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%TCIDATA{<META NAME="Title" CONTENT="Getting Into Polynomial Shape">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Mar 26 00:05:01 1997}
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\input{tcilatex}
\begin{document}


\section{Getting Into Polynomial Shape}

Graphing a function, even with the aid of technology, requires a thoughtful
approach. The default viewing rectangle is rarely the view that provides the
most information. With practice, you will quickly adjust the horizontal and
vertical components of the viewing rectangle to generate a viewing rectangle
that clearly shows important aspects of the function.

\strut

\begin{quotation}
What are the important aspects? Here are a few you might wish to consider:
\end{quotation}

\begin{itemize}
\item  \textsl{Domain:}{\small \ Especially if the domain is restricted
because of such things as square roots or vertical asymptotes.}

\item  \textsl{Intercepts:} {\small Does the graph cross the }$y${\small %
-axis? If so, where? Does the graph cross the }$x${\small -axis? If so,
where?}

\item  \textsl{Symmetry:}{\small \ Is the function even? Odd? Periodic?}

\item  \textsl{Asymptotes:} {\small Are there any horizontal asymptotes? V%
\hspace{-0.02in}ertical asymptotes? Slant asymptotes?}

\item  \textsl{Intervals of Increase or Decrease:} {\small On which
interval(s) is the function increasing? Decreasing?}

\item  \textsl{Local Maximum and Minimum Values:} {\small Are there any
extreme values? If so, where?}

\item  \textsl{Concavity and Points of Inflection:} {\small On which
interval(s) is the graph concave upward? Downward? Where are the points of
inflection?}
\end{itemize}

\begin{quotation}
If the domain is unrestricted, so that the function is defined for all real
numbers, it is not practical to use a viewing window that actually shows $%
-\infty <x<\infty $. However, you should attempt to find a viewing rectangle
that includes at least part of the infinite intervals $-\infty <x<a$ or $%
b<x<\infty $ where the graph is monotone. Of course, for particular
functions, such as periodic functions, this will not be possible.

Occasionally, a single viewing rectangle may not show all of the important
aspects. Zooming in too far may lose large-scale features of the graph, and
zooming out may obscure small-scale features. Two or more views may be
required to adequately describe some functions. The following activities are
designed to give you practice at selecting appropriate viewing rectangles
and gathering information about functions by looking closely at their
graphs.\medskip 
\end{quotation}

Plot a graph of $f(x)=4x^{4}-7x^{2}+4x+6$ using a viewing rectangle that
shows the important aspects. Define the function $f(x)=4x^{4}-7x^{2}+4x+6$
and locate the extreme values precisely by solving exactly or numerically
the equation $f^{\prime }(x)=0$. Locate the inflection points precisely by
solving exactly or numerically the equation $f^{\prime \prime }(x)=0$. Mark
each local high point and low point on your plot using a diamond $\Diamond $%
. Mark each inflection point with a circle $\circ $. State the interval(s)
where $f$ is increasing. State the interval(s) where the graph is concave
upward.

\subsection{Tane's Solution}

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Let $f(x)=4x^{4}-7x^{2}+4x+6$. Solve the equation $f^{\prime }(x)=0$ to get
the critical numbers $-1.\,0545$, $.\,32492$, and $.\,72963$. Solve the
equation $f^{\prime \prime }(x)=0$ to locate the possible inflection points $%
(-.\,54006,f(-.\,54006))$ and $(.\,54006,f(.\,54006))$. In the following
plot of $f$, the extreme values are located with small diamonds and the
inflection points with small circles:\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{%
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\TEXUX{$4x^{4}-7x^{2}+4x+6$};linecolor "green";linestyle 1;linethickness
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-1.\,0545,-1.\,0559,.\,32492,6.\,6053,.\,72963,6.\,3256\right] $};linecolor
"black";linestyle 1;linethickness 1;pointstyle "diamond";pointplot
TRUE;function \TEXUX{$\left[ -.\,54006,2.\,1384,.\,54006,6.\,4589\right]
$};linecolor "black";linestyle 1;linethickness 1;pointstyle
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{\small 
\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{{\small With the insertion point in the equation $%
f(x)=4x^{4}-7x^{2}+4x+6$, choose Define + New Definition. With the insertion
point in the expression $f(x)$, choose Plot 2D + Rectangular. With the
insertion point in the equation $f^{\prime }(x)=0$, choose Solve + Numeric.
Drag to the frame 
\[
\left[ -1.\,0545,f\left( -1.\,0545\right) ,.\,32492,f\left( .\,32492\right)
,.\,72963,f\left( .\,72963\right) \right] 
\]
With the insertion point in the equation $f^{\prime \prime }(x)=0$, choose
Solve + Numeric. Drag to the frame 
\[
\left[ -.\,54006,f\left( -.\,54006\right) ,.\,54006,f\left( .\,54006\right)
\right] 
\]
Choose Edit + Properties, click the Plot Components tab, choose Item Number
2, and set Plot Style to Point and Point Symbol to Diamond. Choose Item
Number 3 and set Plot Style to Point and Point Symbol to Circle. Choose OK%
\textsf{.}}}}

The function $f$ is increasing on $-1.0545\leq x\leq .32492$ and on $%
.72963\leq x<\infty $. The graph is concave upward on $-\infty <x<-.54006$
and on $.54006<x<\infty $.

\subsection{Activities}

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\begin{enumerate}
\item  Find an equation for each of the following graphs of polynomials.
Verify your solution by using a viewing rectangle that displays a graph that
looks like the given graph. (Solutions may not be unique.) Describe the
techniques you used to obtain your equations.
\end{enumerate}

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\end{center}

\begin{enumerate}
\item[2.]  Sketch a graph of $f(x)=\sqrt{x}+\sqrt{1-x}$ and discuss the
important aspects of $f$. In particular, discuss the domain$.$ Drag the
expressions $\sqrt{x}$ and $\sqrt{1-x}$ to the frame and explain how the
curves $y=\sqrt{x}$ and $y=\sqrt{1-x}$ are related to the graph of $y=$ $%
\sqrt{x}+\sqrt{1-x}$.

\item[3.]  Investigate the graph of $f(x)=x^{\sin x}$. What is the domain of 
$f$? In what ways is the graph similar to the graph of $y=x^{x}$? In what
ways is the graph similar to the graph of $y=\sin x$? What is an appropriate
value for $f(0)$? Locate 3 critical numbers. Locate three inflection points. 
\label{Applied max-min}
\end{enumerate}

\end{document}
