Math Is Fun Forum

Potatoe Man

Guest

How do you find the equation for a parabola?

I only know the length and height of my parabola. can u pls tell me how to put it in an equation for finding the focal point?

p.s. im 12 can u make it simple pls this is ALGEBRA 2 and high school stuff DUDE!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: How do you find the equation for a parabola?

hi Potatoe Man

Welcome to the forum.

I'll have a go at explaining.  If there's anything you don't follow, feel free to post back and I'll try to explain that too. 

Here's my diagram:

The parabola has height h and what I guess you're calling the length is 2L across the bottom.  I've called it 2L to keep fractions out of the algebra.

So the coordinates are:

vertex = (0,h)
focus   = (0,f)
points where the parabola cuts the x axis are (-L,0) and (+L,0)

There's an extra (horizontal) line above the vertex.  That line is called the directrix and is an important part of the properties of any parabola.  Every point on the parabola is the same vertical distance from the directrix as it is from the focus.  That property is going to enable us to form the equation you want.

Using the property, the distance from the focus to the vertex (h-f) must be the same as the distance from the vertex to the directrix.

So the height of the directrix must be h + (h-f) = 2h-f

Now use the property on either of those points on the x axis.  The distance of one of those points to the directrix is 2h-f

To get the distance to the focus I will use Pythagoras' theorem. 

The easiest way to make these two distances equal is to square the directrix distance and make it equal to the distance squared.  That way I can avoid any square roots.

Cancel  the f squared

re-arrange

divide by 4h

Hope that helps,

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob