For the sake of convenience in reading
this document, the applet window should be moved to the right
half of the screen, and the two windows should be resized so that
they do not overlap.
When the applet is first run you
will get a screen called "Polynomial Interpolation",
the one to your right. It is a large version of the picture below
.
The column entitled "x y values"
represents five points in the plane: (-2, 2), (-1, 1), (0, -1),
(1, -1), and (2,0). When you click the "Process" button
the applet will construct and graph a fourth degree polynomial
going through these five points:
The "Display Points" toggles
the display of the listed points. If you click on it, the points
on the graph will be removed; if you click on it again the points
will return.
The fourth degree polynomial constructed by the program is
It is displayed in the panel at the bottom of
the screen. The coefficients are displayed to three decimal places,
but trailing 0's are not displayed. That is,-1.150 is written
out as -1.15.
If the coefficients of the polynomial are very
large (> 1000) then they are written out using the Ek notation
for powers of 10: 1532 = 1.532 E3; a negative k is employed for
small numbers, 0.001020 = 1.02E-3.
The panel directly above the polynomial displays
the x and y values of the mouse pointer when it is on the graphing
screen.
The numbers -2 and 2 at the bottom of the graphing
screen are the minimum and maximum x values plotted. The values
-2 and 1.17 at the left side of the graph are the minimum and
maximum of the polynomial on the domain [-2, 2]. The E notation
is also used here for large and small numbers.
The graphs of this program can be pasted into
other documents, e.g., Word. To do so place the mouse on the applet
screen and press Alt-PrintScreen. This places the screen into
the clipboard; it can then be pasted into the other document.
General Ideas
Generally speaking, given n points in the plane
, where 
are
n distinct numbers, then 
, the interpolating
polynomial for these n points is the function of the form
such that
The general ideas about these polynomials are
given in
Lagrange Interpolation.
Clearing the Screen
The screen to the right can be "cleared"
by first marking and deleting all the numbers in the list of (x,
y) values. This is done in the usual way, for the column is a
text field and all the usual word- processor commands can be executed
in it. After it is cleared, press the "Process" button;
this clears the screen.
Inputting Polynomials by the Keyboard
New polynomials are entered by first clearing the screen, typing in one pair of numbers per line in the "x y values" column, and then pressing "Process". For example, clear the (x,y) list and then enter:
Notice that the values can be typed
in any order; they need not be entered with the x values in increasing
order. The program arrange the points so that the x values are
in increasing order. Now, click on the process button; you will
get a parabola going through (-1, 1), (0,0) and (1,1):
There are two rules of syntax you must heed when entering values:
Numbers must be separated by at least one blank. You can, if you like, use several blanks as separators.
You cannot use commas or other symbols
as separators.
Using the Mouse to Enter Points
To use the mouse to enter points move the pointer to the screen and click it. The mouse location will then be added to end of the list of "x y values". Once you have added all the points you wish, enter click on the process button to display the graph. If you clear the screen before you start clicking, your polynomial will pass through the points you entered.
If you do not clear beforehand,
the points you click in will be added to the points already listed.
You may use any mix of keyboard
and mouse commands to construct a list of (x,y) values.
Points can be deleted from the graphs
by deleting then from the list of (x, y) values. The deletion
does not appear on the graph until the process key is pressed.
Graphing two or more polynomials
simultaneously
To see how to graph several polynomials
at the same time, click on the "Examples" menu at the
top, left of the applet screen, click on "Parabolic Arches",
and then, press the process button. A graph of four arches will
appear.
Looking at the (x,y) listing you
will see three appearances of the word "next", which
is used as a signal to stop the construction of one polynomial
and begin the construction of another. In this case we begin with
a parabola, and then translate it to the right three times.
The four different polynomials constructed
for this graph are displayed at the bottom of the screen. You
will have to scroll down to see the third and the fourth.
The colors used for the different
graphs are blue, orange, violet, and rose. If you graph five or
more polynomials the colors will cycle through this sequence.
Note also that if you use the mouse
to click in points, the values are added to the end of the list.
Thus, if you do not add a "next", the values you click
in will be added to the last set of values in the list.
Caution: If you are working with
a text window and the applet window at the same time you might,
in the process of using the mouse to move from one window to the
other, inadvertently click on the graphing window of the applet.
This will add a point to the list of (x, y) values and produce
a strange graph. So, if your graph has a unexpected form, scroll
down the list of (x, y) values and check to see what is there.
Examples
Example 1
Go to "Examples" , and
click in and process "Arches, 4th Degree".
The graphs you get here differ from
the parabolic arches in that two extra points have been added
to the definition of each polynomial, making them polynomials
of the fourth degree. The extra point for each graph, as you can
see, alters the shape of the arch.
Test your knowledge of calculus:
Looking at the definition of p1(x) and its graph, determine how
roots there are to the equation d(p1(x))/dx = 0. Then do the same
for the remaining three polynomials.
Example 2
Go to "Examples", click
in and process "Leaf".
Recall that a polynomial p(x) is
a function of x: for each x in the domain there
is exactly one associated value p(x). Thus the leaf-like figure
in the graph is not the graph of a single polynomial; it was formed
by pasting two polynomials together, using the "next"
command.
Problem: Draw a "stem"
to the leaf at its lower, left corner.
Example 3
Go to "Examples", click in "Primitive Polynomial Art" and then process, and toggle the "Display Points" button to get this happy face:
Problem: By changing one, single
point in the list of (x, y) values change happy face to a sad
face.
Example 4
This program was written under the
assumption that the user would enter two numbers on each line
of the (x, y) list to define a point in the plane. To see what
can happen if this is not done load the "Parabolic Arches"
in "Examples" and then delete the last 0 from each group
of points defining an arch. That is delete the 0s in the third,
seventh, eleventh, and fifteenth lines. Then process.
The following has occurred: The
first five numbers in the (x, y) list
have been replaced by the points
(0, 0), (1, 1) and (0, 0); the graph is the straight line from
the point (0,0) to (1,1) and back to (0,0).
The next five numbers
have been replaced by the points
(2, 0 ), (3, 1) and (0, 0). The graph is the parabola from the
point (0,0) to (2,0) and (3,1). (Remember, the points are arranged
so that the x values are in increasing order.)
A similar change occurs in the last
two parabolas.
Example 5
As another example of the behavior exhibited in Example 4, load the "Leaf" in "Examples".
Delete the 1 in the first row, the
8 in the sixth row, and then process.
Studying what has happened one sees
that the first set of values
have been grouped into three points
(0,1), (2,2), and (4,3) and appear as the blue line in the graph.
The last 8 has been ignored.
The second set of values
have been arranged as points (3,2),
(6.5, 1) (5, 0) plus the point (1,0). The extra 1 in this case
spawned the point at (0, 0).
In short, if the (x, y) list does
not contain two numbers on each line, between "next"s,
then the program groups all the numbers into pairs according to
the order in which they were entered. If there is a number leftover
after this grouping then the it may be ignored or it may be converted
to the point (0,0). The precise action the program takes is a
rather complicated function of the number of polynomials being
graphed and the position of the "missing" numbers.
THE MORAL: Be sure that each point
is defined by two numbers on each line!
An Amusement: Load examples, delete
numbers in the (x,y) list at random; see what you get
Example 6
In this example we begin by loading
Examples|Triangular Frame, which is a picture of three triangles.
The inner and outer
triangles form a frame; the inner
one is a guide to the center. The object is to construct a polynomial
that starts at (0, 0), ends at (2,0), and stays within the frame.
The file "Examples| Triangular Frame, part 2" gives
an example of such a polynomial
Looking at an enlarged picture of this last graph,
you may not be convinced that the
polynomial stays within the frame. If not, you
can get a more convincing picture by
changing the 0.5 and 0.5 in the (x,y) list to
0.6 and 1.4 and the 1.5 and 0.5 to 1.4 and 0.6.
You might also think of the polynomial
as "snaking" along the middle line and ask if you could
get a "better fit" by adding more points to the definition
of the polynomial. Try adding 0.8 and 0.8 as one point and 1.2
and 0.8 as another to see what happens.
Example 7
You try it. Load " Examples|Frame
Problem"
and then find a polynomial that
starts at (0,0), ends at (3, 1.5), and stays within the frame.