FUNCFIT
BRIEF ABSTRACT
FUNCFIT is a fun classroom activity to familiarize students with the shapes of various functions.  Groups write non-technical geometric descriptions of graphs and exchange them with other groups.  Then students try to recreate graphs from the descriptions.  The exercise is appropriate for pre-calculus or beginning calculus students.
GENERAL INFORMATION
FileName:   FUNCFIT
Full title:  Function Fitting to Descriptions of Curves
Last Update: 5/29/96
Developer:    
Lynn Kiaer, Department of Mathematics,  Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA
Contact:  
Aaron Klebanoff, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. 
Phone:  812-877-8151.  Email:  Aaron.Klebanoff@rose-hulman.edu.  
FAX:  812-877-3198.
Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
STATEMENT OF PROBLEM 
This is a cooperative learning exercise.
1.  Each student is given a function definition and asked to graph the function, and then to write a non-mathematical description of the appearance of the function.
2.  Students form pairs, with each partner having the same function.  The partnership refines the written description, and signs it.
3.  Pairs exchange written descriptions (but not functions) with a nearby pair (a pair which started with a different function).  Pairs attempt to find the function that fits the description.  This may be an iterative process.  When a pair proposes a function, the description writers may respond
You've got it.
	
That can't be the function because it doesn't satisfy a particular aspect of the description.
	
Ouch!  We have to modify the description!
Additional rounds give the students practice matching functions to descriptions.
KEYWORDS 
Function, fitting functions.
TEACHER NOTES
ISSUES RELATED TO THE PROBLEM
Prerequisites:
Prerequisites will vary depending on the functions used.  I would envision using polynomials (up to cubic), sines, cosines, exponentials and logarithms.  
It would probably be worth doing this exercise as a part of a general review of functions, and then again after the students have studied derivatives.
Time allotment - time management:
This will depend somewhat on the complexity of the functions being used, but 2-3 minutes for steps 1 and 2, 10-15 minutes for step 3, and 8-10 minutes for additional iterations seems about right.
Students who are stuck looking for a function should be instructed to sketch a graph that matches the description.
Expectations 
Depending on the kinds of functions used, students could be asked to find the exact function or merely a generic version.
Future payoffs
The purpose of this exercise is to build the skills and intuition about functions that students need in order to, for example, choose an appropriate function form for a collection of data points.
Extensions
References and Sources
POSSIBLE SOLUTION(S)
Example 1
In[2]:=
f[x_] = 2 x^3 - 3 x^2 + x + 4;
In[3]:=
Plot[f[x],{x,-3,1.5}];

In[4]:=
Plot[f[x],{x,0,1}];

Description:  This function has a value of 4 when x is 0.  It decreases rapidly when x<0, becoming negative before x gets to -1.  As x increases from 0, the function increases just a little bit (not more than 4.1), then decreases a little bit (not less than 3.9), all before x gets to 1.  When x is 1/2 and when x is 1, the function value is 4 again, and then, for x>1, the function increases rapidly in value.
Example 2
In[5]:=
g[x_] = 100 - 100 Exp[-x/2];
In[6]:=
Plot[g[x],{x,-1,10}];

Description:  This function is always increasing.  Its value is 0 when x is 0, and decreases sharply as x gets more and more negative.  The function has a horizontal asymptote at y=100.
This description would probably have to be refined in order for the function fitters to obtain the coefficient of x in the exponent.
Example 3
In[7]:=
h[x_] = 3 + 2 Sin[2x - Pi/4];
In[8]:=
Plot[h[x],{x,0,2Pi}];

In[9]:=
Plot[h[x],{x,-.01,.01}];

In[10]:=
h[0] //N
Out[10]=
1.58579
Description:  This function oscillates between 1 and 5.  It repeats itself every Pi units.  When x is 0, the value of the function is about 1.58579.
In this case, I would encourage students not to use the exact value of the function at this point (3 - Sqrt[2]), partly because data doesn't look like this, and partly because it gives away the `shape' of the function.
ISSUES IN SOLUTION
The Fitters may have "fits" because the proposers have not given sufficient information to "nail down" a unique function in their mind.   Indeed there is no unique function which satisfies these finite conditions, although there might be a unique function of a given form say one like this: A + B Sin( C x + D).