Given the equation (x + 4)^2 = -12y + 24 determine:

a) The Vertex

b) If the parabola opens up, down, left, right

c) The Focus

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`(x+4)^2=-12y+24`

(a) Vertex

To determine the vertex, express the given equation of the parabola to its vertex form which is:

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

where (h,k) is the vertex.

To convert the equation in this form, subtract both sides by 24.

`(x+4)^2-24=-12y+24-24`

`(x+4)^2-24=-12y`

And divide both sides by -12.

`(x+4)^2-24)/(-12)=(-12y)/(-12)`

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

So the vertex form of the given equation is:

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

**Hence, the vertex of the parabola is `(-4,2)` .**

(b) Direction of the parabola.

Note that when it is the variable x that is squared, the parabola either opens up or down.

And if it is the variable y that is squared, the parabola either opens to the left or right.

In the given equation, it is the variable x that is squared. So, it is either upward or downward. To determine which of these two is the direction of the parabola, consider the value of a.

If a is positive, the parabola opens up. And if it is negative, it open down.

Base on the vertex form, `a=-1/12` .

**Since a is negative, therefore, the parabola opens down.**

(C) Focus

Since the parabola is downward, then its axis of symmetry is vertical. And a parabola with vertical axis, its focus has a coordinates (h, k+p).

To solve for p, use the formula:

`4p=1/a`

Since a=-1/12, the value of p will be:

`4p=1/(-1/12)`

`4p=-12`

`p=-3`

Then, substitute h=-4, k=2 and p=-3 to the coordinate form of the focus.

`(h,k+p)=(-4,2+(-3))=(-4,-1)`

**Hence, the focus of the parabola is `(-4 , -1)` .**

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