Step by step
The question
Question: find the focus and the directrix of y = −(1/12)x².
1 Write it as x² = 4py
Parabolas in the books look like x² = 4py.
Our x² has 1/12 in front.
To remove it, we multiply both sides by −12.
We do it on both sides, so they stay equal.
−12 times minus 1/12 is 1.
So −12y = x².
Swap the sides: x² = −12y.
2 Find p
So 4p is the number in front of y: 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.
3 Focus and directrix
There are no brackets: just x² and y.
So the vertex is (0, 0).
p is negative.
So the parabola opens down.
The focus is 3 below the vertex, inside the curve: (0, −3).
The directrix is the flat line 3 on the other side of the vertex: y = 3.
So the focus is (0, −3) and the directrix is y = 3.
4 Draw it
Draw the grid.
The vertex (0, 0),
the focus (0, −3),
and the directrix y = 3, dashed.
The parabola opens down, around the focus.
5 Check
Let's check with a point on the parabola, level with the focus: (−6, −3).
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: write it as x² = 4py.
Then: find p.
Then: focus and directrix.
Then: draw it.
Last: check.