The Function Visualizer
                   _______________________

    The Function Visualizer (FV) is a software implementation
of mapping diagrams, with the added feature of animation. FV
was created by Mark Bridger (Northeastern University) and
Hubert Hohn (Massachusetts College of Art), with partial
support from the National Science Foundation. FV may be copied
and distributed for non-commercial purposes, and is copyright
1994-1996 by Northeastern University.

Copies of the Visualizer, for either the Mac or PC, can be
downloaded from Northeastern's FTP site at: WWW.MATH.NEU.EDU,
or from the Math Archives. (Northeastern's site will always
have the latest version.)


0. Starting and Exiting

    To start from DOS, run the executible program FV.EXE.
    To start from Macintosh, double click the icon.
    To exit from either, click on the small box in the upper
    left-hand corner.


1. The 2 Graphics Windows

    In the upper right the graph of the function is shown in
purple (increasing) and green (decreasing). The window will
also display a graph of the derivative, in orange, if you
click on the box labeled: Show Derivative. Click on the box
again (which now reads Hide Derivative) to turn off the
derivative plot. When the derivative is turned on, a symbolic
representation of the derivative will be displayed at the
bottom of screen. The plot is always scaled to fit the
function, so the derivative may go out of view.

    On the left is the mapping diagram for the function: the
left vertical bar represents points in the domain, the right
vertical line points in the range. A large number of lines are
drawn in, connecting equally spaced points x in the domain
with their image points f(x) in the range. Lines are draw from
each domain point x; if f(x) does not lie on the
part of the range shown, this "mapping line" is clipped off
where it exits the mapping window. If the function is not
defined at a particular point, the mapping line will not be
drawn. If you move the mouse cursor up and down within the
mapping window, the point x on the horizontal level of the
cursor is highlighted, along with the line joining it to f(x).
Simultaneously, the values of x, f(x) (and f'(x) if Show
Derivative is turned on) are displayed along the top of the
screen, and the lines joining (x, 0) with (x, f(x)) and
(x, f(x)) with (0, f(x)) are shown in the graph at the upper
right.

    If you move the mouse cursor from the very bottom to the
very top of the mapping diagram, at a constant upward
velocity, you can watch how the images of the points vary.
These images will move quickly where the function is either
increasing or decreasing rapidly.

(In the mapping diagram, the vertical line representing the
domain has horizontal coordinate 0; the range line has
horizontal coordinate 1. As you move the mouse cursor, its
horizontal coordinate, a number between 0 and 1, is also
displayed at the top of the screen. This is useful for certain
technical analyses of mapping diagrams; see the author's
paper: Dynamic Function Visualization.)


2. The Function

    The default function is x-->sin(x), which is displayed in
a box just below the mapping diagram. Clicking on this box
enables you to edit (just type, backspace, etc.) or delete
(press [Esc]) and enter a new function. You must use x for the
variable, and * for multiplication. If you make a syntax
error, it will be pointed out for correction. Various special
functions are available for your use, and very complicated
functions can be built from them. You can also type in "pi".
Here is an example of a function that can be entered:
exp(sqrt(2*x-x^2)) - tan(pi+ln(x)/12.779). Arbitrary powers are
entered using "^", but positive integer roots are best entered
using the "rootn" function; for example, the cube root of 1/x
would be entered as: root3(1/x).


3. The Domain and Range

    By clicking on the boxes to the upper left and lower left
of the mapping diagram window, you can reset the domain of the
function by entering new numbers. You can erase the current
number by pressing [Esc]. The upper right and lower right
boxes set the range. In the present version, the range is
restricted to [-1000, 1000] but this will probably be changed
in future versions. (By multiplying your function by a
suitable scaling factor you can get around this restriction.)

    You can also change the domain by dragging the small
horizontal sliders on the domain bar: put the mouse pointer on
one of them, hold down the mouse button and drag it up or
down; release the mouse button at the desired setting.

    The effects of changing the range and domain are not
immediately displayed. To redraw with the changed setting(s),
you click on the commands in the window in the lower right:

                  *************************
                  *<Step * Animate * Step>*
                  *************************
                  * Clear * Point Trail   *
                  *************************
                  *  Redraw   *   Equi    *
                  *************************
                  *  Center   * Rescale   *
                  *************************
                  *Width*--|--*Speed*--|--*
                  *************************
                  *     Screen Dump       *
                  *************************

  * Redraw will replot both graphics windows using the
current function, domain and range. Note that the domain and
range may have different scales, and their origins (0) may not
be at the same horizontal level. Furthermore, for an x in the
domain, f(x) will not appear on the screen if it lies outside
the established range; in the case, the mapping line will be
clipped.

  * Rescaling changes the range interval so that it exactly
encompasses the image of the domain. The program estimates,
for the given domain, the Min and Max of the function on that
domain, sets the range to [Min, Max], and redraws everything.

  * Equi sets the range exacly equal to the domain, then
redraws both windows. This is useful when iterating functions,
or when it is desirable to see the exact amount of scaling and
shifting performed by the function. In this case, domain and
range have exactly the same scale, and each number s in the
domain will align horizontally with s in the range.

  * Centering is a little more complicated: its function is to
preserve scale by making the domain and range intervals have
the exact same length. However, unlike Equi, Centering makes
sure that the new range interval actually contains points in
the image of the function. Here's what Center does. Suppose the
domain is [a, b], L = (b-a)/2, and Min and Max are as just
described. Let Mid = (Min + Max)/2. Center sets the range to
[Mid - L/2, Mid + L/2]. Centering is useful for examining how a
function expands or shrinks distances (scaling). While the
domain and range have the same scale, their actual values do
not necessarily line up horizontally as they do after the Equi
command.

4. Point Trails

    Clicking the Point Trail puts you in point trail mode. If
you now click anywhere in the mapping diagram, the point x on
the horizontal level of the mouse cursor will move toward
f(x), leaving a trail of dots. You can do this over and over.
To erase these trails of dots, click on the Clear box; this
will also get you out of Point Trail mode.

5. Animation

    The most novel feature of FV is its ability to have ALL
the points in the domain move toward their images. What makes
this useful and interesting as that they don't all move at the
same speed. Each point x moves rightward on a line toward
f(x), at a horizontal speed given by: k + K*|f'(x)|. The
constant k, which is a minimal speed, is set by sliding or
dragging the indicator in the Speed box on the right. The
constant K, which represents the spread of speeds is set by
sliding the indicator in the Width box. The bigger the
absolute value of the derivative at x, the faster x moves
rightward toward f(x). Points where |f'| is largest move
fastest, and arrive at the range line soonest. Critical points
(where f'(x) = 0), of course, travel slowest.

    Click on Animate Interval to see this animation. Holding
down on the Mouse Button causes animation to pause, while
clicking on either Step button stops animation. Click on Step
with one of the two arrows, < or >, to see the animation one
frame at a time, backward or forward, or hold down the mouse
button for slower animation. (Backward motion shows the
behavior of the inverse function.)


6. Screen Dump

    If you have a Macintosh, or DOS computer connected to a
LaserJet or compatible printer, you can send a copy of the
current screen image to be printed. MAKE SURE YOUR PRINTER IS
CONNECTED AND ONLINE. Simply click the Screen Dump button.


    These are the main features of FV, the Function
Visualizer. Other pedagogical and mathematical features are
discussed in "Dynamic Function Visualization" (College Math
Journal), a paper by Mark Bridger. Comments and questions
should be addressed to Prof. Mark Bridger, Mathematics
Department, Northeastern University, Boston MA 02115, or
BRIDGER@NEU.EDU.