|Ivars Peterson's MathTrek|
February 17, 1997
That's what mathematician Tom Guglielmo of Mercy College in Dobbs Ferry, N.Y., discovered when he started teaching a calculus course in which graphing calculators play a major role. He described his experiences in a presentation at last month's Joint Mathematics Meetings, held in San Diego.
It was the spring of 1996, and Guglielmo was faced with what was to him a new course, a new textbook, and the Texas Instruments TI-85 graphing calculator. Graphing calculators are widely used in classrooms for plotting curves and performing a variety of mathematical manipulations.
One of the first things that students learn to plot is the trigonometric sine curve: sin x. For values of x from 0 to 2pi, the resulting curve displays a single cycle: the crest and trough of a smooth wave.
For values from 0 to 64pi, the curve shows 32 cycles.
However, the pictures generated by a graphing calculator depend on the number of pixels in the display. The TI-85 has 63 rows and 127 columns in its display (a total of 8,001 pixels). "This configuration of pixels defines what we actually see and, therefore, sometimes can be extremely misleading," Guglielmo says.
In effect, the calculator determines the value of sin x at 127 points in the specified domain, and plots the set of points corresponding to these values. However, if the curve fluctuates too rapidly, the procedure samples the curve at intervals that fail to capture its full complexity, and the resulting images often contain ghost curves and other curious features.
As a result, calculators with different display characteristics have different personalities, as do more sophisticated math programs such as Mathematica and Maple.
Recently, I've been playing around with Mathematica 3.0, which plots curves in all sorts of different ways. By setting parameters appropriately, it's possible to replicate some of the deceptive features of graphing calculator displays.
Of course, as the Mathematica manual points out, plotting curves on a computer always involves some sort of sampling, and it's possible that significant features of plotted functions may be missed in the resulting representations. Indeed, there's often a significant difference between what appears on my computer screen (determined by the screen's resolution), what gets printed out (governed by the printer's resolution), and the "true" shape of a curve! Exploring these limitations offers a wide range of intriguing puzzles.
Here's an example of what I get when I plot sin x for values of x from 0 to 128pi. What you see will depend to some extent on your web browser or printer!
On the TI-85, instead of seeing 64 cycles of the sine curve, you get only one cycle for sin x and one cycle for -sin x. The lines connecting the two curves represent abrupt transitions from one value to the next. Using trigonometry, Guglielmo could demonstrate why these particular ghost curves and transition lines appear.
Based on his experience, Guglielmo has put together a series of demonstrations with the TI-85 to illustrate how various features arise, particularly in the graph of sin x for values from 0 to 64pi.
Try plotting the following curves, and see what you can learn from the sequence: sin x from -pi to pi, sin x from 0 to 126pi, sin x from 0 to 254pi, sin x from 0 to 128pi, sin x from 0 to 63pi, and finally, sin x from 0 to 64pi.
"The trigonometric graphs generated by graphing calculators should be viewed with caution but, at the same time, present us with an interesting opportunity to understand and appreciate their limitations," Guglielmo contends.
Copyright © 1996 by Ivars Peterson.
Taylor, Linda J.C., and Jeri A. Nichols. 1994. Graphing calculators aren't just for high school students. Mathematics Teaching in the Middle School 1(November-December):190-196.
Wolfram, Stephen. 1996. The Mathematica Book (Third Edition). Champaign, Ill.: Wolfram Media.
Tom Gugliemo can be reached at firstname.lastname@example.org.
Graphics generated using Mathematica 3.0.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.