280 CHAPTER 2 Graphs and Functions (c) Use the property of absolute value that states 0 ab0 = 0 a0 # 0 b0 to rewrite 0 2x 0 . k1x2 = 0 2x 0 = 0 20 # 0 x 0 = 20 x 0 Property 3 Therefore, the graph of k1x2 = 0 2x 0 is the same as the graph of g1x2 = 20 x 0 shown in part (a) in Figure 70 and repeated in the margin. This is a horizontal shrink of the graph of ƒ1x2 = 0 x 0 . S Now Try Exercises 17 and 19. Vertical Stretching or Shrinking of the Graph of a Function Suppose that a 70. If a point 1x, y2 lies on the graph of y = ƒ1x2, then the point 1x, ay2 lies on the graph of y =aƒ1x2. (a) If a 71, then the graph of y = aƒ1x2 is a vertical stretching of the graph of y = ƒ1x2. (b) If 0 6a 61, then the graph of y = aƒ1x2 is a vertical shrinking of the graph of y = ƒ1x2. Figure 72 shows graphical interpretations of vertical stretching and shrinking. In both cases, the x-intercepts of the graph remain the same but the y-intercepts are affected. y x 0 (x, y) (x, ay) y = af(x), a > 1 y = f(x) Vertical stretching a > 1 Figure 72 y x 0 (x, y) (x, ay) y = af(x), 0 < a < 1 y = f(x) Vertical shrinking 0 < a < 1 Graphs of functions can also be stretched and shrunk horizontally. Horizontal Stretching or Shrinking of the Graph of a Function Suppose that a 70. If a point 1x, y2 lies on the graph of y = ƒ1x2, then the point Ax a , yB lies on the graph of y =ƒ1ax2. (a) If 0 6a 61, then the graph of y = ƒ1ax2 is a horizontal stretching of the graph of y = ƒ1x2. (b) If a 71, then the graph of y = ƒ1ax2 is a horizontal shrinking of the graph of y = ƒ1x2. See Figure 73 for graphical interpretations of horizontal stretching and shrinking. In both cases, the y-intercept remains the same but the x-intercepts are affected. y x 0 (x, y) y = f(ax), a > 1 y = f(x) Horizontal shrinking a > 1 ( , y) x a Figure 73 y x 0 (x, y) y = f(ax), 0 < a < 1 y = f(x) Horizontal stretching 0 < a < 1 ( , y) x a x y 3 3 –3 0 6 g(x) = 2uxu f(x) = uxu Figure 70 (repeated)
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