Question

The equation of a parabola is `12y=(x-1)^2-48` identify the vertex, focus, and directrix of the parabola?

I just really do not understand this. Please help and explain! Thank you!

Answer 1

The vertex form of the equation of a parabola that opens up or down is:

`y = a(x-h)^2+k`

In this form, we can easily identify the vertex as the point `(h, k)`. Using the formula, `f = 1/(4a)`, allows us to determine the focus:

`(h,k+f)`

and the equation of the directrix:

`y = k-f`

I shall apply this general information to your problem.

Given: `12y=(x-1)^2-48`

We can make the given equation match the vertex form by dividing both sides of the equation by 12:

`y=1/12(x-1)^2-4`

We can easily identify the vertex as `(h, k) = (1,-4)`

We observe that `a = 1/12` and we use the formula to compute `f`:

`f=1/(4a)`

`f = 1/(4(1/12)`

`f = 3`

We can determine the focus:

`(h, k+f) = (1,-4+3)`

`(h, k+f) = (1, -1)`

We can write the equation of the directrix:

`y = k-f`

`y = -4-3`

`y = -7`