which values for h and k are used to write the function f(x) = x² + 12x + 6 in vertex form?\no h = 6, k =…

Explanation:

Step1: Recall vertex - form of a quadratic function

The vertex - form of a quadratic function is (y=a(x - h)^2+k), and for the quadratic function (y = ax^{2}+bx + c), we can complete the square. Given (f(x)=x^{2}+12x + 6), where (a = 1), (b = 12), (c = 6).

Step2: Complete the square for the (x) - terms

[ \begin{align*} x^{2}+12x+6&=(x^{2}+12x)+6\ \end{align*} ] Take half of the coefficient of (x), square it and add and subtract it inside the parenthesis. The coefficient of (x) is (12), half of it is (\frac{12}{2}=6), and its square is (6^{2}=36). So (x^{2}+12x + 6=(x^{2}+12x + 36-36)+6).

Step3: Rewrite the expression as a perfect - square trinomial

((x^{2}+12x + 36-36)+6=(x + 6)^{2}-36 + 6=(x+6)^{2}-30). Comparing ((x + 6)^{2}-30) with (a(x - h)^{2}+k) (here (a = 1)), we have (h=-6) and (k=-30).

Answer:

D. (h = - 6,k=-30)