To determine which graph represents the given parabolic equation [tex]\((x - 2)^2 = 12(y - 3)\)[/tex], we need to follow these steps:
1. Identify the Standard Form: The given equation is written in the form [tex]\((x - h)^2 = 4a(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(a\)[/tex] determines the direction and the width of the parabola.
2. Determine the Vertex:
- By comparing [tex]\((x - 2)^2 = 12(y - 3)\)[/tex] with the standard form [tex]\((x - h)^2 = 4a(y - k)\)[/tex], we can identify that [tex]\(h = 2\)[/tex] and [tex]\(k = 3\)[/tex].
- Therefore, the vertex of the parabola is at the point [tex]\((2, 3)\)[/tex].
3. Determine the Orientation:
- The equation [tex]\((x - 2)^2 = 12(y - 3)\)[/tex] suggests that the parabola opens either upward or downward. This is because the [tex]\(x\)[/tex]-term is squared and it matches the standard form of a vertical parabola [tex]\((x - h)^2 = 4a(y - k)\)[/tex].
- The coefficient on the right side of the equation is positive (i.e., 12), indicating that the parabola opens upward.
4. Find the Value of [tex]\(a\)[/tex]:
- From [tex]\((x - 2)^2 = 12(y - 3)\)[/tex], we have [tex]\(4a = 12\)[/tex].
- Solving for [tex]\(a\)[/tex], we get [tex]\(a = 3\)[/tex]. This value of [tex]\(a\)[/tex] affects the "width" or "stretch" of the parabola, but doesn't impact the shape in a way we need to consider further for identifying the correct graph.
5. Interpret the Characteristics to the Graph:
- We need to look for a graph with a vertex at [tex]\((2, 3)\)[/tex] and a parabola opening upward.
Given these characteristics, the correct graph that represents the parabolic equation [tex]\((x - 2)^2 = 12(y - 3)\)[/tex] is a graph where the vertex is at (2, 3) and the parabola opens upward.
Thus, the correct answer is:
B. [tex]$B$[/tex]
