A quadratic function f (x ) = ax^2 + bx + c, can be expressed in the standard form : a(x-h)^2 + k

by completing the square. The graph of f(x) is a parabola with vertex (h,k); the parabola opens upward if a > 0 or downward if a < 0.

Maximum or Minimum Value of a Quadratic Function

Let f be a quadratic function with standard form f (x) = a( x − h )^2 + k.

The maximum or minimum value of f occurs at x = h

If a > 0, then the minimum value of f is f(h) = k.

If a < 0, then the maximum value of is f (h) = k

We now derive a formula for the maximum or minimum of the quadratic function

F(x) = ax^2 + bx + c.

For either of the two cases (the quadratic having a maxima or a minima), the maxima or the minima,

as the case may be, will occur when x = - b/2a

the maximum or minimum value is f(-b/2a) = c - b^2/4a

remember that - b/2a = sum of roots/2