write the equation in vertex form for the parabola with vertex (0, -7) and directrix y = -5. simplify any fractions.

Question

Accepted Answer

# Explanation:
## Step1: Recall the vertex form of a parabola
The vertex form of a parabola with vertex $(h,k)$ is $(x - h)^2=4p(y - k)$, where $(h,k)$ is the vertex and $p$ is the distance between the vertex and the focus (or the vertex and the directrix).
Given vertex $(h,k)=(0,-7)$, so the equation becomes $x^{2}=4p(y + 7)$.
## Step2: Find the value of $p$
The directrix of a parabola $x^{2}=4p(y - k)$ is $y=k - p$.
We know $k=-7$ and the directrix $y=-5$.
Substitute into $y=k - p$: $-5=-7 - p$.
Solve for $p$:
$$
\begin{align*}
-5&=-7 - p\
p&=-7 + 5\
p&=-2
\end{align*}
$$
## Step3: Substitute $p$ into the vertex - form equation
Substitute $p = - 2$ into $x^{2}=4p(y + 7)$
$$
\begin{align*}
x^{2}&=4\times(-2)(y + 7)\
x^{2}&=-8(y + 7)
\end{align*}
$$

# Answer:
$x^{2}=-8(y + 7)$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the vertex form of a parabola

Step2: Find the value of \(p\)

Step3: Substitute \(p\) into the vertex - form equation

Answer: