\magnification=\magstep1
\centerline{Mathematics 131, section 1 -- Calculus for Physical and Life Sciences}
\centerline{Discussion 1 -- ``New Functions From Old''}
\centerline{\it September 6, 2004}
\bigskip
\noindent
{\it Working in a Group}

\bigskip
Since this is the first of the discussion class of the semester,
a few words about this way of working are probably in order.
In the discussion meetings of this class, we will be aiming
for {\it collaborative learning} -- that is, for an
integrated group effort in analyzing and attacking the discussion 
questions.  The ideal is for everyone in each of the groups to be 
fully involved in the process.  By actively 
participating in the class through talking about the ideas
yourself in your own words, you can come to a better first 
understanding of what is going on than if you simply listen 
to someone else (even me!) talk about it.  

However, to get the most out of this kind of work, 
some of you may have to adjust some of your preconceptions.  
In particular:
\smallskip
\item{$\bullet$} This is {\it not a competition}.  
You and your fellow group members are working as a
team, and the goal is to have everyone understand what the 
group does fully.
\item{$\bullet$} At different times, it is inevitable that
different people within the group will have a more 
complete grasp of what you are working on and
others will have a less complete grasp.  Dealing with
this a group setting is excellent preparation for real
work in a team; it also offers opportunities for significant
educational experiences.  
\item{$\bullet$} If you feel totally``clueless'' at some point
and everyone else seems to be ``getting it,'' 
your job will be to ask questions and even pester your fellow 
group members until the point has been explained to your full 
satisfaction.  (Don't forget, the others may be jumping to 
unwarranted conclusions, and your questions may save the group 
from pursuing an erroneous train of thought!)
\item{$\bullet$}  On the other hand, when you think you do see 
something, you need to be willing to explain it patiently to 
others.  (Don't forget, the absolutely best way to make
sure you really understand something is to try to 
explain it to someone else.  If you are skipping over an
important point in your thinking, it can become very apparent
when you set out to convey your ideas to a team member.)   
\smallskip
\noindent
In short, {\it everyone has something to contribute, and 
everyone will contribute in different ways at different times}.

\bigskip
\noindent
{\it Mathematical Goals}
\bigskip
Once we have some function $f$, we can create a number of 
new functions from it by simple transformations:
\bigskip
\item{$\bullet$} we can multiply $f$ by a constant $c$
to produce a new function $g$ with $g(x) = cf(x)$,
\item{$\bullet$} we can add a constant $c$ to $f$
to produce a new function $g$ with $g(x) = f(x) + c$,
\item{$\bullet$} we can substitute $x + c$ into the function $f$
to produce a new function $g(x) = f(x + c)$,
\item{$\bullet$} we can substitute $cx$ into the function $f$
to produce a new function $g(x) = f(cx)$.
\bigskip
\noindent
Today we want to recall (or perhaps figure out for the first time!)
what these transformations do {\it in geometric terms}.  In 
other words, we want to understand the effect of each on the 
graph $y = f(x)$.  The key question is, given the graph
$y = f(x)$, how is the graph $y = g(x)$ obtained in each of the
four cases above?
\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\noindent
I.  The graph below shows a part of the graph $y = f(x)$ for a 
function with domain equal to the whole set of real numbers.
\vglue 2.0truein
\item{A)} Sketch the graphs $y = f(x) + 1$ and $y = f(x) - 2$, 
showing all $x$- and $y$-axis intercepts and asymptotes.
\item{B)}  Sketch the graphs $y = 2f(x)$ and $y = {-2\over 3} f(x)$, 
showing all $x$- and $y$-axis intercepts and asymptotes.
\item{C)}   Sketch the graphs $y = f(x+1)$ and $y = f(x-1)$, 
showing all $x$- and $y$-axis intercepts and asymptotes.
\item{D)}  Sketch the graphs $y = f(2x)$ and $y = f(-x/4)$, 
showing all $x$- and $y$-axis intercepts and asymptotes.
\bigskip
\noindent
II.  Now in your own words, in complete English sentences,
explain the general geometric relation between each of 
the following.  Note: You should be looking for things
that are true for all functions.  The examples in question
I should be suggestive, but don't focus only on properties of
that $f(x)$.
\item{A)} The graphs $y = f(x)$ and $y = f(x) + c$:  How
is $y = f(x) + c$ obtained from $y = f(x)$, and what difference
does the sign of $c$ (i.e. whether $c > 0$ or $c < 0$) make?
\item{B)} The graphs $y = f(x)$ and $y = cf(x)$:  How
is $y = cf(x)$ obtained from $y = f(x)$, and what difference
do the sign and magnitude of $c$ make?  (That is, how is the 
case $c < 0$ different from $c > 0$?  How is the case $|c| < 1$
different from $|c| > 1$?)
\item{C)} The graphs $y = f(x)$ and $y = f(x + c)$:  How
is $y = f(x + c)$ obtained from $y = f(x)$, and what difference
does the sign of $c$ make?  
\item{D)}  The graphs $y = f(x)$ and $y = f(cx)$:  How
is $y = f(cx)$ obtained from $y = f(x)$, and what difference
do the sign and magnitude of $c$ make?  (That is, how is the 
case $c < 0$ different from $c > 0$?  How is the case $|c| < 1$
different from $|c| > 1$?)
\bigskip
\noindent
{\it Assignment}
\bigskip
Group writeups due in class Wednesday, September 8.
\bye