Step by step
The question
Question: find the focus and the directrix of y² = −8x.
1 Find p
Compare with the parabola rule: y² = 4px.
So 4p is the number in front of x: 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
There are no brackets: just y² and x.
So the vertex is (0, 0).
p is negative.
So the parabola opens left.
The focus is 2 to the left of the vertex, inside the curve: (−2, 0).
The directrix is the upright line 2 on the other side of the vertex: x = 2.
So the focus is (−2, 0) and the directrix is x = 2.
3 Draw it
Draw the grid.
The vertex (0, 0),
the focus (−2, 0),
and the directrix x = 2, dashed.
The parabola opens left, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (−2, −4).
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.