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

Step by step

The question

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

(y + 2)² = −4(x − 1)

1 Find p

Compare with the parabola rule: y² = 4px.

rule: y² = 4px

So 4p is the number in front of the x 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 negative.

So the parabola opens left.

p < 0: opens left

The focus is 1 to the left of the vertex, inside the curve: (0, −2).

focus = (0, −2)

The directrix is the upright line 1 on the other side of the vertex: x = 2.

directrix: x = 2

So the focus is (0, −2) and the directrix is x = 2.

focus (0, −2) and x = 2

3 Draw it

Draw the grid.

The vertex (1, −2),

the focus (0, −2),

and the directrix x = 2, dashed.

The parabola opens left, around the focus.

4 Check

Let's check with a point on the parabola, level with the focus: (0, −4).

Is it on the parabola?

Put it in: 4 and 4.

Yes!

(−4 + 2)² = −4(0 − 1) → 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