Question

A quadratic function is defined by f ( x ) = x 2 − 8 x − 4 . which expression also defines f and best reveals the maximum or minimum of the function.

Answer 1

Explanation:

Since the coefficient of x^2 is positive, this quadratic is a parabola in the shape of a U, hence has a minimum.

We want to end up with the form (x-h)^2 + c. Since (x-h)^2>=0, this form shows that the minimum is achieved when x=h.

Completing the square will put the quadratic in the desired form. Note that:

(x-h)^2=x^2-2hx+h^2

Comparing this with the given form, we must have -8=-2h, or h=4. But we are missing h^2=4^2=16. We can add the missing 16 and subtract it elsewhere without changing the quadratic.

x^2-8x+16 + (16-4) = (x-4)^2 + 12

Now we know that at x=4 the quadratic has a minimum and that the minimum is 12.