Question

Using the completing-the-square method, find the vertex of the function f(x) = 2x2 − 8x + 6 and indicate whether it is a minimum or a maximum and at what point.

Maximum at (2, –2)
Minimum at (2, –2)
Maximum at (2, 6)
Minimum at (2, 6)

Answer 1

Answer: Minimum at (2,-2)

Proof of validity is shown below.

Answer 2

Minimum at (2, -2)

Explanation:

Factor the leading coefficient from the first two terms:

f(x) = 2(x^2 -4x) +6

Inside parentheses, add the square of half the x-coefficient. Add the opposite of that amount outside parentheses.

f(x) = 2(x^2 -4x +4) +6 -8

Rewrite in vertex form.

f(x) = 2(x -2)^2 -2

The positive leading coefficient tells you the vertex is a minimum. Comparing this to the vertex form for a quadratic with vertex (h, k)

f(x) = a(x -h)^2 +k

we find the vertex to be ...

(h, k) = (2, -2)

The function has a minimum at (2, -2).