Let f (x) = x2+4x+12 . What is the vertex form of f (x) ? What is the minimum value of f (x) ?

The answer is y = (x-2) 2 - 16 but the explanation is

The standard form of a quadratic function is y=ax2+bx+c the equation here y = x2 - 4x-12 is of this form by comparison : a = 1, b = - 4 and c = - 12The vertex form of the quadratic function is y=a (x-h) 2 + k

where (h, k) are the coords of the vertex. the x-coord of the vertex = - b 2a = - - 4 2 = 2

substitute x = 2 into original function for y-coord. y = (2) 2 - 4 (2) - 12=4-8-12=-16

hence (h, k) = (2, - 16) and a = 1

⇒y = (x-2) 2 - 16