I expect students to graph functions which are much like the ones they would have had to graph in the past (such as y = (x-1)/[(x-2)(x-3)]) as well as other they would not have been required to in the past. For the example above (provided by Jack Roach) I would require the students to know that the domain of the graph should include 2 and 3 since those will be points of discontinuity. Also the value 1 should be included in the domain of the graph since that is an intercept. After the students decide on a good domain for the graph (say [-1, 5]) they would then use the calculator to determine a good range for the graph in much the same way they would by hand, by plotting points, but this time they can plot a much greater number of points. When the graph is displayed, they should recognize the "bogus" parts of the graph (those almost vertical lines which look like vertical asymptotes.) In other words, the student needs to know about asymptotes, intercepts, and should know what the function should look like to a certain extent BEFORE they begin using the calculator. This is the same information that a student needed to know before the use of the GC technology.
However, I now give the students different problems than before. For instance, ask a student with a TI-85 to graph the line y=1000x+4. If the student does not have an idea of what the function should look like before they graph they are likely to use the standard graphing range ([-10,10]x[-10,10]) and the student will not be able to "see" the graph on this range. I purposefully choose graphs that will not "look good" on the standard range to ensure the students are using their knowledge about graphs of functions to arrive at the answer.
Another example is to graph y=x^3-403x^2+406-1. The student should know how a polynomial should look like when it is graphed. I expect the student to use the polynomial root finder on the TI-85 first to find the roots are close to 402, 1 and 0 so that they will know that [-1,403] is a good domain for the graph and then they must find a good range for the graph. In the past we would only give polynomials that students could factor by hand. The only differences are that we can give polynomials that do not factor over the integers or rationals and the student uses the calculator to plot the points they used to plot by hand. They still need to know what a polynomial "looks like."
As to Jack's "hidden reasons" I have several reasons for asking students to graph rational and polynomial functions. One is to reinforce the definition of a function, that is, that for each input value there is only one output value. In the case of rational functions, can a vertical line be part of the function? No, because it would then not be a function.
Another is to have the students "see" the behavior of functions. We always tried to do this when students graphed by hand, but unfortunately the effort involved in graphing the function often overwhelmed the student and they felt the thinking was done as soon as the picture was on paper. On the TI-85 it is easy to see that as x goes to +/- infinity the value of f(x)=(x-1)/(x+2) gets very close to 1. So the students can look at many different rational functions and make conjectures as to the behavior of the functions at extreme domain values.
We also used to have students graph rational functions which oblique asymptotes and the function of the oblique asymptotes at the same time so that they could compare the behavior of the two functions at extreme values of the domain. However, they ability to "see" this comparison depended on the students ability to graph. Now we can have them graph y1=(2x^3-7x^2+1)/(12x^2-36) and y2=(x/6)-(7/12) on the same graph with the range [-10,10]x[-2.25,1.0833333] (why did I choose y2 and the range the way I did?) Now have them zoom out on the graph using (0,-7/12) as the center of the zoom (it should be the point in the center of the screen). After zooming out by a factor of 16 or more these two graphs begin to look the same on the screen. If they trace along the two graphs they find that y1 is above y2 on one end and y1 is below y2 on the other end. What does all this mean? How could they determine y2 for other rational functions? How could they determine a good range for other rational functions so that they get similar results as this? These are questions I couldn't dare ask before GC technology, and, at the same time, I believe these questions require the student to know more about functions than I used to be able to expect.
Sorry this ended up being such a long post and sorry that it specifically regards the TI-85. We may believe that graphing calculators should be used in the classroom but just giving the calculators to the students is not enough. We also need to develop a curriculum which uses the power of the calculator to help the student understand those concepts they had trouble with in the past but are in fact the concepts we wanted them to understand all along. This means we have to ask new/different questions than before.
From Jack Roach:
>Long before calculators of any type were around, students were given >problems such as "sketch the graph of y = (x-1)/[(x-2)(x-3)]." I don't >think there is any real disagreement that such problems would be handled >differently with a graphing calculator than they were then. Most of the >discussion of whether graphing calculators should be used or banned seems >to hinge on whether such differences are important. This brings up an >interesting question. Just what are these differences? There is another >way to look at this question. In the time before graphing calculators, >there were reasons for assigning such problems (like the one cited above) >beyond the obvious one of obtaining a picture---the one easily done now >with a graphing calculator. What are some of these "hidden reasons"? >Jack
--------------------- Murphy Waggoner Department of Mathematics Simpson College 701 North C Street Indianola, IA 50125 firstname.lastname@example.org ---------------------