To determine the vertex of the parabola defined by the equation [tex]\((x-2)^2 = -12(y-2)\)[/tex], we need to compare this equation to the standard form of a parabola equation:
For a vertical parabola, the standard form is:
[tex]\[
(x-h)^2 = 4a(y-k)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's rewrite our given equation in a form similar to the standard one:
[tex]\[
(x-2)^2 = -12(y-2)
\][/tex]
By comparing [tex]\((x-2)^2 = -12(y-2)\)[/tex] with the standard form [tex]\((x-h)^2 = 4a(y-k)\)[/tex], we can directly see that the terms inside the parentheses indicate the vertex.
Here, [tex]\((x-2)\)[/tex] corresponds to [tex]\((x-h)\)[/tex], and [tex]\((y-2)\)[/tex] corresponds to [tex]\((y-k)\)[/tex]. This tells us that [tex]\(h = 2\)[/tex] and [tex]\(k = 2\)[/tex].
Therefore, the vertex [tex]\((h, k)\)[/tex] of the parabola is:
[tex]\[
(h, k) = (2, 2)
\][/tex]
So the answer is:
B. [tex]\((2, 2)\)[/tex]
