curve IS symmetric zbout the y-axis and IS NOT symmetric with respect to the y-axis 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 & y-coordinates 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 y-axis and also the x-axis. 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 - 5x2 = 4 is symmetric about the origin. original equation: xy - 5x2 = 4 test: (-x)(-y) - 5(-x)2 = 4 simplify: xy - 5x2 = 4 Conclusion: Since the resulting equation is equivalent to the original equation then the graph is symmetrical about the origin. 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 = x3 , for x > or = 0. original equation: y = x3 new equation: x = y3 solve for y: y = x1/3 , 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