To find the equation of a parabola opening upward with a given focus and directrix, we will follow these steps:
1. Identify the focus and directrix:
- The focus is at [tex]\((9, 27)\)[/tex].
- The directrix is the line [tex]\(y = 11\)[/tex].
2. Find the vertex of the parabola:
- For a parabola opening upward, the vertex (h, k) is exactly halfway between the focus and the directrix.
- The x-coordinate of the vertex remains the same as the x-coordinate of the focus: [tex]\(h = 9\)[/tex].
- The y-coordinate of the vertex is the midpoint between the y-coordinate of the focus and the directrix:
[tex]\[
k = \frac{\text{focus\_y} + \text{directrix\_y}}{2} = \frac{27 + 11}{2} = 19.
\][/tex]
- Thus, the vertex is [tex]\((9, 19)\)[/tex].
3. Calculate the distance [tex]\(p\)[/tex] from the vertex to the focus (or the vertex to the directrix):
- The distance [tex]\(p\)[/tex] is the difference in the y-coordinates of the focus and the vertex:
[tex]\[
p = 27 - 19 = 8.
\][/tex]
4. Write the equation of the parabola:
- For a parabola opening upward, the standard form is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
- Substituting [tex]\(h = 9\)[/tex], [tex]\(k = 19\)[/tex], and [tex]\(p = 8\)[/tex]:
[tex]\[
(x - 9)^2 = 4 \times 8 \times (y - 19)
\][/tex]
- This simplifies to:
[tex]\[
(x - 9)^2 = 32(y - 19).
\][/tex]
5. Convert to function form:
- Solve for [tex]\(y\)[/tex] to obtain:
[tex]\[
y = \frac{1}{32}(x - 9)^2 + 19.
\][/tex]
- Therefore, the equation in function form is:
[tex]\[
f(x) = \frac{1}{32}(x - 9)^2 + 19.
\][/tex]
Comparing this result with the given multiple-choice options, the correct answer is:
- D. [tex]\(f(x)=\frac{1}{32}(x-9)^2+19\)[/tex]
