24 CHAPTER 1 Graphs First, find the intercepts.When = x 0, then = y 0; and when = y 0, then = x 0. The origin ( ) 0, 0 is the only intercept. Now test for symmetry. x-Axis: Replace y by −y. Since − = y x3 is not equivalent to = y x ,3 the graph is not symmetric with respect to the x-axis. y-Axis: Replace x by −x. Since ( ) = − = − y x x 3 3 is not equivalent to = y x ,3 the graph is not symmetric with respect to the y-axis. Origin: Replace x by −x and y by −y. Since ( ) − = − = − y x x 3 3 is equivalent to = y x3 (multiply both sides by −1), the graph is symmetric with respect to the origin. To graph = y x3 by hand, use the equation to obtain several points on the graph. Because of the symmetry, we need to locate only points on the graph for which ≥ x 0. See Table 5. Since ( ) 1, 1 is on the graph, and the graph is symmetric with respect to the origin, the point ( ) − − 1, 1 is also on the graph. Plot the points from Table 5 and use the symmetry. Points on the graph can also be obtained using the TABLE feature on a graphing utility. See Table 6. Do you see the symmetry with respect to the origin from Table 6? Figure 37 shows the graph. 3 Know How to Graph Key Equations The next three examples use intercepts, symmetry, and point plotting to obtain the graphs of key equations. It is important to know the graphs of these key equations because we use them later. The first of these is = y x .3 Graphing the Equation = y x3 by Finding Intercepts, Checking for Symmetry, and Plotting Points Graph the equation = y x3 by hand by plotting points. Find any intercepts and check for symmetry first. Solution Figure 37 y x3 = x y (2, 8) 8 (1, 1) (0, 0) (–1, –1) 6 –6 –8 (–2, –8) y x y2 = x y, ( ) 0 0 0, 0 ( ) 1 1 1, 1 ( ) 2 4 4, 2 ( ) 3 9 9, 3 ( ) Table 7 EXAMPLE 7 x y x3 = x y, ( ) 0 0 0, 0 ( ) 1 1 1, 1 ( ) 2 8 ( ) 2, 8 3 27 ( ) 3, 27 Table 5 Table 6 Graphing the Equation = x y2 (a) Graph the equation = x y .2 Find any intercepts and check for symmetry first. (b) Graph = ≥ x y y , 0. 2 Solution (a) The lone intercept is ( ) 0, 0 . The graph is symmetric with respect to the x-axis since ( ) = − x y 2 is equivalent to = x y .2 The graph is not symmetric with respect to the y-axis or the origin. To graph = x y2 by hand, use the equation to obtain several points on the graph. Because the equation is solved for x, it is easier to assign values to y and use the equation to determine the corresponding values of x. Because of the symmetry, start by finding points whose y-coordinates are nonnegative.Then use the symmetry to find additional points on the graph. See Table 7. For example, since ( ) 1, 1 is on the graph, so is ( ) − 1, 1 . Since ( ) 4, 2 is on the graph, so is ( ) − 4, 2 , and so on. Plot these points and connect them with a smooth curve to obtain Figure 38 on the next page. EXAMPLE 8
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