326 CHAPTER 3 Polynomial and Rational Functions (c) Notice that F1x2 = - 1 2 1x - 422 + 3 is related to g1x2 = - 1 2 x 2 from part (b). The graph of F1x2 is the graph of g1x2 translated to the right 4 units and up 3 units. See Figure 7. The vertex is the point 14, 32, which is also shown in the calculator graph in Figure 8, and the axis of the parabola is the line x = 4. The domain is 1-∞, ∞2, and the range is 1-∞, 34. y x y = x2 (0, 0) x = 0 g(x) = – x2 1 2 y = x2 1 2 Figure 5 −5 −8 5 8 g(x) = – x2 1 2 y = x2 1 2 y = x2 Figure 6 Calculator graphs are shown in Figure 6. x y (0, 0) 4 F(x) = – (x – 4)2 + 3 x = 4 (4, 3) 1 2 g(x) = – x2 1 2 Figure 7 −6 −3 4 8 g(x) = – x2 1 2 F(x) = − (x − 4)2 + 3 1 2 Figure 8 S Now Try Exercises 19 and 21. Completing the Square In general, the graph of the quadratic function ƒ1x2 =a1x −h22 +k 1a 302 is a parabola with vertex 1h, k2 and axis of symmetry x =h. The parabola opens up if a 70 and down if a 60. With these facts in mind, we complete the square to graph the general quadratic function ƒ1x2 =ax2 +bx +c. EXAMPLE 2 Graphing a Parabola 1a =12 Graph ƒ1x2 = x2 - 6x + 7. Find the largest open intervals over which the function is increasing or decreasing. SOLUTION We express x2 - 6x + 7 in the form 1x - h22 + k by completing the square. In preparation for this, we first write ƒ1x2 = 1x2 - 6x 2 + 7. Prepare to complete the square. We must add a number inside the parentheses to obtain a perfect square trinomial. Find this number by taking half the coefficient of x and squaring the result.
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