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\centerline{Mathematics 136, Section 3 -- AP Calculus, Section 4}
\centerline{Discussion 1--Matching Formulas to Graphs}
\centerline{September 9, 2003}
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{\it Background}
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We have been reviewing the basic functions studied
in calculus.
Today, in a group discussion exercise, we will conclude 
this initial part of the course.  Given a graph, 
we will try to determine a reasonable formula that might yield a 
graph with that shape.  An additional goal
will be to introduce you to a way of working in a classroom context you may
not have experienced too often before
in mathematics classes -- {\it collaborative learning\/}.  Some 
suggestions:
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\item{$\bullet$} Try to tackle the questions below {\it as a group\/}, not
individually.  For instance,
don't be shy about throwing out partial ideas or suggestions how to proceed.
That's {\it the point} of doing things this way, and you're 
``in this'' together. 
\item{$\bullet$} The goal is for {\it everyone\/} to contribute to, and fully 
understand 
the group's results.  So if you think you ``see'' something, your job is to
{\it explain\/} it to the other members of the group.  On the other hand, if 
you don't understand something someone else says, your job is to {\it ask 
questions\/} until either
you are convinced, or you are able to show the other person that he or she 
was mistaken.
\item{$\bullet$}
Choose one or more ``scribe(s)'' within your group to keep a clean
written record of what you do.  The end product of the exercise will 
be one set of solutions to the problems below, which you will turn in
as a group.    I will assign one grade for each group's solutions. 
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{\it Discussion Questions}
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Find a possible formula for each of the functions 
defined by the graphs in problems 6-17 in the Review Problems
for Chapter 1 on page 49 of our text.

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{\bf Important Directions:}  No calculators, graphing or otherwise,
may be used during the class discussion(!)
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{\it Some suggestions}
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\item{$\bullet$} Start by trying to determine what 
{\it family} of functions fits the properties of each 
graph the best (i.e. is it the graph of a {\it linear} function, 
an {\it exponential} function, a {\it polynomial} function, a 
{\it trigonometric function}, a {\it rational} function?)
\item{$\bullet$}  Once you have decided what type of function
you are dealing with, you may also need to determine constants, and {\it shift and/or scale} 
the basic functions we have studied to get a graph like the one shown.
\item{$\bullet$} NOTE: There are many different correct answers
for each of these, but some correct answers are also
{\it very complicated}. There are relatively ``simple'' formulas that will
work in each case!
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{\it Due:}  In class, Wednesday, September 10.
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