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What is the directrix? How can we use this to help us derive the parabola?
Honestly I have no clue, Can you give me like a quick run down on how to do any of this?
The directrix is a special line to the parabola. Consider some point on the parabola, then the distance from the diretrix to this point and the distance from the focus to this point are the same. Mathematically, we can write this as \[ (x-h)^2 + (y-k)^2 = (y-h_1)^2\], where (h,k) is the focus of the parabola and y = h_1 is the equation of the directrix. In this case, h = 0, k = 1, and h_1 = -1.
Solve for y in this equation and you will get the equation of your parabola.
you can answer this in like two seconds if you know two things first, what does this look like?
a simple plug in problem.
there is a crude picture with a focus and directrix labelled do you know what the parabola looks like?
I'm guessing it would look vertical??
copy my picture and draw it
|dw:1443317495455:dw|(like I said I have no clue)
yeah i can see that
the vertex is half way between \((0,1)\) and \(y=-1\) like this
the point half way between \((0,1)\) and \(y=-1\) is \((0,0)\) so the vertex is at the origin, and the parabola opens up
that means it looks like \[4py=x^2\] where \(p\) is the distance between the vertex and the focus, or the vertex and the directrix in this case \(p=1\)
your answer is therefore \[4y=x^2\]
Didn't I need to solve for Y or something like that?
whatevs thanks guys!