160 CHAPTER 3 Linear and Quadratic Functions The method used in Example 1 can be used to graph any quadratic function f x ax bx c a , 0, 2 ( ) = + + ≠ as follows: ( ) ( ) ( ) ( ) ( ) = + + = + + = + + + − ⋅ = + + − = + + − f x ax bx c a x b a x c a x b a x b a c a b a a x b a c b a a x b a ac b a 4 4 2 4 2 4 4 2 2 2 2 2 2 2 2 2 2 2 These results lead to the following conclusion: Properties of the Graph of a Quadratic Function (a Parabola) f x ax bx c a 0 2 ( ) = + + ≠ • ( ) ( ) = − − b a f b a Vertex 2 , 2 • Axis of symmetry: the vertical line = − x b a2 • A parabola is concave up if a 0; > the vertex is the minimum point. • A parabola is concave down if a 0; < the vertex is the maximum point. Factor out a from ax bx 2 + . Complete the square by adding b a4 2 2 in the parentheses, and subtracting a⋅ b a4 2 2 . Factor; simplify. − = ⋅ − = − c b a c a a b a ac b a 4 4 4 4 4 4 2 2 2 DEFINITION Vertex Form of a Quadratic Function Suppose f is the quadratic function f x ax bx c. 2 ( ) = + + If f is written as f x a x h k 2 ( ) ( ) = − + (1) where h b a 2 = − and k ac b a 4 4 , 2 = − then the quadratic function f is in vertex form . The graph of f x a x h k 2 ( ) ( ) = − + is the parabola y ax2 = shifted horizontally h units (replace x by x h − ) and vertically k units (add k ). The vertex of the parabola is h k , ( ) , and the function is concave up if a 0 > and is concave down if a 0. < The axis of symmetry is the vertical line x h. = For example, compare equation (1) with the solution given in Example 1. f x x x a x h k 2 2 3 2 2 3 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) = + − = − − + − ↑ ↑ ↑ = − + Because a 2, = the graph is concave up. Also, because h 2 = − and k 3, = − its vertex is 2, 3 ( ) − − , the lowest point on the graph. 2 Identify the Vertex and Axis of Symmetry of a Parabola We do not need to complete the square to identify the vertex of a parabola. It is usually easier to obtain the vertex by remembering that its x -coordinate is = − h b a2 . The y -coordinate k is then found by evaluating f at − b a2 . That is, ( ) = − k f b a2 .
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