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
1994-1995 by Northeastern University.
 
1. The 2 Graphics Windows
 
    In the upper right the graph of the function is shown in
purple (increasing) and green (decreasing), along with a plot
of the derivative (orange). It is 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 line 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. If you move the
mouse cursor up and down within the mapping diagram, 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) are displayed along the top of the
screen, and the points (x,0), (x, f(x)) and (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 applications.)
 
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, Rescaling and Zooming
 
    By clicking on the appropriate boxes, you can manually set
the domain and range of the function. 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.)
 
    On the right of the screen are boxes labeled "Zoom" and
"Rescale". Rescaling is simple: 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. Zooming
is a little more complicated: its function is to preserve
scale by making the domain and range intervals have the exact
same length. Here's what Zoom does. Suppose the domain is
[a, b], L = (b-a)/2, and Min and Max are as just described.
Let Mid = (Min + Max)/2. Zoom sets the range to [Mid - L/2,
Mid + L/2]. Zooming is useful for examining how a function
expands or shrinks distances.
 
4. Point Trails
 
    Clicking the Point Trail button on the right 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. To get out of Point Trail mode, click the Point
Trail button again.
 
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. Click
outside the control boxes to stop animation. Click on Stepped
Interval, then on one of the two arrows: < or >, to see the
animation one frame at a time, backward or forward.
 
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", 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.