`y=-7` Write the standard form of the equation of the parabola with the given directrix and vertex at (0,0)

Expert Answers
marizi eNotes educator| Certified Educator

A parabola with directrix at `y=k` implies that the parabola may open up towards upward or downward direction.

The position of the directrix with respect to the vertex point can be used to determine in which side the parabola opens up. 

If the directrix is above the vertex point then the parabola opens up in downward direction.

If the directrix is below the vertex point then the parabola opens up in upward direction.

The parabola indicated in the problem has directrix of `y=-7` which is located below the vertex `(0,0)` .

Thus, the parabola opens up in upward direction and follows the standard formula: `(x-h)^2=4p(y-k)` . We consider the following properties:

vertex as `(h,k)`

focus as `(h, k+p) `

 directrix as `y=k-p`

Note: `p` is the distance of between focus and vertex or distance between directrix and vertex.

From the given vertex point `(0,0)` , we determine `h =0 ` and `k=0` .

Applying directrix `y =-7` and `k=0 ` on `y=k-p` , we get:

`-7= 0-p`

`-7=-p `

`-1*-7=-1*-p `

`7=p or p=7`

Plug-in the values: `h=0` ,`k=0` , and `p=7` on the standard formula, we get:

`(x-0)^2=4*7(y-0).`

`x^2=28y `  as the standard form of the equation of the parabola with vertex `(0,0) ` and directrix `y=-7` .