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Finding the Equation of a Curve


Date: 10/02/2000 at 08:23:14
From: Erik 

Subject: Equation of a curve

Dr. Math,

I'm hoping you can help me find the equation to a certain curve. I 
have looked through several Internet sites listing graphs of common 
curves, but can't find the one I'm looking for.

The curve I would like to describe mathematically starts at some 
point on the positive y-axis, grows increasingly positive in the 
x-direction, and perhaps decreases in y only slightly. At some point 
in the x-direction, the slope of the curve becomes sharply negative 
and approaches the x-axis. The curve should flatten out as it 
approaches the x-axis, which is an asymptote for the curve. I'll try 
to provide an illustration:

     |* * * * * * *
     |              * * * * *
     |                         *
     |                           *
     |                            *
     |                             *
     |                              *
     |                               *
     |                                *
     |                                 *
     |                                  *
     |                                   *
     |                                     * * * * * * *
     +--------------------------------------------------

Generally, I would like to know how one might go about finding the 
equation of any curve. What process does one go through to find an 
equation for a simple curve like this one?

Thanks for the help,
Erik


Date: 10/02/2000 at 14:30:08
From: Doctor Ian
Subject: Re: Equation of a curve

Hi Erik,

There is no 'standard' way to figure out what kind of function 
corresponds to a given graph, except to think of the different kinds 
of functions that you know about and see whether any of them are 
approximately the right shape. 

When you can't come up with anything off the top of your head, there 
are three other ways to proceed:

1) Think about whether the graph describes the behavior of any 
physical object or process that you know about. For example, an 
inverse-square rule (like the one that governs gravity and 
electrostatic processes) falls off asymptotically to zero with 
increasing distance, which describes one part of the curve, but it 
rises very quickly as distance approaches zero, so it doesn't describe 
the other part.

2) Look for two or more functions, each of which matches one part of 
the curve, but goes to zero everywhere else.

One thing to look out for is a curve that has the right shape, but is 
shifted slightly, or inverted in some way. In your case, you might be 
able to find a curve that looks like this:

      *
              *      |
                  *  |
                    *|
  -------------------+---------------
                     |*
                     |  *
                     |      *
                     |              *

If you move this up and to the right, you get something that looks a 
lot like your curve. Or something like:

                                    *
                     |      *
                     |  *
                     |*
  -------------------+---------------
                    *|
                  *  |
              *      |
       *             |


Which looks like your curve inverted and shifted. Now the part on the 
right looks like a logarithmic function. Unfortunately, log(x) isn't 
defined for negative x. But this is the general idea. 

A lot depends on what you want to do with the mathematical 
description. If you're trying to describe a causal relationship, then 
you want to hold out for a single function. But if you just want to 
replace a lookup with a computation, then it might suffice to find a 
polynomial that has the right shape over the interval of interest, 
without worrying about what it will look like outside that interval. 
For example, you might be able to find a cubic equation that looks 
like this:

       
        *     *    
    *               *                            *
                      *                         *
                       *                       *
                        *                     *
                          *                 *
                                *      *
                                      
             |---------------------------|


In a limited domain, this mimics your curve, which might be good 
enough. 

Having said all that, the curve you've drawn looks a lot like the 
right half of a normal distribution curve, pictured here in the 
Internet Glossary of Statistical Terms:

   Normal Distribution
   http://www.animatedsoftware.com/statglos/sgnormdi.htm   

and even more like the mirror image of a cumulative distribution 
function for a normal distribution. See section 0.2 in 

   The Thickets Approach to P-Bounds   
   http://class.ee.iastate.edu/berleant/home/me/vita/papers/thicketsChapter.htm   

for a side-by-side look at graphs of these two functions. To turn 
cumulative distribution function, F(X), into your curve, you would 
plot 

     y = 1 - F(x)

instead of just y = F(x).

If you'd like to write back and say a little more about where you got 
the curve you're trying to describe, and what you plan to do with the 
description, I could be of more help. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   


Date: 10/04/2000 at 17:04:06
From: Erik
Subject: Re: Equation of a curve

Dr. Ian,

Thanks a lot for the very detailed and thorough explanation! Now that 
you mention it, the mirror of the cumulative normal distribution does 
look like what I am trying to describe very well.

As far as what I am trying to do, I would like to find a way to 
describe the way people store and discard pending information. For 
instance, if the sitting president happens to have impeachment charges 
brought against him, an interested party will have this fact in the 
forefront of his mind for the time the impeachment is announced until 
the time the hearing is held (the flat part of the curve.) After the 
impeachment vote is taken and the results announced, the psychological 
impact of the impeachment proceedings will fall off fairly quickly but 
will probably never reach 0 (the negative slope and the asymptote on 
the x-axis.) 

Memory researchers usually model the memory decay portion of the curve 
as x^2, where 0 < x < 1 and f(x) represents a "memory percentage" of 
sorts. This fits the decay to the asymptote part of the curve, but not 
the flat part of the curve.

I have looked on the Internet for some model that would be similar to 
the one I have described, but have not been able to find anything 
similar. I did find one site about mathematical modeling of human 
memory, but these models seemed to be dealing much more with a human's 
ability to remember specific facts for a given amount of time (which 
may be a little different from what I would like to measure -- the 
emotional impact of a pending fact).

Has this explanation been clear? The cumulative normal distribution's 
mirror does seem to fit the general shape, but I'm not sure how to 
control the length of the relatively flat portion near the y-axis 
(i.e., how to distinguish between an event that is to be pending for 3 
units of time versus one that is to be pending for 4 units of time). 
I'm also not really clear as to what the mathematical formula would be 
for this function. "The mirror of the cumulative normal distribution" 
makes sense to me, but I don't know what this would look like in an 
equation.

Thanks again for the help. I appreciate you spending brainpower on 
this funny little problem! I also appreciated your general comments 
about how to figure what an equation should look like and was happy 
that I did the 'right' thing when I first started this problem (I just 
wish I'd thought of the normal distribution myself...)

Best regards,
Erik


Date: 10/05/2000 at 17:57:52
From: Doctor Ian
Subject: Re: Equation of a curve

Hi Erik,

I'm glad I was able to help.

One of the URLs that I mentioned had an equation for the normal 
distribution function; the cumulative distribution function is the 
integral of that function. And the mirror would be one minus the 
integral.

But you've hit on the crux of the problem, which is that the flat part 
of the curve can stretch out indefinitely. This is the case (I would 
argue) because the person is continually being reminded of the pending 
event.

So what you're trying to model is something like the charge of a 
battery that can sit in a recharger indefinitely before being used. 
There are two different behaviors (being charged, and discharging), so 
it shouldn't be a big surprise to find that it's difficult to find one 
equation that describes them both.

I had fun thinking about the problem, and for what it's worth, I was 
about to send the general part of the answer (i.e., how to go about 
solving a problem like this one), when the similarity to the 
distribution function finally occurred to me.

Once you see it, you wonder how you could have missed it, but I 
wouldn't say it's obvious at all. (I've shown it to a few other 
people, and none of them recognized it right away, either. And if you 
look at the form of the normal distribution function, it becomes clear 
why you wouldn't be likely to arrive at it by playing around 
symbolically.)

Anyway, thanks for writing. Write back if you have any other 
questions, about this or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Equations, Graphs, Translations

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