Symmetry TestsDate: 01/12/99 at 15:50:16 From: Lindsey Scardino Subject: Functions I need an explanation on how to find out whether a function is symmetric to the x-axis, the y-axis, or the origin. We have information stating how to test it, but I don't understand what it means. The information we have is: for the function to be symmetric to the y-axis f(x) = f(-x); for the function to be symmetric to the x-axis f(x) = -f(x); and for something to be symmetric to the origin then f(-x) = -f(x). Any information you can provide on this would be greatly appreciated. Thanks for your time. Lindsey. Date: 01/12/99 at 17:51:45 From: Doctor Pat Subject: Re: Functions Lindsey, First, a function on x can NOT be symmetric in the x-axis. If it were, it would not pass the vertical line test (there would be two outcomes for one input). Relations and equations can be symmetric in x however. Usually it helps to understand these better if you write them as equations using x and y rather than f(x) (just put y in place of f(x) in the expression. Then here are some rules: Symmetric in Y-axis TEST ------------------------ Replace all x values with (-x) and simplify. If the equation is the same as you started with, it is symmetric about the y-axis. Example: y = x^2 - 5 Replacing x with -x, we have y = (-x)^2 - 5 = x^2 - 5 Since it is the same, y = x^2 - 5 is symmetric about the y-axis Symmetric in X-axis TEST ------------------------ Replace all y values with (-y) and simplify. If the equation is the same then it is symmetric about the x-axis. Example: x^3 +y^2 = 4 Replacing y with -y, we have x^3 + (-y)^2 = 4, which simplifies to x^3 + y^2 = 4. Since it is the same, x^3 + y^2 = 4 is symmetric about the x-axis Symmetric in Origin TEST ------------------------ Replace BOTH x and y by their opposites and simplify. If they are the same then the graph is symmetric about the origin. Example: y = x^3 Replacing both x and y, we get (-y) = (-x)^3, which simplifies to -y = -x^3, and then y = x^3. Since we get the same equation, y = x^3 is symmetric about the origin. For another example, is 3xy = x^2 y^2 symmetric in x, y, or the origin? Test y-axis: 3(-x)y = (-x)^2 y^2 => -3xy = x^2 y^2 Thus, 3xy = x^2 y^2 is not symmetric about the y-axis. Test x-axis: 3x(-y) = x^2 (-y)^2 => -3xy = x^2 y^2 Thus, 3xy = x^2 y^2 is not symmetric about the x-axis. Test origin 3 (-x)(-y)= (-x)^2 (-y)^2 => 3xy = x^2 y^2 Thus, 3xy = x^2 y^2 is not symmetric about the origin. But what do these symmetries mean? Well, if a graph is symmetric about something, all it means is that you can fold the graph in some way, and the different parts of the graph will overlap. A graph that is symmetric about the y-axis will overlap if you fold the graph along the y-axis. Similarly, a graph that is symmetric about the x-axis will overlap if you fold the graph along the x-axis. Finally, a graph that is symmetric about the origin will overlap if you fold the graph along the x-axis and the y-axis. Here's a graph of y = x^3 to help you see the folds. First, here is the graph folded along the y-axis: The graph folded along the x-axis: And finally, the graph folded along both the x-axis and the y-axis: Hope this helps. This is the same test your teacher uses, but worded a little differently. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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