How to put my parabola in standard form.

Question

I found three points of my parabola: (2.5, 1), (7, 4), and (11.5, 1). I need to put my parabola into standard form, but I am not sure how to do it with the information that I have.

Answer 1

If you plot the three points, you can see the parabola opens downward.  Assuming the axis is vertical we have

that the standard form for a parabola with a vertical axis is (x-h)²=4p(y-k), where the vertex is at (h,k) and the focal length is p.  Substituting the three points, in turn, into the equation of the standard form gives

(2.5-h)² = 4p(1-k)

(7-h)² = 4p(4-k)

(11.5-h)² = 4p(1-k)

 

The right sides of equations 1 and 3 are the same.  Setting their left sides equal to each other and solving

we have h=7. 

Using this result in the second equation we have, since p is not zero, that k=4.

Using this in the first equation we get p=-25/12.

I leave the labor of solution to you.

 

Answer 2

Hi Alex,

 

Well, not much to go on, right?

 

Different textbooks teach different ways of writing the equation of a parabola, so I'm going to have to let you decide some of the details.  But no matter what, and no matter which equation you use, there are a couple of key things you need to know about a parabola in order to graph it.

 

1.  Where is its vertex?

2.  What is the distance from the vertex to the focus...or to the directrix...or, if you don't know either of those things, I need other information to figure it out.

 

So, in your problem, you have three points.  In other words, just looking at what you have, it doesn't appear that you actually know anything about the vertex or anything about the focus.  Crud!

 

If that's true, then this is a reeeeeally tough problem.  :)  So hopefully, we know more than we think.

 

- The problem gives you two points that both have a y-value of 1.  (2.5, 1) and (11.5, 1).  Aren't parabolas symmetric?

- If parabolas are symmetric, then can we figure out something about this vertex?  Can you draw a dotted line down the "middle" of where the parabola should be, knowing those two points?  Hint:  what is half-way between 2.5 and 11.5?  (It's called the "line of symmetry" if you're looking for it in your book.) 

- Key Question #1:  is (7, 4) the vertex of the parabola you're looking for?  that'd be convenient, wouldn't it?  The dotted line you drew splits your parabola in half.  The vertex is at the "middle" of a parabola, so the vertex sits on that line of symmetry.  Is (7,4) on the line of symmetry?

- Once you have found the vertex, you can now start to write the equation of your parabola.  In your class or book, maybe you use (h, k) as the coordinates of the vertex, and now you can write part of the equation of the parabola, in vertex form.  Or, if your book likes "y=ax² + bx + c" as a parabola, then you can figure out a or b or c based on h and k.

 

If you can get to that point in the problem, you're almost home.  But you still have a problem.  You don't know where the focus of the parabola is, because they haven't told you where it is.  So, you can get close to writing the equation, but there's a constant you can't figure out yet.  Maybe it's "a," or in your book, maybe it's something called "p."  Either way, you're going to be stuck.

 

One more hint to get you un-stuck, and then we'll need to wait and see how far you get: 

 

- Whatever you're missing, it's a constant.  Whatever points are on the parabola, your missing value stays the same.  So, if you know another point on your parabola besides the vertex, like an "x" and a "y," then is that enough information to find the missing constant? 

 

I hope this helps, and let me know if you need another nudge in the right direction...

 

-- Michael