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\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Lab \#1 Solutions} \\
Professor Hwang \\
\end{center}

\subsection*{Function Transformations}

The first three lines of the Maple worksheet define functions $f$,
$f_1$~and $f_2$, while the next two lines plot these functions in
pairs. 

\prob{1.} On the grid below, sketch the graphs of the three functions
$f$, $f_1$,~and $f_2$, clearly labeling each.
\begin{center}
\input{lsoln01a.eepic}
\end{center}

\prob{2.} Let $c$~be a constant. Give a brief verbal description of
the graph $y=f(x-c)$ in terms of the graph $y=f(x)$. If necessary, try
modifying the definition of~$f_1$ and re-plotting to see the effect of
``shifting'' the input variable.

\bigskip

\soln The graph $y=f(x-c)$ is the graph $y=f(x)$ shifted to the right
by~$c$. (If $c<0$, this statement may be interpreted as ``shifted to
the left by $|c|=-c$''.)

\vfil\eject

The next few lines of the worksheet define and plot functions
$f_3$~and $f_4$ obtained by adding or subtracting constants from~$f$.

\prob{3.} On the grid below, sketch the graphs of $f$, $f_3$,~and $f_4$,
clearly labeling each.
\begin{center}
\input{lsoln01b.eepic}
\end{center}

\prob{4.} Let $c$~be a constant. Give a brief verbal description of
the graph $y=f(x)+c$ in terms of the graph $y=f(x)$.
\bigskip

\soln The graph $y=f(x)+c$ is the graph $y=f(x)$ shifted up
by~$c$. (If $c<0$, this statement may be interpreted as ``shifted down
by $|c|=-c$''.)

\vfil\eject

The next few lines of the worksheet define and plot functions
$f_5$~and $f_6$ obtained by multiplying~$f$ by a constant.

\prob{5.} On the grid below, sketch the graphs of $f$, $f_5$,~and $f_6$,
clearly labeling each.
\begin{center}
\input{lsoln01c.eepic}
\end{center}

\prob{6.} Let $c$~be a constant. Give a brief verbal description of
the graph $y=cf(x)$ in terms of the graph $y=f(x)$.

\bigskip

The graph $y=cf(x)$ is the graph $y=f(x)$ stretched vertically by a
factor of~$c$ when $c>0$. If $0<c<1$, ``stretching'' means
``squashing''. When $c<0$, the effect is a vertical stretch by a
factor of~$|c|=-c$ followed by reflection through the $x$-axis.

\vfil\eject

\subsection*{Sinusoidal Graphs}

Use the next few lines of the Maple worksheet to plot $y=\cos x$ and
$y=\cos^2 x$ on the same set of axes. 

\prob{7.} Using the fact that both graphs are sinusoidal, find a
formula for $\cos^2 x$ in terms of $\cos x$~and $\sin x$. Emailed:
Express your answer in the form $y=D\pm A\cos(Bx)$ or $y=D\pm A\sin(Bx)$.

\bigskip

\soln $\cos^2 x=\frac{1}{2}+\frac{1}{2}\cos 2x$.

\prob{8.} The next lines of the worksheet define and plot the
functions $y=\sin x$ and $y=\sin^2 x$. Using the fact that both graphs
are sinusoidal, find a formula for $\sin^2 x$ in terms of $\cos x$~and
$\sin x$. Use this formula together with the previous question to
verify that $\sin^2 x + \cos^2 x=1$. Emailed: Express your answer in
the form $y=D\pm A\cos(Bx)$ or $y=D\pm A\sin(Bx)$.


\bigskip

\soln $\sin^2 x=\frac{1}{2}-\frac{1}{2}\cos 2x$. When this is added to
$\cos^2x$, the terms $\pm\frac{1}{2}\cos 2x$ cancel, leaving
$\frac{1}{2}+\frac{1}{2}=1$. 

\subsection*{Large-Scale Behavior of Polynomials}

The next line of the worksheet defines a function~$g$:
\[
f(x) = 0.125x(x+1)(x-2) = 0.125(x^3-x^2-2x),\qquad g(x) = 0.125x^3.
\]

\prob{9.} Use the worksheet to plot $f$~and $g$ on the
interval~$[-1,1]$. Change the plot interval to $[-3,3]$, then to
$[-10,10]$~and $[-30,30]$. What do you observe as the intervals get
larger? Briefly explain why this is happening.

\bigskip

\soln The graphs become more and more nearly the same. The reason is
that the difference $f(x)-g(x)$ is of degree~2, so for large~$|x|$ is
small compared to $f(x)$~and $g(x)$, and is not readily visible when
$f(x)$~and $g(x)$~are large.

Now that we have the language of limits, the previous paragraph may
be summarized by saying
\[
\lim_{x\to\pm\infty}\frac{f(x)}{g(x)}=1\qquad\text{or}\quad
\lim_{x\to\pm\infty}\frac{f(x)-g(x)}{f(x)}=0.
\]

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