288 CHAPTER 2 Graphs and Functions CAUTION Errors frequently occur when horizontal shifts are involved. To determine the direction and magnitude of a horizontal shift, find the value that causes the expression x - h to equal 0, as shown below. F1x2 = 1x −522 Because +5 causes x - 5 to equal 0, the graph of F1x2 illustrates a shift to the right 5 units. F1x2 = 1x +522 Because −5 causes x + 5 to equal 0, the graph of F1x2 illustrates a shift to the left 5 units. x y y = uxu h(x) = z2x – 4z = 2zx – 2z 4 2 2 4 0 Figure 88 –3 3 –3 4 x y g(x) = – x2 + 4 1 2 0 Figure 89 (c) The graph of g1x2 = -1 2x 2 + 4 has the same shape as that of y = x2, but it is wider (that is, shrunken vertically), reflected across the x-axis because the coefficient -1 2 is negative, and then translated up 4 units. See Figure 89. S Now Try Exercises 75, 77, and 79. EXAMPLE 8 Using MoreThan OneTransformation Graph each function. (a) ƒ1x2 = -0 x + 30 + 1 (b) h1x2 = 0 2x - 40 (c) g1x2 = - 1 2 x2 + 4 SOLUTION (a) To graph ƒ1x2 = -0 x + 30 + 1, the lowest point on the graph of y = 0 x 0 is translated to the left 3 units and up 1 unit. The graph opens down because of the negative sign in front of the absolute value expression, making the lowest point now the highest point on the graph, as shown in Figure 87. The graph is symmetric with respect to the line x = -3. (b) To determine the horizontal translation, factor out 2. h1x2 = 0 2x - 40 h1x2 = 0 21x - 22 0 Factor out 2. h1x2 = 0 20 # 0 x - 20 0 ab 0 = 0 a 0 # 0 b 0 h1x2 = 20 x - 20 0 2 0 = 2 The graph of h is the graph of y = 0 x 0 translated to the right 2 units, and vertically stretched by a factor of 2. Horizontal shrinking gives the same appearance as vertical stretching for this function. See Figure 88. x y 3 –2 –6 –3 0 f(x) = – zx + 3z + 1 Figure 87
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