Question

find the equation of the parabola with the vertex at (0,0) and the focus at (-1,0)

Answer 1

Answer:

Step-by-step explanation:

You have to know some basics of parabolas to get this far in your math class, so hopefully you do, and what I'm about to do and say is not new or surprising to you.

If you plot the center (0, 0) which is your h and k, and then plot the focus, you should know that, since the parabola wraps itself around the focus, this parabola opens to the left. There is a work form and a standard form for a left-opening parabola. The work form will be

$x=-a(y-k)^2+h$ and the standard form will be

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

We will figure out what we need and will fill in both of these forms.

In general, the value of p is a distance; namely, it is the distance from the center to either the focus or the directrix (which is the same for both since the focus and the directrix are the same distance from the center, one on either side of it). From this we can determine that, since the focus is one unit away from the center, p = 1. Now that we know p we can find |a|:

$|a|=\frac{1}{4p}$

Since p = 1, then $|a|=\frac{1}{4}$

Now we can write the work form:

$x=-\frac{1}{4}(y-0)^2+0$ which of course simplifies to

$x=-\frac{1}{4}y^2$

And remember we have the p value for the standard form:

$-(y-0)^2=4(1)(x-0)$ which simplifies a bit to

$-y^2=4x$ and when we solve for x:

$x=-\frac{1}{4}y^2$ which is the exact same as the work form!!! Look at that! : )