329 3.1 Quadratic Functions and Models ƒ1x2 = ax2 + bx + c General quadratic form = aax2 + b a x b + c Factor a from the first two terms. = aax2 + b a x + b2 4a2b + c - aa b2 4a2b = aax + b 2ab 2 + c - b2 4a Factor and simplify. ƒ1x2 = aJx - a- b 2ab R 2 + 4ac - b2 4a Vertex form of ƒ1x2 = a1x -h22 +k (11)11* (111)111* h k Thus, the vertex 1h, k2 can be expressed in terms of a, b, and c. It is not necessary to memorize the expression for k because it is equal to ƒ1h2 =ƒA −b 2aB . The Vertex Formula We can generalize the earlier work to obtain a formula for the vertex of a parabola. Add C 1 2 A b aB D 2 = b2 4a2 inside the parentheses. Subtract a A b2 4a2B outside the parentheses. LOOKING AHEAD TO CALCULUS An important concept in calculus is the definite integral. If the graph of ƒ lies above the x-axis, the symbol L b a ƒ1x2 dx represents the area of the region above the x-axis and below the graph of ƒ from x = a to x = b. For example, in Figure 10 with ƒ1x2 = -3x2 - 2x + 1, a = -1, and b = 1 3 , calculus provides the tools for determining that the area enclosed by the parabola and the x-axis is 32 27 (square units). Graph of a Quadratic Function The quadratic function ƒ1x2 = ax2 + bx + c can be written as y =ƒ1x2 =a1x −h22 +k, with a≠0, where h = − b 2a and k =ƒ1h2. Vertex formula The graph of ƒ has the following characteristics. 1. It is a parabola with vertex 1h, k2 and the vertical line x = h as axis. 2. It opens up if a 70 and down if a 60. 3. It is wider than the graph of y = x2 if a 61 and narrower if a 71. 4. The y-intercept is 10, ƒ1022 = 10, c2. 5. The x-intercepts are found by solving the equation ax2 + bx + c = 0. • If b2 - 4ac 70, then the x-intercepts are Q-b { 2b2 - 4ac 2a , 0R . • If b2 - 4ac = 0, then the x-intercept is A - b 2a , 0B. • If b2 - 4ac 60, then there are no x-intercepts. EXAMPLE 4 Using the Vertex Formula Find the axis and vertex of the parabola having equation ƒ1x2 = 2x2 + 4x + 5. SOLUTION The axis of the parabola is the vertical line x = h = - b 2a = - 4 2122 = -1. Use the vertex formula. Here a = 2 and b = 4. The vertex is 1-1, ƒ1-122. Evaluate ƒ1-12. ƒ -12 = 21-122 + 41-12 + 5 = 3 The vertex is 1-1, 32. S Now Try Exercise 29(a).
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