Step by step
The question
Question: find the focus and the directrix of (x + 2)² = 12(y + 3).
1 Find p
Compare with the parabola rule: x² = 4py.
So 4p is the number in front of the y bracket: 4p = 12.
Careful!
12 is 4p, not p.
We still divide by 4.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just p is left.
12 divided by 4 is 3.
So p = 3.
2 Focus and directrix
The vertex comes from the brackets.
The sign turns: x + 2 gives −2, y + 3 gives −3.
So the vertex is (−2, −3).
p is positive.
So the parabola opens up.
The focus is 3 above the vertex, inside the curve: (−2, 0).
The directrix is the flat line 3 on the other side of the vertex: y = −6.
So the focus is (−2, 0) and the directrix is y = −6.
3 Draw it
Draw the grid.
The vertex (−2, −3),
the focus (−2, 0),
and the directrix y = −6, dashed.
The parabola opens up, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (4, 0).
Is it on the parabola?
Put it in: 36 and 36.
Yes!
Its distance to the focus: 6.
Its distance to the directrix: 6 too.
The same distance.
That is what makes a parabola.
Right!
Let's look back.
First: find p.
Then: focus and directrix.
Then: draw it.
Last: check.