Monotonicity: Algebraically and Graphically

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  • As the curve is traced the monotonicity at points is shown.

     
    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.

    • Show the graph of a function like Figure 3.

    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.

    The Quicktime movie provides start/stop features so that the instructor can pause during the course of the lesson to discuss what is happening in the animation.  The functionality of the movie control bar is shown in Figure 6.

     

    Figure 6.

    For a gallery of examples using Excel as described above and illustrated in Figures 4 and 5 (see also the Quicktime file below Figure 5) click here.

    A Java applet using the same function examples as in the Excel gallery is available by clicking here. The applet does not superimpose + or - signs on the graph as it is traced. So the Excel files maybe better suited for classroom demonstrations.
     

    References

    1.  Crouch, Catherine H., Fagen, Adam P., Callan, J. Paul, and Mazur, Eric. "Classroom Demonstrations:  Learning tools or entertainment?" Am. J. Physics, 72(6), June 2004, pp 835-838.


    Credits:  This demo was submitted by 

    David R. Hill
    Department of Mathematics 
    Temple University

    and is included in Demos with Positive Impact with his permission. He also constructed the Excel files and implemented the Java applet using the Java Components for Math, developed by David Eck under NSF grant number DUE-9950473.


     

    DRH    9/24/2004    Last updated 5/23/2006

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