287 2.7 Graphing Techniques EXAMPLE 7 Translating a Graph Horizontally Graph g1x2 = 0 x - 40 . SOLUTION Comparing the tables of values given with Figure 84 shows that for corresponding y-values, the x-values of g are each 4 more than those for ƒ. The graph of g1x2 = 0 x - 4 0 is the same as that of ƒ1x2 = 0 x 0 , but translated to the right 4 units. The lowest point is at 14, 02. As suggested by the graphs in Figure 84, this graph is symmetric with respect to the line x = 4. x ƒ1x2 = ∣ x∣ g1x2 = ∣ x −4∣ -2 2 6 0 0 4 2 2 2 4 4 0 6 6 2 x y (4, 0) 8 0 4 g(x) = zx – 4z f(x) = zxz Figure 84 S Now Try Exercise 71. The graphs in Example 7 suggest the following generalization. Horizontal Translations Given a function g defined by g1x2 =ƒ1x −c2, where c is a real number: • For every point 1x, y2 on the graph of ƒ, there will be a corresponding point 1x + c, y2 on the graph of g. • The graph of g will be the same as the graph of ƒ, but translated to the right c units if c is positive or to the left 0 c 0 units if c is negative. The graph of g is a horizontal translation of the graph of ƒ. Figure 85 shows a graph of a function ƒ and two horizontal translations of ƒ. Figure 86 shows two horizontal translations of y1 = x 2 on a TI-84 Plus calculator screen. Vertical and horizontal translations are summarized in the table, where ƒ is a function and c is a positive number. Summary of Translations To Graph 1 c +02 : Shift the Graph of y =ƒ1x2 by c Units: y = ƒ1x2 + c up y = ƒ1x2 - c down y = ƒ1x + c2 left y = ƒ1x - c2 right x y –3 –1 1 2 3 4 Horizontal translation to the left 2 units y = f(x + 2) Horizontal translation to the right 3 units Original graph y = f(x) y = f(x – 3) Figure 85 −5 −10 10 10 y1 =x 2 y2 = (x + 2) 2 y 3 = (x − 6) 2 Figure 86 y2 is the graph of y1 = x 2 translated to the left 2 units. y3 is that of y1 translated to the right 6 units.
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