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%TCIDATA{<META NAME="Title" CONTENT="Functions and Their Graphs">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Feb 14 00:08:17 1997}
%TCIDATA{CSTFile=Math.cst}
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\begin{document}


\section{Functions and Their Graphs}

These paragraphs offer experience in techniques for doing mathematics with 
\textsl{Scientific Notebook} that you will use in most of the activities.
Additional hints will be given within each activity, as needed. Press 
\textsf{ctrl} and click the \textsf{Hint} box.

\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
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{%
We will describe \textsl{Scientific Notebook} commands in terms of the menu
choices.
\par
For keyboard and toolbar shortcuts for choosing commands, choose \textsf{%
Help.}}

\subsection{Entering Mathematics}

\begin{quote}
$\blacktriangleright $ To enter the polynomial $3x^{2}+x-5$
\end{quote}

\begin{itemize}
\item  Choose \textsf{Insert + Math} to toggle to mathematics mode. (On the 
\textsf{Math Toolbar}, you will see the button \FRAME{itbpF}{0.3001in}{%
0.3001in}{0.0701in}{}{}{math.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
0.3001in;height 0.3001in;depth 0.0701in;original-width
20.3125pt;original-height 18.8125pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Math.wmf';file-properties "XNPEU";}} when in
mathematics mode.)

\item  Type $3x$ (This should appear in red on your screen.)

\item  Choose \textsf{Insert + Superscript}. Type $2$ in the input box.

\item  After entering the superscript, press the \QTR{KeyName}{spacebar} to
return to the line and type the rest of the expression: $+\;x-5$.
\end{itemize}

\ 

When you enter mathematics, \textsl{Scientific Notebook} handles the spacing
for you. Pressing the spacebar while in mathematics may change the level of
the insertion point, but it will not make spaces.

\begin{quote}
$\blacktriangleright $ To enter the rational expression $\frac{3x^{2}+x-5}{%
2x^{2}-7}$
\end{quote}

\begin{itemize}
\item  Choose \textsf{Insert + Fraction} (This automatically places you in
mathematics mode.)

\item  Type the polynomial $3x^{2}+x-5$ in the upper input box.

\item  Press \QTR{KeyName}{tab} or an arrow key to move to the lower input
box.

\item  Type the polynomial $2x^{2}-7$ in the lower input box.

\item  Press \QTR{KeyName}{spacebar} to exit the fraction.
\end{itemize}

\begin{quote}
$\blacktriangleright $ To enter the expression $\sqrt{\frac{3x^{2}+x-5}{%
2x^{2}-7}}$
\end{quote}

\begin{itemize}
\item  Select the expression $\frac{3x^{2}+x-5}{2x^{2}-7}$ with the mouse.

\item  Choose \textsf{Insert + Radical}.
\end{itemize}

\subsection{Plotting Expressions}

\begin{quote}
$\blacktriangleright $ To plot a polynomial
\end{quote}

\begin{itemize}
\item  Enter a polynomial expression (say $3x^{2}-5x+4$).

\item  Leave the insertion point inside or to the right of the polynomial
expression (or select the expression with the mouse).

\item  Choose \textsf{Maple + Plot2D + Rectangular} (or click the \textsf{2D
Plot} button on the compute toolbar).
\end{itemize}

The result should be:

$3x^{2}-5x+4$\FRAME{itbpF}{2.2917in}{1.5281in}{0in}{}{}{}{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.2917in;height 1.5281in;depth
0in;display "PICT";function \TEXUX{$3x^{2}-5x+4$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";xviewmin "-5";xviewmax "5";yviewmin "-0.1249";yviewmax "106.1";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}

\strut

\begin{quote}
$\blacktriangleright $ To apply a command to a mathematical expression
\end{quote}

\begin{itemize}
\item  Place the insertion point to the right of, or inside of, the
expression (or select the expression with the mouse) and choose the command
from the Maple menu or from a toolbar.
\end{itemize}

From now on we will often assume this to be understood, abbreviating the
entire list of instructions in the previous example as

\begin{itemize}
\item  \textsf{Plot2D + Rectangular} $3x^{2}-5x+4$
\end{itemize}

Remember that to be recognized as a mathematical expression, the contents of
the expression must all be in mathematics (i.e. appear in red or gray on the
screen) and not be separated by text spaces.

\begin{quote}
$\blacktriangleright $ To change settings in a plot
\end{quote}

Select the plot and choose \textsf{Edit + Properties} (or click the
miniature dialog box icon that appears in the lower right-hand corner of the
plot) to obtain a tabbed dialog. Make changes in the pages of the dialog and
choose \textsf{OK}.

\begin{quote}
$\blacktriangleright $ \label{Zoom tools}To change plot settings in a 2D
plot with the plotting tools
\end{quote}

\begin{itemize}
\item  Double click the plot. Three plotting tools (\QTR{Terminology}{Pan}, 
\QTR{Terminology}{Zoom in}, and \QTR{Terminology}{Zoom out}) appear in the
upper right corner of the plot.

\begin{itemize}
\item  \textsf{Pan}: with the hand selected, place the mouse cursor over the
plot, and press and hold the mouse button. Move the mouse to pan left or
right. This will change the domain coordinates of the plot setting.

\item  \textsf{Zoom in}: with the large mountain selected, place the mouse
cursor over a point in the plot, press and hold the mouse button and drag
across and down. This will create a box and the contents of the box will
expand to fill the frame.

\item  \textsf{Zoom out}: with the small mountain selected, place the mouse
cursor over a point in the plot, press and hold the mouse button and drag
across and down. This will create a box and the contents of the frame will
shrink to fill the box, bringing in additional parts of the graph to fill
the frame.
\end{itemize}
\end{itemize}

The number of degrees of freedom for the zoom boxes depends on the plot
settings and the type of plot. For a rectangular plot, the default is one
degree of freedom (the domain interval) with the range interval determined
automatically by the underlying system. You can change this default on the 
\textsf{Axis \& View} page of the \textsf{Plot Properties} dialog.

\begin{quote}
$\blacktriangleright $ To plot the expression $\frac{3x^{2}+x-5}{2x^{2}-7}$
\end{quote}

\begin{itemize}
\item  With the insertion point in the expression\textsf{\ }$\frac{3x^{2}+x-5%
}{2x^{2}-7}$, choose \textsf{Maple + Plot2D + Rectangular}.
\end{itemize}

The result should be: $\frac{3x^{2}+x-5}{2x^{2}-7}$\FRAME{itbpF}{3in}{%
2.0003in}{0in}{}{}{Plot }{\special{language "Scientific Word";type
"MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "PICT";function
\TEXUX{$\frac{3x^{2}+x-5}{2x^{2}-7}$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-5";xmax "5";xviewmin
"-5";xviewmax "5";yviewmin "-42740";yviewmax "1006";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};}}\bigskip

\begin{quote}
$\blacktriangleright $ To get a better view of this plot
\end{quote}

\begin{itemize}
\item  Leave the insertion point to the right of the plot, and choose 
\textsf{Edit + Properties} (or click the frame and then click the dialog box
tool that appears in the lower right corner of the frame).

\item  Choose the \textsf{Axis \& View} page, click the \textsf{View
Intervals Default} box to remove the check, and change the settings on the
y-interval to $-15<y<15$.
\end{itemize}

This gives the following view.\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{Plot }{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";function
\TEXUX{$\frac{3x^{2}+x-5}{2x^{2}-7}$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-5";xmax "5";xviewmin
"-5";xviewmax "5";yviewmin "-15";yviewmax "15";viewset"XY";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}

\begin{quote}
$\blacktriangleright $ To add an expression to a plot
\end{quote}

\begin{itemize}
\item  From the \textsf{Plot Components} page of the \textsf{Plot Properties}
dialog, choose \textsf{Add Item} and type the expression in the \textsf{%
Expressions and Relations} box
\end{itemize}

- or -

\begin{itemize}
\item  Select the expression and drag it to the plot with the mouse.
\end{itemize}

\begin{quote}
$\blacktriangleright $ To plot several expressions at once
\end{quote}

\begin{itemize}
\item  Start with the expressions in a mathematical list. The expressions in
the list must be separated by red commas, with no text spaces.

\item  Place the insertion point in the mathematics expression $%
x^{2},2x+1,x^{3}-2x$ and choose \textsf{Plot2D + Rectangular.}
\end{itemize}

$x^{2},2x+1,x^{3}-2x$ \FRAME{itbpF}{3in}{2.0003in}{0in}{}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0in;display "PICT";function
\TEXUX{$x^{2},2x+1,x^{3}-2x$};linecolor "black";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$x^{2},2x+1,x^{3}-2x$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";xviewmin "-5";xviewmax "5";yviewmin "-50";yviewmax
"50";viewset"XY";rangeset"X";phi 45;theta 45;plottype 4;numpoints
49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\subsection{Defining Expressions and Functions}

Some functions, such as $\sin x$, $\ln x$, and $e^{x}$ are automatically
recognized by \textsl{Scientific Notebook} when entered in mathematics mode.
You can also create temporary user-defined expressions and functions.

\begin{quote}
$\blacktriangleright $ To give an expression a name
\end{quote}

\begin{itemize}
\item  Place the insertion point in the equation $z=ax+b$ (or select the
equation with the mouse).

\item  From the \textsf{Maple} menu, choose \textsf{Define + New Definition}
(or choose the \textsf{New Definition} button from the \textsf{Compute}
toolbar).
\end{itemize}

\begin{quote}
$\blacktriangleright $ To create a user-defined function
\end{quote}

\begin{itemize}
\item  Place the insertion point in the equation $p(x)=ax+b$ (or select the
equation with the mouse).

\item  From the \textsf{Maple} menu, choose \textsf{Define + New Definition}.
\end{itemize}

\begin{quote}
$\blacktriangleright $ To evaluate a defined function
\end{quote}

\begin{itemize}
\item  Place the insertion point in an expression of the form $p(3)$.

\item  From the \textsf{Maple} menu, choose \textsf{Evaluate}.
\end{itemize}

If you have defined $p(x)=ax+b$, then the result should be $p(3)=3a+b$.

\begin{quote}
$\blacktriangleright $ To define a \label{Generic function}generic function
\end{quote}

\begin{itemize}
\item  Place the insertion point in the expression $f(x)$.

\item  From the \textsf{Maple} menu, choose \textsf{Define + New Definition}.
\end{itemize}

The expression $f(x)$ will then be recognized as a mathematical function
rather than simply a sequence of four characters. For example, evaluating $%
p(f(x))$ and $f(p(x))$ with $p$ and $f$ defined as above, gives the results 
\[
p(f(x))=\allowbreak af\left( x\right) +b\text{ and }f(p(x))=\allowbreak
f\left( ax+b\right) 
\]

\begin{quote}
$\blacktriangleright $ To view the list of active user-defined expressions
and functions
\end{quote}

\begin{itemize}
\item  Choose \textsf{Maple + Define + Show Definitions} (or click \textsf{%
Show Definitions} on the \textsf{Compute} toolbar).
\end{itemize}

\begin{quote}
$\blacktriangleright $ To remove all active user-defined definitions
\end{quote}

\begin{itemize}
\item  Choose \textsf{Maple + Define + Clear Definitions}
\end{itemize}

\begin{quote}
$\blacktriangleright $ To remove one user-defined definition
\end{quote}

\begin{itemize}
\item  Select the function or expression and choose \textsf{Maple + Define +
Undefine.}
\end{itemize}

-or-

\begin{itemize}
\item  Make a new definition using the same function or expression name.
\end{itemize}

\begin{quote}
$\blacktriangleright $ To make a table of values\label{Table of values}
\end{quote}

\begin{itemize}
\item  Select the function or expression and choose \textsf{Maple + Define +
New Definition.}

\item  Choose \textsf{Insert + Matrix} and specify $6$ rows (or whatever
number of values you want) and $1$ column. Fill it in with $x$-values near $%
0 $. For this example we use $0.1$, $0.01$, $0.001$, $-0.001$, $-0.01$, $%
-0.1 $. (As you type these in the matrix, you can \textsf{tab} to move
between the entries.)

\item  Select the matrix with the mouse and choose \textsf{Insert + Brackets}
(or click a brackets button); then place the function name (in mathematics
mode) just to the left of the brackets. $f(x)=x^{2}$

\item  With the insertion point in the expression $f\left( 
\begin{array}{l}
\;0.1 \\ 
\;0.01 \\ 
\;0.001 \\ 
-0.001 \\ 
-0.01 \\ 
-0.1
\end{array}
\right) $, choose \textsf{Evaluate}.
\end{itemize}

\begin{quote}
$\blacktriangleright $ To plot a defined function
\end{quote}

\begin{itemize}
\item  \textsf{Define + New Definition }$f(x)=3x^{2}-5x-1$

\item  \textsf{Plot 2D + Rectangular }$f(x)$
\end{itemize}

-or-

\begin{itemize}
\item  \textsf{Plot 2D + Rectangular }$f$
\end{itemize}

\FRAME{dtbpF}{2.0937in}{1.3681in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 2.0937in;height 1.3681in;depth 0pt;display
"PICT";function \TEXUX{$3x^{2}-5x-1$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-5";xmax "5";xviewmin
"-5.2";xviewmax "5.204";yviewmin "-5.125";yviewmax "101.1";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}

\emph{Note}{\small \quad You can use either an expression, such as }$f(x)$%
{\small \ or }$3x^{2}-5x-1${\small , or the function name }$f${\small \ to
obtain a plot of the function }$f(x)=3x^{2}-5x-1.$

If you place the insertion point in an equation, such as $f(x)=3x^{2}-5x-1$
or $y=3x^{2}-5x-1$, and ask for a 2D rectangular plot, \textsl{Scientific
WorkPlace} will select the expression that is on the same side of the equals
sign as the insertion point, not the entire equation. \textsl{Scientific
WorkPlace} selects equations only for special types of plots, such as
implicit plots.\medskip

\begin{itemize}
\item  \textsf{Plot 2D + Rectangular} (insertion point to the left of the
equals sign)
\end{itemize}

$y=3x^{2}-5x-1$\FRAME{itbpF}{1.7573in}{1.1545in}{0in}{}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 1.7573in;height
1.1545in;depth 0in;display "PICT";function \TEXUX{$y$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";xviewmin "-5";xviewmax "5";yviewmin "-5.2";yviewmax "5.204";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{y};var1name
\TEXUX{$x$};}}

\begin{itemize}
\item  \textsf{Plot 2D + Rectangular} (insertion point on the right-hand
side)
\end{itemize}

$y=3x^{2}-5x-1$\FRAME{itbpF}{1.7573in}{1.1545in}{0in}{}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 1.7573in;height
1.1545in;depth 0in;display "PICT";function \TEXUX{$3x^{2}-5x-1$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";xviewmin "-5";xviewmax "5";yviewmin "-5.125";yviewmax "101.1";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}

\end{document}
