SECTION 3.3 Quadratic Functions and Their Properties 161 3 Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts The location of the vertex and intercepts, along with knowledge of whether the graph is concave up or concave down, is usually enough information to graph f x ax bx c a , 0. 2 ( ) = + + ≠ • The y -intercept is the value of f at x 0; = that is, the y -intercept is f c 0 . ( ) = • The x -intercepts, if there are any, are found by solving the quadratic equation ax bx c 0 2 + + = A quadratic equation has two, one, or no real solutions, depending on whether the discriminant b ac 4 2 − is positive, 0, or negative. Now Work PROBLEM 45(a) Locating the Vertex of a Parabola without Graphing Without graphing, locate the vertex and axis of symmetry of the parabola defined by f x x x 3 6 1. 2 ( ) = − + + Is it concave up or concave down? Solution EXAMPLE 2 For this quadratic function, a b 3, 6, = − = and c 1. = The x -coordinate of the vertex is h b a 2 6 2 3 1 ( ) = − = − − = The y -coordinate of the vertex is ( ) ( ) = − = = − ⋅ + ⋅ + = k f b a f 2 1 3 1 6 1 1 4 2 The vertex is located at the point 1, 4 ( ) . The axis of symmetry is the line x 1. = Because a 3 0, = − < the parabola is concave down. The x -Intercepts of a Quadratic Function • If the discriminant b ac 4 0, 2 − > the graph of f x ax bx c 2 ( ) = + + has two distinct x -intercepts so it crosses the x -axis in two places. • If the discriminant b ac 4 0, 2 − = the graph of f x ax bx c 2 ( ) = + + has one x -intercept so it touches the x -axis at its vertex. • If the discriminant b ac 4 0, 2 − < the graph of f x ax bx c 2 ( ) = + + has no x -intercepts so it does not cross or touch the x -axis. NOTE If the vertex lies in Quadrants I or II and the parabola is concave up, then the graph of the quadratic function has no x -intercepts. Similarly, if the vertex lies in Quadrants III or IV and the parabola is concave down, then the graph of the quadratic function has no x -intercepts. j Figure 18 illustrates these possibilities for parabolas that are concave up. x x -intercept x-intercept , f ( ( )) x-intercept y (a) b2 – 4ac . 0 Two x-intercepts – – b 2a b 2a , f ( ( )) – – b 2a b 2a (– b One x-intercept x y (b) b2 – 4ac = 0 , 0 2a ) No x-intercepts (c) b2 – 4ac < 0 x y Axis of symmetry b 2a x = – b 2a x = – b 2a x = – Axis of symmetry Axis of symmetry Figure 18 ( ) = + + > f x ax bx c a , 0 2
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