Expert: Ahmed Salami - 2/25/2006

Question
Hello, I am having a hard time creating functions for data since functions have always been a weak aspect of math for me. The question is that I have data for braking distances and thinking distances for the various speeds of a vehicle and I have to find functions which model the behvaiors of each. The data for each group is listed below.  
Speed(kmh)  Thinking (meter)  Braking(meter)
32            6              6
48            9              14
64            12              24
80            15              38
96            18              55
112            21              75

The question is quite straightforward. How would you develop functions that model the behavior of this or any data?
Your assistance will be greatly appreciated.

Answer
Hi Harshal,
Sorry for the time it took.
What you require here is knowledge on the topic of
'curve fitting. You'll need to get a good textbook and
read about it to get the best understanding of it. Its
quite involved and so i won't be doing it enough justice
trying to explain it all to you here.
But i guess i still have to give out something.
Firstly you'll have to determine the kind of
relationship that exists between the variables.
It could be linear, quadratic, cubic, exponential,
logarithmic, trigonometrical etc.
They are of varying complexity and so some knowledge
of the particular situation is required.
The easiest way to determine this relationship is by
drawing a graph with the values you have and comparing
with that of standard functions, and see if it matches
any or requires modifications. For instance, if you end
up with a quadratic graph you know then that the
relationship is of the form
y = ax^2 + bx + c
The next problem would now be to determine the values of
the constants a,b,c that would adequately model the
situation in most cases. This is the objective of the
study of curve fitting.
As for this particular situation, S against T looks
linear enough and would be of the form
S = mT + c
where m is the slope of the graph of S against T and c
is the intercept on the S axis.
Experience lets me know that an adequate relationship is
S = (16/3)T
There are, however, standard ways of determining the
value of the constants.
S against B is not linear by any means and so is more
complicated. It could well be quadratic, or even
something else.
Please get back to me, i want to know how you fare. I
might always have other things to add for you.

I hope i have helped. You can always get back to me.
Regards.