282 CHAPTER 2 Graphs and Functions y x 0 (–x, y) (x, y) y = f(x) y-axis symmetry (a) Figure 76 y x 0 (x, y) (x, –y) x-axis symmetry (b) Symmetry The graph of ƒ shown in Figure 76(a) is cut in half by the y-axis, with each half the mirror image of the other half. Such a graph is symmetric with respect to the y-axis. In general, for a graph to be symmetric with respect to the y-axis, the point 1 −x, y2 must be on the graph whenever the point 1x, y2 is on the graph. Similarly, if the graph in Figure 76(b) were folded in half along the x-axis, the portion at the top would exactly match the portion at the bottom. Such a graph is symmetric with respect to the x-axis. In general, for a graph to be symmetric with respect to the x-axis, the point 1x, −y2 must be on the graph whenever the point 1x, y2 is on the graph. Symmetry with Respect to an Axis The graph of an equation is symmetric with respect to the y-axis if the replacement of x with -x results in an equivalent equation. The graph of an equation is symmetric with respect to the x-axis if the replacement of y with -y results in an equivalent equation. Graphs of the squaring and absolute value functions in the previous section are symmetric with respect to the y-axis. EXAMPLE 3 Testing for Symmetry with Respect to an Axis Test for symmetry with respect to the x-axis and the y-axis. (a) y = x2 + 4 (b) x = y2 - 3 (c) x2 + y2 = 16 (d) 2x + y = 4 SOLUTION (a) In y = x2 + 4, replace x with -x. y = x2 + 4 y = 1-x22 + 4 The result is equivalent to the original equation. y = x2 + 4 Use parentheses around -x. Multiply by -1. Thus the graph, shown in Figure 77, is symmetric with respect to the y-axis. The y-axis cuts the graph in half, with the halves being mirror images. Now replace y with -y to test for symmetry with respect to the x-axis. y = x2 + 4 -y = x2 + 4 The result is not equivalent to the original equation. y = -x2 - 4 The graph is not symmetric with respect to the x-axis. See Figure 77. x y y-axis symmetry 4 2 –2 0 y = x2 + 4 Figure 77
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