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

- anonymous

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- anonymous

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

- Hero

Just a second...

- anonymous

no problem

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## More answers

- Hero

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.

- anonymous

ok so what am i plugging in?

- Hero

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

- anonymous

yea thats it

- Hero

There's a process to find the directrix.

- anonymous

thats where im lost

- Hero

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

- Hero

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

- anonymous

so how?

- Hero

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.

- Hero

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 \((x, 4)\)

- anonymous

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

- Hero

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.

- anonymous

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

- Hero

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

- anonymous

oh ok

- Hero

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.

- anonymous

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

- Hero

Use the draw button or use \(LaTeX\)

- anonymous

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

- Hero

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

- anonymous

just give me a couple minutes

- Hero

There's a draw button you can click BTW.

- anonymous

i know im just in the middle of doing the work

- Hero

|dw:1439499465843:dw|

- Hero

You should post what you've done now, that way you don't get too far ahead because one wrong mistake and then you'll have so much re-work to do.

- anonymous

ok ill start off with showing you what i plugged in:
(x-0)^2 + (y-(-4))^2 = (x-x)^2 + (y-4)^2

- Hero

Looks good so far.

- anonymous

x^2 + y^2 + 8y + 16 = y^2 - 8y +16

- Hero

Yes you can do it that way, but there's a way to do it that avoids expanding.

- Hero

What do you get for your final simplified result?

- anonymous

give me a sec

- Hero

Once you get the correct result, I'll show you the other way to do it.

- anonymous

y=x^2/16

- anonymous

my answer would be D

- Hero

D is correct, but you forgot the negative in your answer.

- anonymous

yea i forgot to type it

- Hero

So \(LaTeX\) does not load for you?

- anonymous

no.... :(

- Hero

If not, I can do it by hand and upload it that way.

- anonymous

sure

- Hero

Hang on, I'm about to upload it

- anonymous

ok

- Hero

|dw:1439501181613:dw|

- anonymous

ok i see. i understand it. thanks for all the help.

- Hero

You're most welcome :) You're a great student.

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