symmetry

The audio controller is at the bottom of the web page. (Look down!) This gives easy access while reading. The yellow highlighting follows my voice! Depending on your connection speed, the audio may take some time to load. (It will display 0:00 until it's ready.) Of course I'm biased—but I think it's worth the wait! A knowledge of symmetry can increase your efficiency when working with graphs. This section discusses: If you ‘fold’ the coordinate plane along the x-axis, the top and bottom portions of the graph overlap. That is, the x-axis acts as a ‘mirror’ for the graph. If b≠0 and (a,b) is a point on the graph, then there must be a second, different, point on the graph, which has the same x-value but opposite y-value. Most graphs with x-axis symmetry fail the vertical line test, and hence are not functions. (Recall that functions have the property that every input must have exactly one output.) The zero function, f(x) =0, has x-axis symmetry, but is pretty boring! In-a-nutshell test for x-axis symmetry: Replace every y by −y, and see if the same equation results. (The example below gives more logical details.) If you ‘fold’ the coordinate plane along the y-axis, the left and right portions of the graph overlap. That is, the y-axis acts as a ‘mirror’ for the graph. If a≠0 and (a,b) is a point on the graph, then there must be a second, different, point on the graph, which has the same y-value but opposite x-value. In-a-nutshell test for y-axis symmetry: Replace every x by −x, and see if the same equation results. (The example below gives more logical details.) If you ‘fold’ the coordinate plane twice: once along the x-axis, and once along the y-axis; then the graph will overlap itself. If (a,b) is a point on the graph where at least one of the coordinates is not zero, then there must be a second, different, point on the graph, which has opposite x and y values. If a graph has both x-axis and y-axis symmetry, then it must have origin symmetry: NEW! An offline version of my Algebra I course is now available. It includes all lessons, worksheets, randomly-generated practice, audio read-throughs, Algebra Pinball—everything—but with no internet connection required. The following example illustrates the logic and procedure for testing an equation in two variables for symmetry about the x-axis, y-axis, and origin. [Note: WolframAlpha has discontinued support for embedded widgets, so the widget described here is no longer available. The discussion, however, remains as originally written.] 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 randomly-generated 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! To get a randomly-generated practice problem, click the ‘New problem’ button above. Think about your answer, and then press ‘Enter’ or ‘Check your answer’.