Minimum and Maximum Values of Quadratic Functions :

Consider the function  with vertex .

1. If a > 0, f(x) has a minimum at . The minimum value is .

2. If a < 0, f(x) has a maximum at . The maximum value is .

The polynomial function is .

Compare the polynomial function with general form of quadratic function .

a = - 1, b = 0 and c = 9.

Since a = - 1 < 0, f(x) has a maximum at  .

The maximum value is and this is called global maximum.

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(Note: Quadratics open upward or downward, therefore they will never have any local maximum or minimums)

The leading coefficient (a = - 1) is negative hence f(x) has a maximum at (0 , 9) and f(x) is increasing on (- infinity, 9) and decreasing on (9, + infinity). See graph below of f(x) below.

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