286 CHAPTER 2 Graphs and Functions x y (0, –4) –4 4 0 g(x) = uxu – 4 f(x) = uxu Figure 81 x ƒ1x2 = ∣ x∣ g1x2 = ∣ x∣ −4 -4 4 0 -1 1 -3 0 0 -4 1 1 -3 4 4 0 S Now Try Exercise 73. The graphs in Example 6 suggest the following generalization. EXAMPLE 6 Translating a Graph Vertically Graph g1x2 = 0 x 0 - 4. SOLUTION Comparing the tables of values shown with Figure 81, we see that for corresponding x-values, the y-values of g are each 4 less than those for ƒ. The graph of g1x2 = 0 x 0 - 4 is the same as that of ƒ1x2 = 0 x 0 , but translated down 4 units. The lowest point is at 10, -42. The graph is symmetric with respect to the y-axis and is therefore the graph of an even function. Vertical Translations Given a function g defined by g1x2 =ƒ1x2 +c, where c is a real number: • For every point 1x, y2 on the graph of ƒ, there will be a corresponding point 1x, y + c2 on the graph of g. • The graph of g will be the same as the graph of ƒ, but translated up c units if c is positive or down 0 c 0 units if c is negative. The graph of g is a vertical translation of the graph of ƒ. Figure 82 shows a graph of a function ƒ and two vertical translations of ƒ. Figure 83 shows two vertical translations of y1 = x 2 on a TI-84 Plus calculator screen. x y y = f(x) + 2 Vertical translation down 3 units Vertical translation up 2 units Original graph y = f(x) – 3 y = f(x) 3 2 1 –1 –2 –3 –4 0 Figure 82 −8 −5 8 5 y1 = x 2 y 2 = x 2 + 2 y3 = x 2 − 6 y2 is the graph of y1 = x 2 translated up 2 units. y3 is that of y1 translated down 6 units. Figure 83 Translations The next examples show the results of horizontal and vertical shifts, or translations, of the graph of ƒ1x2 = 0 x 0 .
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