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\begin{center}
\textbf{Graphing - January 30, 2002}
\end{center}

Do the following either with your calculator or by hand, make sure you put your calculator in radian mode.
For each exercise, graph the functions together on the same graph and answer the questions.

\begin{enumerate}
% vertical shift
\item  \label{vshift1}
	Note that $(\sin x + 2) \neq \sin(x+2)$ (why??)
	\[ 
	y= \sin x \qquad y= \sin x +2 \qquad y= \sin x -2 
	\]
	What does the graph of $y=\sin x + D$ look like?

\item  \label{vshift2}
	\[ 
	y= \cos x \qquad y= \cos x +2 \qquad y= \cos x -2 
	\]
	What does the graph of $y=\cos x + D$ look like?

%vertical stretch
\item  \label{vstretch1}
	\[ 
	y= \sin x \qquad y= 2 \sin x \qquad y= -2 \sin x 
	\]
	What does the graph of $y=A\sin x$ look like?

\item  \label{vstretch2}
	\[ 
	y= \cos x \qquad y= 2 \cos x \qquad y= -2 \cos x 
	\]
	What does the graph of $y=A\cos x$ look like?

% horizontal shift
\item  \label{hshift1}
	\[ 
	y= \sin x \qquad y= \sin (x-\pi/3) \qquad y= \sin (x + \pi/3)
	\]
	What does the graph of $y=\sin (x-C)$ look like?
\item  \label{hshift2}
	\[ 
	y= \cos x \qquad y= \cos (x-\pi/3) \qquad y= \cos (x + \pi/3)
	\]
	What does the graph of $y=\cos (x-C)$ look like?

% horizontal stretch
\item  \label{hstretch1}
	\[ 
	y= \sin x \qquad y= \sin (2x) \qquad y= \sin (-2x)
	\]
	What does the graph of $y=\sin (Bx)$ look like?
\item  \label{hstretch2}
	\[ 
	y= \cos x \qquad y= \cos (2x) \qquad y= \cos (-2x)
	\]
	What does the graph of $y=\cos (Bx)$ look like?


% combinations
\item  For each of the graphs above (and below), identify the following:
	\begin{enumerate}
	\item  Domain
	\item  Range
	\item  Amplitude
	\item  Period
	\item  Phase shift (this is the horizontal shift)
	\end{enumerate}

\item  Try to relate the next graphs to the graphs above
	\[
	y = \sin(3x-\pi/4)
	\]
	(It is usually easier to see what happens if you rewrite this as $y=\sin(3(x-\pi/12))$,
	try to rewrite these in this manner.)
\item 
	\[
	y = 2\cos(4x+\pi)
	\]
\item  
	\[
	y = -2\cos(\pi x)
	\]
\item 
	\[
	y = 4 \sin(2\pi x - \pi/2)
	\]
	

\end{enumerate}


Hints and things to notice:
\\
Try to ``break the function down'' into the different things happening.
For example, consider:
\[
y = -4 \sin(3x+\pi)
\]
We first rewrite it as:
\[
y =  -4 \sin(3(x+\pi/3))
\]
Starting at the inner parenthesis, we have $(x+\pi/3)$, this is similar to problems~\ref{hshift1} and~\ref{hshift2}.
Then, the next thing out is the $3$, which is similar to problems~\ref{hstretch1} and~\ref{hstretch2}.
Then we have the $\sin$ and the $4$ which lands us at problems~\ref{vstretch1} and~\ref{vstretch2}.
The graph of $y=-4 \sin(3(x+\pi/3))$ should be a combination of the transformations made.

With some practice you should be able to do these graphs without a calculator.













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