To find the vertex of the parabola defined by the equation [tex]\((x - 2)^2 = -12(y - 2)\)[/tex], we need to compare it with the standard form of a parabola that opens vertically. The standard form for a vertically oriented parabola is [tex]\((x - h)^2 = 4p(y - k)\)[/tex].
In this form:
- [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
- The sign and value of [tex]\(4p\)[/tex] determine the direction and "width" of the parabola. If [tex]\(p\)[/tex] is positive, the parabola opens upward; if [tex]\(p\)[/tex] is negative, it opens downward.
Now let's rewrite and compare the given equation step-by-step:
1. Given Equation: [tex]\((x - 2)^2 = -12(y - 2)\)[/tex]
2. Standard Form of Vertical Parabola: [tex]\((x - h)^2 = 4p(y - k)\)[/tex]
By comparing the given equation [tex]\((x - 2)^2 = -12(y - 2)\)[/tex] with the standard form [tex]\((x - h)^2 = 4p(y - k)\)[/tex]:
- We identify that [tex]\(h = 2\)[/tex] and [tex]\(k = 2\)[/tex].
Thus, the vertex [tex]\((h, k)\)[/tex] of the parabola is at the point [tex]\((2, 2)\)[/tex].
The correct answer is:
B. [tex]\((2, 2)\)[/tex]
