To determine which graph represents the parabolic equation [tex]\((x-2)^2 = 12(y-3)\)[/tex], we need to understand the standard form of a parabola and its implications.
The given equation [tex]\((x-2)^2 = 12(y-3)\)[/tex] is in the form:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(4p\)[/tex] gives information on the distance between the vertex and the focus or the directrix.
### Step-by-Step Solution:
1. Identify the vertex:
The equation [tex]\((x-2)^2 = 12(y-3)\)[/tex] indicates that the vertex [tex]\((h, k)\)[/tex] of the parabola is [tex]\((2, 3)\)[/tex].
2. Determine the orientation:
Since the squared term is on the [tex]\(x\)[/tex]-side of the equation [tex]\((x-2)^2\)[/tex], this parabola opens vertically. Specifically, if the squared term is on the left side of the equation, and [tex]\(y\)[/tex] is on the right, it opens either up or down.
3. Calculate [tex]\(p\)[/tex]:
In the form [tex]\((x - h)^2 = 4p(y - k)\)[/tex], we can see that [tex]\(4p = 12\)[/tex]. Solving for [tex]\(p\)[/tex]:
[tex]\[
4p = 12 \implies p = \frac{12}{4} = 3
\][/tex]
Since [tex]\(p\)[/tex] is positive, the parabola opens upwards.
4. Focus and Directrix:
To further confirm, the focus will be [tex]\((h, k + p)\)[/tex] which translates to [tex]\((2, 3 + 3) = (2, 6)\)[/tex]. The directrix will be [tex]\(y = k - p\)[/tex] which is [tex]\(3 - 3 = 0\)[/tex].
5. Sketch and match:
With the vertex at [tex]\((2, 3)\)[/tex] and the parabola opening upward with a relatively steep curvature (since [tex]\(12\)[/tex] is quite a large coefficient), we can sketch or match the parabola to one of the given options.
Given these characteristics, you should look for the graph that fits these parameters:
- Vertex at [tex]\((2, 3)\)[/tex]
- Opens upwards
- Wider gap due to a larger vertical stretch factor for a parabola (since [tex]\(12\)[/tex] makes the parabola wider).
### Conclusion:
Using this understanding, you would examine each of the graphs provided (A, B, C, D) and select the one that fits the criteria. Be sure to check each graph against the points and orientation described. This should lead you to identify the correct graph.
