To find the equation of the parabola with the given vertex [tex]\((-6, 2)\)[/tex] and focus [tex]\((-7.25, 2)\)[/tex], we need to follow a systematic approach.
First, let's recall that the general equation for a parabola with a horizontal axis of symmetry (vertex form) is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
where:
- [tex]\((h, k)\)[/tex] is the vertex of the parabola,
- [tex]\(p\)[/tex] is the distance from the vertex to the focus.
### Step-by-Step Solution:
1. Identify the vertex and the focus:
The vertex of the parabola is [tex]\((-6, 2)\)[/tex].
The focus of the parabola is [tex]\((-7.25, 2)\)[/tex].
2. Determine the distance [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] is the horizontal distance between the vertex and the focus. Since the focus is to the left of the vertex:
[tex]\[
p = \text{distance from } (-6, 2) \text{ to } (-7.25, 2) = |-7.25 - (-6)| = |-7.25 + 6| = |-1.25| = 1.25
\][/tex]
However, since the focus is to the left of the vertex, [tex]\(p\)[/tex] is negative:
[tex]\[
p = -1.25
\][/tex]
3. Substitute the values into the standard form equation:
We know [tex]\( (h, k) = (-6, 2) \)[/tex] and [tex]\(p = -1.25\)[/tex]. Substitute these values into the formula:
[tex]\[
(y - 2)^2 = 4(-1.25)(x + 6)
\][/tex]
4. Simplify the expression:
[tex]\[
(y - 2)^2 = 4 \times -1.25 \times (x + 6)
\][/tex]
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
Therefore, the correct equation for the parabola with vertex [tex]\((-6, 2)\)[/tex] and focus [tex]\((-7.25, 2)\)[/tex] is:
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
Thus, the correct choice is:
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
