Graph (x - 3)^2 = -8(y - 1) Step by Step

Step by step

The question

Question: find the focus and the directrix of (x − 3)² = −8(y − 1).

(x − 3)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 − 3 gives 3, y − 1 gives 1.

So the vertex is (3, 1).

vertex = (3, 1)

x − 3 → h = 3

y − 1 → k = 1

p is negative.

So the parabola opens down.

p < 0: opens down

The focus is 2 below the vertex, inside the curve: (3, −1).

focus = (3, −1)

The directrix is the flat line 2 on the other side of the vertex: y = 3.

directrix: y = 3

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

focus (3, −1) and y = 3

3 Draw it

Draw the grid.

The vertex (3, 1),

the focus (3, −1),

and the directrix y = 3, dashed.

The parabola opens down, around the focus.

4 Check

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

Is it on the parabola?

Put it in: 16 and 16.

Yes!

(−1 − 3)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