Graph (x + 1)^2 = 4(y - 2) Step by Step

Step by step

The question

Question: find the focus and the directrix of (x + 1)² = 4(y − 2).

(x + 1)² = 4(y − 2)

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 = 4.

4p = 4

Careful!

4 is 4p, not p.

We still divide by 4.

Careful: not p = 4 ✗

To remove times 4, we divide by 4.

We do it on both sides, so they stay equal.

4p4 = 44

4 divided by 4 is 1.

So just p is left.

4 divided by 4 is 1.

So p = 1.

p = 1

2 Focus and directrix

The vertex comes from the brackets.

The sign turns: x + 1 gives −1, y − 2 gives 2.

So the vertex is (−1, 2).

vertex = (−1, 2)

x + 1 → h = −1

y − 2 → k = 2

p is positive.

So the parabola opens up.

p > 0: opens up

The focus is 1 above the vertex, inside the curve: (−1, 3).

focus = (−1, 3)

The directrix is the flat line 1 on the other side of the vertex: y = 1.

directrix: y = 1

So the focus is (−1, 3) and the directrix is y = 1.

focus (−1, 3) and y = 1

3 Draw it

Draw the grid.

The vertex (−1, 2),

the focus (−1, 3),

and the directrix y = 1, dashed.

The parabola opens up, around the focus.

4 Check

Let's check with a point on the parabola, level with the focus: (1, 3).

Is it on the parabola?

Put it in: 4 and 4.

Yes!

(1 + 1)² = 4(3 − 2) → 4 = 4

Its distance to the focus: 2.

Its distance to the directrix: 2 too.

The same distance.

That is what makes a parabola.

Right!

to the focus 2, to the directrix 2

Let's look back.

First: find p.

Then: focus and directrix.

Then: draw it.

Last: check.

Answer