To determine the equation of the parabola with vertex [tex]\((-6, 2)\)[/tex] and focus [tex]\((-7.25, 2)\)[/tex], follow these steps:
1. Identify the type of parabola:
- The vertex is [tex]\((-6, 2)\)[/tex] and the focus is [tex]\((-7.25, 2)\)[/tex].
- Both the vertex and focus have the same [tex]\(y\)[/tex]-coordinate ([tex]\(2\)[/tex]), indicating that the parabola is horizontal (it opens either to the left or right).
2. Calculate the value of [tex]\(p\)[/tex]:
- The distance [tex]\(p\)[/tex] between the vertex and focus determines the direction and width of the parabola.
- [tex]\(p\)[/tex] is the horizontal distance between [tex]\((-6, 2)\)[/tex] and [tex]\((-7.25, 2)\)[/tex]:
[tex]\[
p = -|(-7.25 - (-6))| = -(7.25 - 6) = -1.25
\][/tex]
Here, [tex]\(p\)[/tex] is negative because the focus is to the left of the vertex, indicating that the parabola opens to the left.
3. Write the standard form of the parabola’s equation:
- For a horizontal parabola that opens to the left (or right), the standard form is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the distance from the vertex to the focus.
4. Substitute the vertex [tex]\((-6, 2)\)[/tex] and [tex]\(p = -1.25\)[/tex] into the equation:
[tex]\[
(y - 2)^2 = 4(-1.25)(x + 6)
\][/tex]
Simplifying this:
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
Thus, the equation for the graph with vertex [tex]\((-6, 2)\)[/tex] and focus [tex]\((-7.25, 2)\)[/tex] is:
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
So, the correct answer is:
[tex]\[
(y - 2)^2 = -5(x + 6)
\][/tex]
