Question

write the equation in vertex form for the parabola with focus (0, -10) and directrix y = -6. simplify any fractions.

Explanation:

Step1: Find vertex (h,k)

The vertex is the midpoint between focus $(0,-10)$ and directrix $y=-6$.
$h = 0$, $k = \frac{-10 + (-6)}{2} = -8$

Step2: Find p value

$p$ is the distance from vertex to focus:
$p = -10 - (-8) = -2$

Step3: Plug into vertex form

Vertex form for vertical parabola: $(x-h)^2 = 4p(y-k)$
Substitute $h=0$, $k=-8$, $p=-2$:
$(x-0)^2 = 4(-2)(y - (-8))$

Step4: Rearrange to solve for y

$x^2 = -8(y + 8)$
$y + 8 = \frac{x^2}{-8}$
$y = -\frac{1}{8}x^2 - 8$

Answer:

$y = -\frac{1}{8}x^2 - 8$