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

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

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Answer 1 · 2018-05-17T03:22:11

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:

$y = k-f$

$y = -4-3$

$y = -7$