Step by step
The question
Question: find the focus and the directrix of x² = 16y.
1 Find p
Compare with the parabola rule: x² = 4py.
So 4p is the number in front of y: 4p = 16.
Careful!
16 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.
16 divided by 4 is 4.
So p = 4.
2 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 4 above the vertex, inside the curve: (0, 4).
The directrix is the flat line 4 on the other side of the vertex: y = −4.
So the focus is (0, 4) and the directrix is y = −4.
3 Draw it
Draw the grid.
The vertex (0, 0),
the focus (0, 4),
and the directrix y = −4, dashed.
The parabola opens up, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (8, 4).
Is it on the parabola?
Put it in: 64 and 64.
Yes!
Its distance to the focus: 8.
Its distance to the directrix: 8 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.