Objective:
Students in precalculus and calculus need to be
proficient in recognizing and using a variety of properties of functions.
Functions are presented in a variety of ways: graphical, numerical,
analytic, or verbal. This demo provides a set of interactive graphical
visualizations designed to help students better understand what it means for
a function to be increasing/decreasing over an interval. The demos ask that the student
first predict the behaviors by inspecting the graph and then follow this
with an activity that illustrates the behavior. Recent research [1]
indicates that this mode of demonstration leads to significantly greater
understanding.
Level:
The visualizations and activities in this demo
are appropriate for high school or college level precalculus or calculus
classes.
Prerequisites:
Platform: The
some movies in the demo require Quicktime
player, however, additional platforms are described and where appropriate,
accompanying files and/or links are provided. The Quicktime movies may be
downloaded by right clicking and saving the movie to your computer. Excel files
that accompany this demo are also freely downloadable and can be used in class
or for individual investigations by students. The Java applet use the
Java Components for Math, developed
by David Eck under NSF grant number
DUE-9950473.
Instructor's
Notes: Understanding and interpreting the meaning
of functions requires a comprehension of variation. To develop an
understanding of change requires that students not only deal with the basic
notion of a function, but also such properties as slope, monotonicity,
concavity, and asymptotic behavior. Here we are concerned only with
the topic of monotonicity. Often these concepts are given
algebraically and we expect students to transfer such formulations to
graphs. Some students have difficulties in making such transitions.
Monotonicity
It is common in precalculus and calculus texts to see
a definitions like the following.
ALGEBRAIC DEFINITIONS
A function f is called increasing on
an interval [a, b] if
f(x1) < f(x2)
whenever x1 < x2 in [a, b].
A function f is called decreasing on
an interval [a, b] if
f(x1) > f(x2)
whenever x1 < x2 in [a, b].
In such formulations it is important to
emphasize that the inequality f(x1) < f(x2) must be
satisfied for every pair of numbers x1 and x2
with x1 < x2 in [a, b] so that we can say f is
increasing on interval [a, b]. Correspondingly for decreasing on interval [a, b].
To transfer the algebraic formulations
stated above to graphs we often use statements like the following to provide a
bridge between the written and visual representations of the properties of
increasing and decreasing.
GRAPHICAL EXPLANATIONS
A function f is called
increasing on an interval [a, b] if the graph rises
from left to right.
A function f is called
decreasing on an interval [a, b] if the graph falls
from left to right.
The concepts are then illustrated by a
few figures like those in Figures 1 and 2 with accompanying explanations.
-
The function in Figure 1 is
increasing on [-2, 0] and decreasing on [0, 3].
-
The function in Figure 2 is
increasing on intervals [a, b] and [c, d], while decreasing on interval
[b, c].
Figure 1.
Figure 2.
The type of demo we propose that connects
the algebraic definitions with the graphical explanation for increasing and
decreasing has the following format.
Figure 3.
-
Have students predict intervals over
which the function y = f(x) is increasing and those over which is it is
decreasing and record the information on a sheet of paper.
-
Use an interactive program that
displays pairs of points like (x, f(x)) and (x+h, f(x + h)), h > 0, as
students use a device to trace the curve. This provides an opportunity
to connect the algebraic definition to graphical explanation.
-
Use a feature of the interactive
program that displays the monotonicity at points along the curve.
-
Have students compare their
predictions with the display generated.
-
Follow with a discussion as needed.
Figure 4 shows the screen of an Excel
program that has the features list above. Note the ACTIONs described in the
boxed regions.
Figure 4. |
Figure 5 shows the result of tracing the curve and
then changing the NO to YES. Note the use of + to indicate increasing and -
for decreasing at points along the curve.
|
Figure 5. |
Below is a Quicktime animation of the ideas outlined
above.
