Step by step
The question
Question: find the focus and the directrix of y = (1/8)x².
1 Write it as x² = 4py
Parabolas in the books look like x² = 4py.
Our x² has 1/8 in front.
To remove it, we multiply both sides by 8.
We do it on both sides, so they stay equal.
8 times 1/8 is 1.
So 8y = x².
Swap the sides: x² = 8y.
2 Find p
So 4p is the number in front of y: 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.
3 Focus and directrix
There are no brackets: just x² and y.
So the vertex is (0, 0).
p is positive.
So the parabola opens up.
The focus is 2 above the vertex, inside the curve: (0, 2).
The directrix is the flat line 2 on the other side of the vertex: y = −2.
So the focus is (0, 2) and the directrix is y = −2.
4 Draw it
Draw the grid.
The vertex (0, 0),
the focus (0, 2),
and the directrix y = −2, dashed.
The parabola opens up, around the focus.
5 Check
Let's check with a point on the parabola, level with the focus: (4, 2).
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: write it as x² = 4py.
Then: find p.
Then: focus and directrix.
Then: draw it.
Last: check.