283 2.7 Graphing Techniques (b) In x = y2 - 3, replace y with -y. x = 1-y22 - 3 = y2 - 3 Same as the original equation The graph is symmetric with respect to the x-axis, as shown in Figure 78(a). It is not symmetric with respect to the y-axis. (c) In x2 + y2 = 16, substitute -x for x and then -y for y. 1-x22 + y2 = 16 and x2 + 1-y22 = 16 Both simplify to the original equation, x2 + y2 = 16. The graph, which is a circle of radius 4 centered at the origin, is symmetric with respect to both axes. See Figure 78(b). LOOKING AHEAD TO CALCULUS The tools of calculus enable us to find areas of regions in the plane. To find the area of the region below the graph of y = x2, above the x-axis, bounded on the left by the line x = -2 and on the right by x = 2, draw a sketch of this region. Because of the symmetry of the graph of y = x2, the desired area is twice the area to the right of the y-axis. Thus, symmetry can be used to reduce the original problem to an easier one by simply finding the area to the right of the y-axis and then doubling the answer. (d) In 2x + y = 4, replace x with -x and then replace y with -y. 2x + y = 4 21-x2 + y = 4 Not equivalent -2x + y = 4 2x + y = 4 2x + 1-y2 = 4 Not equivalent 2x - y = 4 The graph is not symmetric with respect to either axis. See Figure 78(c). S Now Try Exercise 45. Symmetry with Respect to the Origin The graph of an equation is symmetric with respect to the origin if the replacement of both x with -x and y with -y at the same time results in an equivalent equation. Another kind of symmetry occurs when a graph can be rotated 180° about the origin, with the result coinciding exactly with the original graph. Symmetry of this type is symmetry with respect to the origin. In general, for a graph to be symmetric with respect to the origin, the point 1 −x, −y2 is on the graph whenever the point 1x, y2 is on the graph. Figure 79 shows two such graphs. x y 0 (–x, –y) (x, y) Origin symmetry Figure 79 x y 0 (–x, –y) (x, y) Graphs of the cubing and cube root functions in the previous section are symmetric with respect to the origin. x y –3 –2 2 0 x = y2 – 3 x-axis symmetry x y 4 4 –4 –4 x2 + y2 = 16 0 x-axis and y-axis symmetry (b) x y 2 4 0 No x-axis or y-axis symmetry (c) Figure 78 (a)
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