Much more than a toy. Graphing calculators in secondary school calculus.

The power of zooming

To be sure, there are many instances where we simply wish to generate a graph quickly for our inspection and consideration in the mathematics classroom. For these ends alone, the graphing calculator is an invaluable aid, allowing all the students to participate actively in producing their own examples and not just passively accepting those of the instructor. Lest this point go by as so much introductory boilerplate, let me say that this is not a trivial difference in the graphing calculator classroom. I use many of the same graphical examples in my own teaching of calculus that I have always used. What I have noticed now with graphing calculators is a real change in the language of the students. For example, if I draw a graph on the blackboard of the function

and prominently display the "hole" in the graph represented by the removable discontinuity at x =1, it is the teacher's graph. But if all the students in the class graph exactly the same function in a window that exhibits the "hole" one hears a remarkable shift in tone:

"Hey, why does my graph have a hole in it?"
(turning to another student) "Does your graph have a hole in it, too?"

The graphing calculator bestows a sense of personal ownership on graphs, and that phenomenon alone can make a tremendous difference in the dynamics of the classroom.

Zooming to investigate function behavior --limits

There is no argument that limits are part of the basic language of calculus (the debate is on what level of rigor is most appropriate for an introductory calculus course). If we think of the limit of a function at a point as a mathematical description of its local behavior, then the graphing calculator is our microscope for investigation.

Here's a basic collection of functions (none exotic) exhibiting classic types of local function behaviors, each of which can be investigated nicely using a graphing calculator:
A removable discontinuity at x =1
A vertical asymptote at x =1
A jump discontinuity at x =0
Another essential discontinuity at x =0
Another removable discontinuity at x =0

The use of a graphing calculator to explore function behavior is a powerful pedagogical tool, but its discrete nature places limitations on us. For example, a hole in the graph of a function will not be seen on a graphing calculator's graph unless i) the position of the hole is in the viewing window, and ii) the dimensions of the viewing window place the location of the hole precisely on a pixel location. Similarly, having a calculator on connected or dot mode greatly influences how a discontinuity may appear.

The epsilon's and delta's of scaling

If we could imagine an infinitely precise graphing machine, we could rigorously recast many of the definitions of calculus into the terms of the graphics window. For example, here is a perfectly rigorous definition of the limit of a function at a point: we can write

to mean that given any vertical scaling (y - range) from to , we can find a corresponding horizontal scaling (x - range) from to such that the graph of stays on screen from left to right (except possibly at x = a itself).

The height of the viewing window plays the role of our epsilon neighborhood of L, and the width of the viewing window plays the role of our delta neighborhood of a. Imagine a game in which one player sets the epsilon tolerance by setting the y -range bounds to and . The other player is not allowed to touch these settings and must find some suitable x -range bounds and to keep the graph of y = f (x) on screen (the one exception allowed is at itself). If this second player always has a winning strategy, then the function f has limit L as x approaches a. Indeed, if this is the case, then a small enough choice of will result in the graph appearing horizontal.

A natural consequence of this definition is a commonly observed phenomenon on graphing calculators: Horizontal zoom-ins flatten out the graphs of continuous functions. By a horizontal zoom-in, I mean a rescaling of the viewing window so that the vertical range remains the same but the horizontal range represented on the calculator screen becomes a smaller interval. In terms of the global behavior of a function, a similar phenomenon can be observed: If a function has a horizontal asymptote, horizontal zoom-outs make the graph look like the asymptote. To be sure, graphing calculators are not infinitely precise. To see these limitations play out in particularly spectacular fashion, try zooming in horizontally by repeated factors of 10 on the graph of .

Zooming to investigate function behavior --limits
The epsilon's and delta's of scaling
Dick - 4 OCT 1996



There is a PDF file available for this paper.


Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.

Mail comments to Pedro Gómez: pgomez@uniandes.edu.co