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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{More Practice on Sinusoids -- September 18, 2017}
\vskip 10pt
\noindent
{\it Suggestion}
\vskip 10pt
Try at least a few of the following problems \emph{before}
consulting the solutions that will be posted on the course 
homepage.
\vfill\eject
\noindent
I.  Plot each of the following sinusoid graphs by determining
the amplitude, period, and vertical shift:
\begin{enumerate}
\item[(A)]  $y = 4 \sin(10 x) - 1$.  Show the portion of the graph between
$x = 0$ and $x = 2\pi/5$.

\item[] {\it Solution:}  Amplitude is 4, vertical shift is $-1$ and
 $\frac{2\pi}{\rm period} = 10$, so the period is $\frac{\pi}{5}$.  
\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidIA.png}
\end{center}!
\caption{The graph $y = 4\sin(10 x) - 1$ from I (A).}
\end{figure}

\item[(B)]  $y = 2\cos(\pi x) + 2$.  Show the portion of the graph between
$x = 0$ and $x = 4$.

\item[] {\it Solution:} Amplitude is $2$, period is $\frac{2\pi}{\pi} = 2$, vertical shift
is 2.  There are two complete periods for $0 \le x \le 4$.  
\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidIB.png}
\end{center}
\caption{The graph $y = 2\cos(\pi x) + 2$ from I (B).}
\end{figure}

\item[(C)]  $y = -3\sin(x) + 4$.  Show the portion of the graph between $x=-2\pi$
and $x = 2\pi$. 

\item[] {\it Solution:}  The amplitude is $3$, but the $-3$ means the graph is
also \emph{reflected across the $x$-axis} before it is shifted up by $4$.  The period
is $2\pi$, as for the standard $\sin(x)$ graph.
\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidIC.png}
\end{center}
\caption{The graph $y = -3\sin(x) + 4$ from I (C).}
\end{figure} 
\end{enumerate}
\vskip 10pt
\noindent
II.  Find possible formulas for each of the following sinusoidal graphs (starting on back):
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\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidA.png}
\end{center}
\caption{Sinusoid II (A) is $y = 2\sin(4x) + 1$}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidB.png}
\end{center}
\caption{Sinusoid II (B) is $y = 3\cos(\pi x) - 5$}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidC.png}
\end{center}
\caption{Sinusoid II (C) is $y = -4\sin(3x)$}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidD.png}
\end{center}
\caption{Sinusoid II (D) is $y = \sin(2\pi x) + 3$ }
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{SinusoidE.png}
\end{center}
\caption{Sinusoid II (E) -- more challenging(!) is $y = 3\sin(2(x - \pi/4)) + 1$}
\end{figure}

\end{document}
