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.

Answer 1

minimum at (2, - 2 )

Explanation:

The equation of a parabola in vertex form is

f(x) = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

If a > 0 then the vertex is a minimum

If a < 0 then the vertex is a maximum

Given

f(x) = 2x² - 8x + 6 ← factor out 2 from the first 2 terms

= 2(x² - 4x) + 6

To complete the square

add/ subtract ( half the coefficient of the x- term)² to x² - 4x

f(x) = 2(x² + 2(- 2)x + 4 - 4) + 6

= 2(x - 2)² - 8 + 6

= 2(x - 2)² - 2 ← in vertex form

with vertex = (2, - 2) and a > 0

Thus minimum at (2, - 2 )