Question

Derive the equation of the parabola with a focus at (−5, 5) and a directrix of y = -1.

Answer 1

Hint:
the vertex is half-way between the focus and the directrix.
Can you sketch the parabola?

Answer 2

It's not like i need to sketch the graph, i just need the equation but I have o idea what to do to find it

Answer 3

You need to know if the parabola opens up or down. Sketching in this case will clarify this.

Answer 4

Is that right?

Answer 5

@mathmate

Answer 6

From the given information, does the parabola open up or down?

Answer 7

up?

Answer 8

How can you tell?

Answer 9

I don't know i'm just guessing

Answer 10

Guessing is bad for you when you are getting help.
You would have been considered to have understood that if you guessed it right.

Back to opening up or down.
First, I am not sure what the point (-5,3) was for. Perhaps you can explain.
V(-5,2) is the vertex and is half way between the focus F(-5,5) and the directrix y=-1.
What is the distance between F and V ?

Answer 11

@mathmate

Answer 12

The distance between F(-5,5) and V(-5,2) is _________?

Answer 13

(0,3)

Answer 14

The distance is 3 because sqrt(0^2+3^2)=3. Do you agree?

Answer 15

yes

Answer 16

This distance is usually called c or p, in the definition of parabolas.
This is also the distance between the vertex and the directrix.
You can take a minute to double check that.

Answer 17

Ok

Answer 18

Now remember I wanted to know if the parabola opens up or down?
You can tell that once you know the focus F and the vertex V.

Answer 19

But we already know that... right?

Answer 20

Because the axis of the parabola passes both points, and the parabola "wraps" around the focus. This is what you need to know to sketch it and to find the sign of c (or p).

Answer 21

Yes, we already know that. But we need to \(use\) that information.

Answer 22

I dont understand how those two points can determine it...

Answer 23

Do you know what a "vertex" of a parabola means?

Answer 24

That's the middle of the parabola right?

Answer 25

Right.
Can you draw the parabola on the graph that we started? Recall that the vertex is at V, and the axis of the parabola is vertical (since it passes through both V and F), AND that the parabola "wraps" around F.