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Topic: Re: Problem: "Sketch the graph of ..."
Replies: 1   Last Post: Oct 28, 1995 6:34 PM

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Murphy Waggoner

Posts: 52
Registered: 12/6/04
Re: Problem: "Sketch the graph of ..."
Posted: Oct 28, 1995 8:35 AM
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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

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

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"?

Murphy Waggoner
Department of Mathematics
Simpson College
701 North C Street
Indianola, IA 50125

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