Parabola Equation

Date: 15 May 1995 14:38:55 -0400
From: Eva Jason
Subject: parabola, equations, directed distance

        Hi-I hope you can help me. I'm trying to learn equations of 
parabolas. The math text I'm using is very confusing. The example given 
is find the equation of the parabola with Vertex (0,0) and Focus (0,-3). 
(the graph I've been given opens left and the focus pt. is (-3,0)). Axis 
of symmetry is x. Then it says that p=directed distance from 
V to F = -3-0=-3? Shouldn't V to F be 0-(-3)=3?
Next they tell me x=-(-3), x=3, and y2=4(-3)x, y2=-12x. Can you 
straighten me out? Thanx for any help.

Date: 16 May 1995 23:05:34 -0400
From: Dr. Sydney
Subject: Re: parabola, equations, directed distance

Hello!

I'm glad you wrote to us!  I hope we can be of some help to you.  

I'm a little confused about the problem.  First you say they want you to
find the equation of the parabola with vertex (0,0) and Focus (0.-3), but
then you describe a graph whose focus is (-3, 0), and the rest of the
problem seems to describe this second graph.  Maybe it was a typo.  Why
don't we look at both cases.

Let's first examine the case where the focus is (0, -3).  That means the
focus is on the negative side of the x-axis, to the left of the origin.
Indeed, as you say the parabola will then open left, and the axis of
symmetry will be the x-axis (Are you clear on why this is true?  You didn't
express any confusion about this, so I won't say anything here, but if you
do have questions, feel free to write back).  Now, what they mean by
"directed distance" is what is the distance between V and F keeping
direction in mind.  Have you worked with vectors before? If so, then think
of p as a vector that starts at V and ends at F.  If not, think of p as a
line that starts at V and goes to F (in that direction!).  You could
indicate that the direction is V to F by putting an arrow on the line
pointing toward F.  

It looks something like this:
                    |
                    |
                    |
          |<--------|
          F         |V
       (-3, 0)      |
                    |
                    |

There are 3 units between F and V, so the distance between F and V is 3,
right?  Well, then we know that the DIRECTED distance is going to be +3 or
-3.  All the word DIRECTED means is that you will indicate if we are going
to the left or to the right with the sign.  Since p is the directed distance
from V to F, that means we are starting at V and going to F.  We want to
know how many units from V must we travel to get to F.  Well, the answer
becomes clear when we put it this way, doesn't it?  We want to travel -3
units from V to get to F (in other words, we want to travel 3 units in the
leftward direction).  If we were to travel 3 units from V on the x-axis,
that would put us at the point (3, 0), which is not where we want to be.
Does that makes sense?  Another way to look at it is that you are taking the
x-coordinate of F and subtracting the x-coordinate of V from it.  So, p = -3

Now, I have no idea what they are talking about when the say x = - (-3) =
3.  ?????????????  What x are they referring to?

As far as coming up with the equation y^2 = -12x, here is an explanation:

For hyperbolas that open up to the left (with p<0 and focus on the x-axis),
the equation of the hyperbola is:

(y-k)^2 = 4p(x-h)

where (h,k) is the vertex.  Since, here (h,k) = (0,0), our equation is:

y^2 = 4(-3)x = -12x

If you want to know how they derived the general formula above, write back,
and we'll be happy to help out.  

Now, given all of this information, do you think you could try to do the
case where the vertex is the origin and the focus is (0, -3)?  

If you have any more questions or if I've said something confusing, please
write back!  Have fun.

--Sydney, "dr.math"