Question

A parabola can be drawn given a focus of (-4, -7)(−4,−7) and a directrix of y=-3y=−3. What can be said about the parabola?

The parabola has a vertex at ( , ), has a p-value of
and it
-opens left
-opens right
-opens up
-opens down

Answer 1

Vertex = (-4,-5)

P-value = -2

Opens Downward

Explanation:

Given:

• Focus = (-4,-7)
• Directrix = -3

Since focus is less than directrix, the parabola obviously opens downward.

To find vertex (h,k), for downward parabola, focus is (h, k + p) and directrix is y = k - p

We have:

`\displaystyle \large{k+p=-7 \to (1)}\\\displaystyle \large{k-p=-3 \to (2)}`

First equation being focus and second being directrix, solve the simultaneous equation:

`\displaystyle \large{2k=-10}\\\displaystyle \large{k=-5}`

Substitute k = -5 in any equation - I’ll choose (1) for this:

`\displaystyle \large{-5+p=-7}\\\displaystyle \large{p=-2}`

Therefore vertex is at (h,k) = (-4,-5) with p-value being -2 since p < 0 then the parabola opens downward.

Attachment added for visual reference