Question

Find the standard form of the equation of the parabola with a vertex at the origin and a focus at (0, -4).

Answer 1

y = -1/4 x^2
y^2 = -4x
y^2 = -16x
y = -1/16 x^2

Answer 2

Just a second...

Answer 3

no problem

Answer 4

If given two points, the focus \((x_1, y_1)\) and the directrix \((x_2, y_2)\), you can insert them into the following formula:

\((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\) then simplify afterwards to get the standard form of the parabola.

Answer 5

ok so what am i plugging in?

Answer 6

Oh, wrong question. You have the vertex and the focus, but not the directrix.

Answer 7

yea thats it

Answer 8

There's a process to find the directrix.

Answer 9

thats where im lost

Answer 10

Did you know that any point on the parabola is equidistant from the focus and the directrix?

Answer 11

This means that we can easily find the directrix since we have the vertex.

Answer 12

so how?

Answer 13

Well, basically, we know that the focus is (0,-4) and the vertex is (0,0). Since the focus is 4 units below the vertex, that means the directrix is four units above it.

Answer 14

We also know that the directrix is a horizontal line. In this case it will be y = 4. When we express the directrix as a point, it becomes \((x, 4)\)

Answer 15

oh ok i understand. so then from there i would plug it in to the equation you gave earlier?

Answer 16

So now we have the focus (0,-4) and the directrix (x,4). Plug those points into the formula above to find the standard form of the equation of the parabola. Yes.

Answer 17

do i need to solve for the missing x in the directrix

Answer 18

You don't. You insert the x into the formula in place of \(x_2\)

Answer 19

oh ok

Answer 20

At this point, it's probably a good idea to show the work you've done so far that way I can make sure you've done this step correctly.

Answer 21

how would i do that? im on the computer and im doing my work on paper...

Answer 22

Use the draw button or use \(LaTeX\)

Answer 23

ok ill have to draw it because my computer doesn't support latex

Answer 24

You can still type the LaTeX. It won't stop it from showing up on my end.

Answer 25

just give me a couple minutes

Answer 26

There's a draw button you can click BTW.

Answer 27

i know im just in the middle of doing the work