Step by step
The question
Question: find the focus and the directrix of (x − 2)² = 8(y + 1).
1 Find p
Compare with the parabola rule: x² = 4py.
So 4p is the number in front of the y bracket: 4p = 8.
Careful!
8 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.
8 divided by 4 is 2.
So p = 2.
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 up.
The focus is 2 above the vertex, inside the curve: (2, 1).
The directrix is the flat line 2 on the other side of the vertex: y = −3.
So the focus is (2, 1) and the directrix is y = −3.
3 Draw it
Draw the grid.
The vertex (2, −1),
the focus (2, 1),
and the directrix y = −3, dashed.
The parabola opens up, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (6, 1).
Is it on the parabola?
Put it in: 16 and 16.
Yes!
Its distance to the focus: 4.
Its distance to the directrix: 4 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.