Question

Find the standard form of the equation of the parabola with a focus at (0, -2) and a directrix at y = 2

Answer 1

If the focus is at (0,-2) and the directrix is the line y = 2, then the vertex is right between them, at (0,0). Because the directrix is above the focus the graph opens downward, so it is a y = x^2 parabola.

Answer 2

mhmm

Answer 3

Thank you

Answer 4

but

Answer 5

So then, so far we have \[x ^{2}=4py\]because the origin is the vertex. Now we need to find p.

Answer 6

p is the number of units the focus is away from the vertex. I graphed the points you provided and it appears that the focus is down 2 units from the vertex, so p = -2.

Answer 7

Let me correct myself, quick-like...this opens downward, so the equation is this:\[x ^{2}=-4py\]i failed to put the - sign above.

Answer 8

-2 still right :)

Answer 9

So anyways, now we know that p = -2 (this is why the parabola opens downward, cuz of the negative), so filling in our equation with what we know...

Answer 10

(x-h)^2=4p(y-k)

Answer 11

\[x ^{2}=4(-2)y\]or\[x ^{2}=-8y\]

Answer 12

That's it.

Answer 13

TY for the medal.

Answer 14

:) thats not an answer choice @IMStuck

Answer 15

y2 = -2x

y2 = -8x

y equals negative 1 divided by 8 x squared

y equals negative 1 divided by 2 x squared

Answer 16

It would be equivalent to
\(y=-\frac{x^2}{8} \)

Answer 17

ahh because you isolate the y

Answer 18

Wish I could give you a medal thanks you

Answer 19

No, that's perfectly alright.
You're most welcome! :)

Answer 20

I medalled mathmate for you. You use my answer and solve it for y. If I would have known the form your answers were in, I could have done that part, too! Sorry!

Answer 21

You can always do it from the definition
\[
x^2 + (2 + y)^2 = (y - 2)^2\\
x^2 =-8 y\\
y=-\frac x 8
\]
Since the parabola is the set of points equidistant from the focus and the directrix.

Answer 22

How would I find the vertex of this qeuation?

Answer 23

(0,0)

Answer 24

To find it what steps would I use

Answer 25

Is it i between the focus and directrix/?

Answer 26

Yes, it is midpoint between the two

Answer 27

Thank you that clarifies a lot,

Answer 28

YW

Answer 29

So if I have a focus at (-8, 0) and a directrix at x = 8 then te midpoint is (0,0) and it's a horizontal parabola opening the the left correct?

Answer 30

Yes

Answer 31

Thank you I fully understand it now. I found the answer to the next problem :D

Answer 32

Great