164 CHAPTER 3 Linear and Quadratic Functions 4 Find a Quadratic Function Given Its Vertex and One Other Point Finding the Quadratic Function Given Its Vertex and One Other Point Determine the quadratic function whose vertex is 1, 5 ( ) − and whose y -intercept is 3. − The graph of the parabola is shown in Figure 22. Solution EXAMPLE 6 The vertex is 1, 5 , ( ) − so h 1 = and k 5. = − Substitute these values into equation (2). f x a x h k f x a x 1 5 2 2 ( ) ( ) ( ) ( ) = − + = − − To determine the value of a , use the fact that f 0 3 ( ) = − (the y -intercept). ( ) ( ) ( ) = − − − = − − − = − = f x a x a a a 1 5 3 0 1 5 3 5 2 2 2 ( ) = = =− x y f 0, 0 3 The quadratic function we seek is f x a x h k x x x 2 1 5 2 4 3 2 2 2 ( ) ( ) ( ) = − + = − − = − − Figure 22 ( ) = − − f x x x 2 4 3 2 x y –2 –1 1 2 4 –4 8 –8 12 3 4 (0, –3) (1, –5) (2, –3) If the vertex h k , ( ) and one additional point on the graph of a quadratic function f x ax bx c a , 0, 2 ( ) = + + ≠ are known, then the vertex form of f , f x a x h k 2 ( ) ( ) = − + (2) can be used to obtain the quadratic function. Figure 23 = − − Y x x 2 4 3 1 2 12 26 22 4 Equation (2) = =− h k 1, 5 Check: Figure 23 shows the graph of Y x x 2 4 3 1 2 = − − using a TI-84 Plus CE. Now Work PROBLEM 61 5 Find the Maximum or Minimum Value of a Quadratic Function The graph of a quadratic function f x ax bx c a 0 2 ( ) = + + ≠ is a parabola with vertex b a f b a 2 , 2 . ( ) ( ) − − The vertex is the highest point on the graph if a 0 < and the lowest point on the graph if a 0. > If the vertex is the highest point a 0 , ( ) < then f b a 2 ( ) − is the maximum value of f . If the vertex is the lowest point a 0 , ( ) > then f b a 2 ( ) − is the minimum value of f . Finding the Maximum or Minimum Value of a Quadratic Function Determine whether the quadratic function f x x x4 5 2 ( ) = − − has a maximum or a minimum value. Then find the maximum or minimum value. EXAMPLE 7
RkJQdWJsaXNoZXIy NjM5ODQ=