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\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Discussion 1}
\centerline{September 4, 2009}
\vskip 10pt
\noindent
A)  The graph $y = f(x) = 4 - x^2$ for $x$ in the interval $[-1,2]$ 
is given in Figure~\ref{figforA} on the back, together with several
transformed graphs.   Label the graphs with the letters $A,B,C,D,E$ 
in some way.  Then match each 
equation with its graph by letter and give reasons for your choices.  
(Note that one of the graphs is $y = f(x)$ and you will need to identify
that one as well.)
\begin{itemize}
\item[1.]  $y = f(2x) - 4$
\item[2.]  $y = 2f(x + 6)$
\item[3.]  $y = -f(x + 4)$
\item[4.]  $y = \frac{1}{2}f(x) - 1$ 
\end{itemize}
\vskip 10pt
\noindent
B)  Say $y = f(x) = \cos(x)$ gives the usual graph of the 
cosine function for $x$ in the domain $[0,4\pi]$.  For each 
of the following, write equations for the graph that is 
described and sketch.
\begin{itemize}
\item[1.]  The graph $y = f(x)$ shifted three units down.
\item[2.]  The graph $y = f(x)$ reflected across the $x$ axis,
then shifted 2 units up.
\item[3.]  The graph $y = f(x)$ shifted $\frac{\pi}{2}$ units
to the left.
\item[4.]  The graph $y = f(x)$ stretched vertically by a factor of 4, 
then shifted up one unit.
\item[5.]  The graph $y = f(x)$ compressed, or shrunk horizontally 
by a factor of $2$.
\end{itemize}  
\vskip 10pt
\noindent
C) Imagine that each of the scatter plots 1, 2, 3 in Figure~\ref{figforC}
were obtained as data from some experiment.  
\begin{itemize}
\item[a.]  Decide what type
of function you might choose as a model for the data.  That is, what
type of formula would yield a graph with something like this shape?  Explain your
choice.  
\item[b.] From the information given in the plot, determine a possible formula of
a function of the type you decided on for part 1. that would fit the data
reasonably well.
\item[c.]  Plot your function.
\end{itemize}
 
\vskip 10pt
\noindent
\emph{Assignment}
\vskip 10pt
Group writeups due in class on Monday, September 7.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth,angle=270]{Disc1B.ps}
\end{center}
\caption{Plots for question A}
\label{figforA}
\end{figure}
\begin{figure}
\centering
\subfloat[Plot 1]{\includegraphics[width=0.3\textwidth,angle=270]{Disc1Ca.ps}}\hspace{3mm}
\subfloat[Plot 2]{\includegraphics[width=0.3\textwidth,angle=270]{Disc1Cb.ps}}\hspace{3mm}
\subfloat[Plot 3]{\includegraphics[width=0.3\textwidth,angle=270]{Disc1Cc.ps}}\hspace{3mm}
\caption{Scatter plots for question C}
\label{figforC}
\end{figure}
\end{document}
