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4.1.1 4.1: Quadratic Functions Definition: A quadratic function is a function which can be written in the form f ( x) ax 2 bx c where a 0 . The graph of a quadratic function is a parabola. Definition: The maximum value of a function is the largest y-value on the graph. The minimum value of a function is the smallest y-value on the graph. Standard form for a quadratic function: Every quadratic function f ( x) ax 2 bx c ,where a 0 , can be written in the form f ( x ) a ( x h) 2 k . This is called standard form for a quadratic function. To write a quadratic function f ( x) ax 2 bx c in standard form, we need to complete the square which results the following b formulation of h and k f (h) . 2a b The vertex of the parabola is (h, k ) where h and k f (h) . 2a The parabola is symmetric with respect to the vertical line x h . This line is called axis of symmetry for the parabola. If a 0 , the graph of f ( x) opens down and f f ( x) has a maximum value. If a 0 , the graph of f ( x) opens up and f ( x) has a minimum value. The larger a , is the narrower the parabola is. Example 1: Sketch the graph of f ( x) 5 4 x x 2 . State the vertex and intercepts. What is the maximum or minimum value and where they are attained? Find the domain and range. 4.1.2 Example 2: Sketch the graph of f ( x) 2 x 2 12 x 16 . State the vertex and intercepts. What is the maximum or minimum value and where they are attained? Find the domain and range. Example 3: Sketch the graph of f ( x) 3x 2 15 x 18 . State the vertex and intercepts. What is the maximum or minimum value and where they are attained? Find the domain and range. 4.1.3 Example 4: Sketch the graph of f ( x) 2 x 2 16 x 29 . State the vertex and intercepts. What is the maximum or minimum value and where they are attained? Find the domain and range. Example 5: Sketch the graph of f ( x) 2 x 2 12 x 16 . State the vertex and intercepts. What is the maximum or minimum value and where they are attained? Find the domain and range. 4.1.4 Example 6: Write f ( x) x 2 x 2 in standard form. What is its maximum or minimum value? What input produces this maximum or minimum? Example 7: Find the maximum or minimum value of f ( x) 2 x 2 5 x 1 . Example 8: Find the quadratic function such that f (3) 6 and the vertex is (2, 3) . HOMEWORK 4.1: 4.1 Pg. 265: 24, 28,32, 36-50 EVEN ,54-62 EVEN , 76