Worked examples
The question
Question: find the focus and the directrix of x² = 8y.
x2 = 8y
1 Find p
Compare with the parabola rule: x² = 4py.
rule: x² = 4py
So 4p is the number in front of y: 4p = 8.
4p = 8
Careful!
8 is 4p, not p.
We still divide by 4.
Careful: not p = 8 ✗
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4p4 = 84
4 divided by 4 is 1.
So just p is left.
8 divided by 4 is 2.
So p = 2.
p = 2
2 Focus and directrix
There are no brackets: just x² and y.
So the vertex is (0, 0).
vertex = (0, 0)
p is positive.
So the parabola opens up.
p > 0: opens up
The focus is 2 above the vertex, inside the curve: (0, 2).
focus = (0, 2)
The directrix is the flat line 2 on the other side of the vertex: y = −2.
directrix: y = −2
So the focus is (0, 2) and the directrix is y = −2.
focus (0, 2) and y = −2
3 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.
4 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!
4² = 8 × 2 → 16 = 16
Its distance to the focus: 4.
Its distance to the directrix: 4 too.
The same distance.
That is what makes a parabola.
Right!
to the focus 4, to the directrix 4
Let's look back.
First: find p.
Then: focus and directrix.
Then: draw it.
Last: check.
Answer focus (0, 2) and y = −2
▶ Watch it solved
The question
Question: find the focus and the directrix of y² = 12x.
y2 = 12x
1 Find p
Compare with the parabola rule: y² = 4px.
rule: y² = 4px
So 4p is the number in front of x: 4p = 12.
4p = 12
Careful!
12 is 4p, not p.
We still divide by 4.
Careful: not p = 12 ✗
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4p4 = 124
4 divided by 4 is 1.
So just p is left.
12 divided by 4 is 3.
So p = 3.
p = 3
2 Focus and directrix
There are no brackets: just y² and x.
So the vertex is (0, 0).
vertex = (0, 0)
p is positive.
So the parabola opens right.
p > 0: opens right
The focus is 3 to the right of the vertex, inside the curve: (3, 0).
focus = (3, 0)
The directrix is the upright line 3 on the other side of the vertex: x = −3.
directrix: x = −3
So the focus is (3, 0) and the directrix is x = −3.
focus (3, 0) and x = −3
3 Draw it
Draw the grid.
The vertex (0, 0),
the focus (3, 0),
and the directrix x = −3, dashed.
The parabola opens right, around the focus.
4 Check
Let's check with a point on the parabola, level with the focus: (3, 6).
Is it on the parabola?
Put it in: 36 and 36.
Yes!
6² = 12 × 3 → 36 = 36
Its distance to the focus: 6.
Its distance to the directrix: 6 too.
The same distance.
That is what makes a parabola.
Right!
to the focus 6, to the directrix 6
Let's look back.
First: find p.
Then: focus and directrix.
Then: draw it.
Last: check.
Answer focus (3, 0) and x = −3
▶ Watch it solved
The question
Question: find the focus and the directrix of (x − 2)² = 8(y + 1).
(x − 2)2 = 8(y + 1)
1 Find p
Compare with the parabola rule: x² = 4py.
rule: x² = 4py
So 4p is the number in front of the y bracket: 4p = 8.
4p = 8
Careful!
8 is 4p, not p.
We still divide by 4.
Careful: not p = 8 ✗
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4p4 = 84
4 divided by 4 is 1.
So just p is left.
8 divided by 4 is 2.
So p = 2.
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).
vertex = (2, −1)
x − 2 → h = 2
y + 1 → k = −1
p is positive.
So the parabola opens up.
p > 0: opens up
The focus is 2 above the vertex, inside the curve: (2, 1).
focus = (2, 1)
The directrix is the flat line 2 on the other side of the vertex: y = −3.
directrix: y = −3
So the focus is (2, 1) and the directrix is y = −3.
focus (2, 1) and 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!
(6 − 2)² = 8(1 + 1) → 16 = 16
Its distance to the focus: 4.
Its distance to the directrix: 4 too.
The same distance.
That is what makes a parabola.
Right!
to the focus 4, to the directrix 4
Let's look back.
First: find p.
Then: focus and directrix.
Then: draw it.
Last: check.
Answer focus (2, 1) and y = −3
▶ Watch it solved
