Question

Finding the standard form of an equation of a parabola given the focus and directrix?

Answer 1

Decide whether the parabola opens up or down, or to the left or to the right.

Note that the vertex is always halfway between the directrix and the focus.

You might want to graph this situation and then find the distance between vertex and focus by inspection. That distance is denoted "p"

Hope this helps!

Answer 2

That doesn't particularly help. I'm having trouble finding P and (h,k).

Answer 3

You can actually find the parabola given a foci \(F( \alpha, \beta)\) and a directrix \(d)x=k \) Where \[k, \alpha, \beta \in \mathbb{R} \]
by using the definition of parabola which is:
"The parabola is the geometric body composed by all the points equidistant from a fixed point called foci and a line called directrix".
Of course, we can use this definition with the equations for the distance between points and from point to line:
\[\sqrt{(x-\alpha)+(y-\beta)}=\frac{ \left| x_o-k \right| }{ \sqrt{a^2+b^2} }\]

Answer 4

Words to the wise: Don't respond to others' efforts to help with "Thant doesn't help." Motivation goes down the drain if you do that.

Just what do you mean by P? In the context of parabolas, it's usually p, not P.

I stick by my previous advice. Sketch this situation. Locate the focus and directrix on your graph. Locate the point halfway between the focus and the directrix; that's your vertex.