Question

question 25 (1 point)
(02 08 mc)
which of the following is the equation for a parabola with a focus at (0, -8) and a directrix at y = 8?
a $f(x)=\frac{1}{32}x^{2}$
b $f(x)=\frac{1}{8}x^{2}$
c $f(x)=-\frac{1}{32}x^{2}$
d $f(x)=-\frac{1}{8}x^{2}$

Explanation:

Step1: Recall parabola formula

The standard - form equation of a parabola with its vertex at the origin and opening either up or down is $x^{2}=4py$, where $(0,p)$ is the focus and $y = - p$ is the directrix.

Step2: Determine the value of $p$

Given the focus is $(0,-8)$ and the directrix is $y = 8$. The value of $p$ is the distance from the vertex (origin in this case) to the focus. Here, $p=-8$.

Step3: Write the equation of the parabola

Substitute $p = - 8$ into the equation $x^{2}=4py$. We get $x^{2}=4\times(-8)y=-32y$. Then, solve for $y$: $y =-\frac{1}{32}x^{2}$.

Answer:

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