284 CHAPTER 2 Graphs and Functions –2 –1 2 –8 1 8 x y y = x3 Origin symmetry Figure 80 EXAMPLE 4 Testing for Symmetry with Respect to the Origin Determine whether the graph of each equation is symmetric with respect to the origin. (a) x2 + y2 = 16 (b) y = x3 SOLUTION (a) Replace x with -x and y with -y. x2 + y2 = 16 1-x22 + 1-y22 = 16 Equivalent x2 + y2 = 16 Use parentheses around -x and -y. The graph, which is the circle shown in Figure 78(b) in Example 3(c), is symmetric with respect to the origin. (b) Replace x with -x and y with -y. y = x3 -y = 1-x23 Equivalent -y = -x3 y = x3 The graph, which is that of the cubing function, is symmetric with respect to the origin and is shown in Figure 80. S Now Try Exercise 49. Notice the following important concepts regarding symmetry: • A graph symmetric with respect to both the x- and y-axes is automatically symmetric with respect to the origin. (See Figure 78(b).) • A graph symmetric with respect to the origin need not be symmetric with respect to either axis. (See Figure 80.) • Of the three types of symmetry—with respect to the x-axis, with respect to the y-axis, and with respect to the origin—a graph possessing any two types must also exhibit the third type of symmetry. • A graph symmetric with respect to the x-axis does not represent a function. (See Figures 78(a) and 78(b).) Summary of Tests for Symmetry Symmetry with Respect to: x-axis y-axis Origin Equation is unchanged if: y is replaced with -y x is replaced with -x x is replaced with -x and y is replaced with -y Example: x y 0 x y 0 x y 0
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