208 Chapter 2 Linear Equations in Two Variables and Functions The shapes of these six graphs will be developed in the homework exercises. These functions are used often in the study of algebra. Therefore, we recommend that you associate an equation with its graph and commit each to memory. 3. Definition of a Quadratic Function In Example 1 we graphed the function defined by f 1x2 x2 by plotting points.This function belongs to a special category called quadratic functions. Definition of a Quadratic Function A quadratic function is a function defined by f where a, b, and c are real numbers and a 0. 1x2 ax2 bx c The graph of a quadratic function is in the shape of a parabola.The leading coefficient, a 7 0 • If , then the parabola opens upward, and the vertex is the minimum point f 1x2 x2 on the parabola. For example, the graph of is shown in Figure 2-40. a 6 0 • If , then the parabola opens downward, and the vertex is the maximum point on the parabola. For example, the graph of f is shown in Figure 2-41. 1x2 x2 5 4 3 2 Vertex 1 y a, determines the direction of the parabola. 54 3 1 2 3 4 5 21 1 2 3 4 5 x f(x) x2 Figure 2-41 5 4 3 y 2 1 Vertex 54 3 1 2 3 4 5 21 1 2 3 4 5 x f (x) x2 Figure 2-40 Identifying Functions Example 3 Identify each function as linear, constant, quadratic, or none of these. a. b. c. f 1x2 f 7 2x 1x2 f x2 3x 2 1x2 4 4x 8 f 2 1x2 d. e. f 1x2 6 x 8 Solution: a. is a constant function. It is in the form where b. is a quadratic function. It is in the form f b 4. 1x2 f b, 1x2 4 f 1x2 x2 3x 2 f 1x2 ax2 bx c, where a 0. f 1x2 f mx b, 1x2 7 2x c. is linear. Writing it in the form we get f where m 2 and b 7. 1x2 2x 7,
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