327 3.1 Quadratic Functions and Models f(x) = x2 − 6x + 7 f(x) = (x − 3)2 − 2 −4 −3 10 9 This screen shows that the vertex of the graph in Figure 9 is the point 13, -22. Because it is the lowest point on the graph, we direct the calculator to find the minimum. x f(x) = x2 – 6x + 7 f(x) = (x – 3)2 – 2 y 0 3 7 6 –2 (3, –2) x = 3 Figure 9 The domain of this function is 1-∞, ∞2, and the range is 3-2, ∞2. Because the lowest point on the graph is the vertex 13, -22, the function is decreasing on 1-∞, 32 and increasing on 13, ∞2. S Now Try Exercises 31 and 41. x y 0 7 1 2 3 -2 5 2 6 7 Find using symmetry about the axis. y-intercept Vertex C 1 2 1-62D 2 = 1322 = 9 Take half the coefficient of x. Square the result. ƒ1x2 = 1x2 - 6x + 9 - 92 + 7 Add and subtract 9. ƒ1x2 = 1x2 - 6x + 92 - 9 + 7 Regroup terms. ƒ1x2 = 1x - 322 - 2 Factor and simplify. The vertex of the parabola is the point 13, -22, and the axis is the line x = 3. We find additional ordered pairs that satisfy the equation, as shown in the table, and plot and join these points to obtain the graph in Figure 9. This is the same as adding 0. NOTE In Example 2 we added and subtracted 9 on the same side of the equation to complete the square. This differs from adding the same number on each side of the equation, as is sometimes done in the procedure. We want ƒ1x2:that is, y—alone on one side of the equation, so we adjusted that step in the process of completing the square here. EXAMPLE 3 Graphing a Parabola 1a 312 Graph ƒ1x2 = -3x2 - 2x + 1. Identify the intercepts of the graph. SOLUTION To complete the square, the coefficient of x2 must be 1. ƒ1x2 = -3ax2 + 2 3 x b + 1 Factor -3 from the first two terms. ƒ1x2 = -3ax2 + 2 3 x + 1 9 - 1 9b + 1 C 1 2 A 2 3B D 2 = A1 3B 2 = 1 9 , so add and subtract 1 9 . ƒ1x2 = -3ax2 + 2 3 x + 1 9b - 3a- 1 9b + 1 Distributive property ƒ1x2 = -3ax + 1 3b 2 + 4 3 Factor and simplify. The vertex is the point A - 1 3 , 4 3B . The intercepts are good additional points to find. The y-intercept is found by evaluating ƒ102. ƒ102 = -31022 - 2102 + 1 Let x = 0 in ƒ1x2 = -3x2 - 2x + 1. ƒ102 = 1 The y-intercept is 10, 12. Be careful here.
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