Question

select the correct answer. what is the equation of the parabola opening upward with a focus at (9,27) and a directrix of y = 11? a. f(x) = (1/16)(x - 9)² + 19 b. f(x) = (1/16)(x + 9)² - 19 c. f(x) = (1/32)(x + 9)² - 19 d. f(x) = (1/32)(x - 9)² + 19

Explanation:

Step1: Find vertex $(h,k)$

The vertex is midway between focus $(9,27)$ and directrix $y=11$.
$h=9$, $k=\frac{27+11}{2}=19$

Step2: Calculate $p$ (distance vertex to focus)

$p=27-19=8$

Step3: Write parabola equation

For upward-opening parabola: $f(x)=\frac{1}{4p}(x-h)^2+k$
Substitute $p=8, h=9, k=19$:
$f(x)=\frac{1}{4\times8}(x-9)^2+19=\frac{1}{32}(x-9)^2+19$

Answer:

D. $f(x) = \frac{1}{32}(x - 9)^2 + 19$