279 2.7 Graphing Techniques 2.7 Graphing Techniques ■ Stretching and Shrinking ■ Reflecting ■ Symmetry ■ Even and Odd Functions ■ Translations Graphing techniques presented in this section show how to graph functions that are defined by altering the equation of a basic function. EXAMPLE 1 Stretching or Shrinking Graphs Graph each function. (a) g1x2 = 20 x 0 (b) h1x2 = 1 2 0 x 0 (c) k1x2 = 0 2x 0 SOLUTION (a) Comparing the tables of values for ƒ1x2 = 0 x 0 and g1x2 = 20 x 0 in Figure 70, we see that for corresponding x-values, the y-values of g are each twice those of ƒ. The graph of ƒ1x2 = 0 x 0 is vertically stretched. The graph of g1x2, shown in blue in Figure 70, is narrower than that of ƒ1x2, shown in red for comparison. x y 3 3 –3 0 6 g(x) = 2uxu f(x) = uxu Figure 70 x ƒ1x2 = ∣ x∣ g1x2 =2∣ x∣ -2 2 4 -1 1 2 0 0 0 1 1 2 2 2 4 (b) The graph of h1x2 = 1 2 0 x 0 is also the same general shape as that of ƒ1x2, but for corresponding x-values, the y-values of h are each half those of ƒ. The coefficient 1 2 is between 0 and 1 and causes a vertical shrink. The graph of h1x2 is wider than the graph of ƒ1x2, as we see by comparing the tables of values. See Figure 71. x y 4 2 –2 –4 0 4 2 h(x) = uxu f(x) = uxu 1 2 Figure 71 x ƒ1x2 = ∣ x∣ h1x2 =1 2 ∣ x∣ -2 2 1 -1 1 1 2 0 0 0 1 1 1 2 2 2 1 Stretching and Shrinking We begin by considering how the graphs of y =aƒ1x2 and y =ƒ1ax2 compare to the graph of y = ƒ1x2, where a 70.
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