Step by step
The question
Question: find the focus and the directrix of (y − 1)² = 12(x + 2).
1 Find p
Compare with the parabola rule: y² = 4px.
So 4p is the number in front of the x 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 − 1 gives 1.
So the vertex is (−2, 1).
p is positive.
So the parabola opens right.
The focus is 3 to the right of the vertex, inside the curve: (1, 1).
The directrix is the upright line 3 on the other side of the vertex: x = −5.
So the focus is (1, 1) and the directrix is x = −5.
3 Draw it
Draw the grid.
The vertex (−2, 1),
the focus (1, 1),
and the directrix x = −5, dashed.
The parabola opens right, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (1, 7).
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.