Judy Roitman pointed out that I used the word "domain" in a nonmathematical fashion. In my previous post when I said the "domain of the graph" I meant the interval of values which sets the screen limits for the independent variable on the graph. Not the domain of the function in the mathematical sense. I hope this was not confusing.
Chi-Tien Hsu responded to this portion of my post: >> 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 > >To certain point, I will expect the student to be able to find this wihtout >GC, that will strengthen their analytical ability and understanding a great >deal. So, when are you going to give them this problem with GC? >Before or after they are able to find the approximate roots without GC?
I would like the students to find the roots of many polynomials before I gave them something like this. However, those polynomials are usually restricted to having rational or relatively simple radical roots of index 2. This particular polynomial is not one I would want them to find the roots by hand even though there is a cubic formula for roots.
Depending on the level of the student I might even expect them to understand that the TI-85 finds solutions to polynomial equations using numeric methods. If the student understand that also, they understand MORE than the traditional methods for finding rational and radical roots of polynomials.
Chi-Tien Hsu also asks this question: >> 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] ...Now have them zoom out on the graph >>using >>(0,-7/12) as the center >> of the zoom ... After >> zooming out by a factor of 16 or more these two graphs begin to look the >> same on the screen. > >I don't understand what do you mean by look the same. I don't have the GC with >me, but, just a quick look at it, I find it hard to visualize that they will >look the same if the graph center is (0,-7/12).... Now, if the resolution >of >the GC is too bad >to see this difference, then I wonder how could they see the difference >at two ends where y1 and y2 deffer only by 0.5/x.
Of course, the resolution of a graphing calculator is poor. This is another reason the students must rely on their mathematical skills and not the picture the calculator gives them. I graphed this immediately prior to my previous post and with a window of [-160,160]xwhatever the two graphs were almost identical except for a small blip in the center of the graph. the point (0,-7/12) was chosen because that was the y-intercept of the oblique asymptote. The window was chosen so that the oblique asymptote went from corner to corner of the window even after zooming. The students can tell the difference between the graphs because the calculator will give the coordinates of the points on the screen for each function (the rational function and its oblique asymptote) and they can compare the ordinates of these points to see which is higher or lower and that they are very close for extreme values of x.
--------------------- Murphy Waggoner Department of Mathematics Simpson College 701 North C Street Indianola, IA 50125 firstname.lastname@example.org ---------------------