The Graphs

A variety of functions will fool a graphing calculator ranging from the obvious, which many students will be familiar with, to the subtle, which will be less familiar. In the pictures below, the graph on the left is, approximately, what would be generated by a TI-85 and the picture on the right is a, somewhat, more accurate picture. You should have a basic understanding of graphs and how a graphing calculator works in order to understand what's going on in these pictures and why the graphing calculator makes the mistakes it does.

The discontinuity of the function induces one of the simplest mistakes that a graphing calculator will make.

Many students will be aware that the vertical line is an artifact produced by the graphing calculator and will, therefore, not draw it on a test or quiz. However, if a class is asked to graph this function on a test or quiz and the students have access to a graphing calculator, chances are that several students will draw that line.

If we let the discontinuity problem becomes more subtle.

The problem here is that the graph is undefined at x=2. Of course, we have a convention of drawing a small open circle centered at that point to indicate that the function is not defined there, but the graphing calculator will not do this. Looking at the more accurate graph on the right, we see another problem - the slopes are slightly different. This is because the aspect ratio in the graphing calculator's standard [-10,10]x[-10,10] viewing rectangle is slightly askew.

Next, consider the semi-circle given by . Many students will have noticed, and will have been confused by, the dangling behavior of the calculator's graph.

In this example, the aspect ratio problem arises again. Also, the roots at 6 and -6 are not shown, since those points are not among those plugged into the function by the calculator. Combined with the infinite slope at those points, this leads to the dangling affect shown. As the next example shows, this behavior can appear in a less familiar setting.

Next, let . This is a tricky one where the calculator's graph is just completely off. This comes off of an actual calculus test I graded while at Ohio State. The problem was broken up into a number of parts asking the students to identify the domain of the function, to take first and second derivatives, to identify intervals where the function was increasing or decreasing and convex or concave, and, finally, to graph the function. Many students answered all parts correcly, except the final part, where they drew a picture as seen to the lower left. Graphing calculators were allowed on the exam.

The most glaring error (at the calculus level) is the omision of a large portion of the domain. Also, 5 is not seen to be a root. The infinite slope at 5 leads to the dangling affect. A subtle inflection point at 50/7 is not seen in the calculator's graph. An all around terrible performance by the calculator.

Addendum: Paul King of TI says to write to get the calculator to show the whole domain. In fact the calculator is using the principal branch of the complex logarithm.

When is graphed in the standard [-10,10]x[-10,10] viewing rectangle, the essential features of the graph are missed. This is not so much a fault of the graphing calculator, but care should be taken by the student to find a better viewing rectangle.

The viewing rectangle on the right can be found with a little experimentation combined with a knowledge of what the graph should look like.

The oscillatory behavior of leads to an interesting problem. There is no way for the calculator to indicate that the graph oscillates infinitely many times in a neighborhood of zero.

Of course, the graph on the right is really no better. This example shows why writing is an important part of mathematical communication. No picture can oscillate infinitely many times in a neighborhood of zero and so some words must be used to indicate that this happens. It's fun to have students zoom in on this example.

Functions of the form sin(nx) oscillate only finitely many times on a bounded interval, but they can still cause problems. The actual graph oscillates 40n * Pi times over the interval [-10,10] (which is alot for large n). Different values of n, however, give rise to strikingly different behavior on the graphing calculator depending on the relationship between the period of the function and the spacing of the points plotted by the graphing calculator.

The above graphs emulate a TI-85, which is 127 pixels across. Other graphing calculators will display similar behavior, but for different values of n, depending on how many pixels span the screen.