Polynomial & Rational Function Gallery

The following is a gallery of demos for illustrating selected families of polynomials and rational functions. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the change of parameters in expressions for functions. Two file formats, gif and mov are available.

1. The gif animations should run on most systems and the file sizes are relatively small.

2. The mov animations require the QuickTime Player (version 6) which is a free download available by clicking here; these files are also small. (The mov files may not execute properly in older versions of QuickTime.)

3. The collection of animations in gif and mov format can be downloaded; see the zipped download category at the bottom of the following table.

4. We use abbreviations, GSP for Geometer's Sketch Pad 4 and EX for Excel spread sheet, VB for visual basic, and J for java to indicate the software used to generate the frames of the animation. If alternate corresponding software is also available there will be more than one abbreviation in that category. The VB software routines can be executed and the others can be downloaded.

Polynomials, Power Functions, & Rational Functions

 Constant functions, y = c GSP also EX and VB files Vary c: Gif Animation              Mov Animation Click to execute the VB file Linear functions, y = mx + b GSP also EX and VB files Vary m: Gif Animation                Mov Animation Vary b: Gif Animation               Mov Animation Click to execute the VB file Absolute value functions, y = a |x - b| + c GSP also EX, VB, and J files Vary a: Gif Animation              Mov Animation Vary b: Gif Animation              Mov Animation Vary c: Gif Animation               Mov Animation Click to execute the VB file Click to execute the java applet Greatest integer functions, (or step functions) y = a [bx - c] GSP also an EX file Vary a: Gif Animation               Mov Animation Vary b: Gif Animation               Mov Animation Vary c: Gif Animation              Mov Animation Quadratic functions, (or parabolas) y = ax2 + bx + c GSP also EX, VB, and J files Vary a: Gif Animation              Mov Animation Vary b: Gif Animation              Mov Animation Vary c: Gif Animation              Mov Animation Click to execute the VB file Click to execute the java applet Quadratic functions, (or parabolas) y = a(x - h)2 + k GSP also EX, VB, and J files Vary a: Gif Animation              Mov Animation Vary h: Gif Animation                Mov Animation Vary k: Gif Animation                Mov Animation Click to execute the VB file Click to execute the java applet Cubic function y = ax3+bx2+cx+d GSP also EX, VB, and J files Vary a: Gif Animation              Mov Animation Vary b: Gif Animation              Mov Animation Vary c: Gif Animation                Mov Animation Vary d: Gif Animation              Mov Animation Click to execute the VB file Click to execute the java applet Square Root GSP also an EX and a J file Vary a: Gif Animation             Mov Animation Vary b: Gif Animation              Mov Animation Vary c: Gif Animation               Mov Animation Click to execute the java applet Cube Root GSP also an EX file Vary a: Gif Animation               Mov Animation Vary b: Gif Animation              Mov Animation Vary c: Gif Animation                Mov Animation Power function GSP also a J file Vary b: Gif Animation              Mov Animation Click to execute the java applet Reciprocal function GSP also an EX file Vary a: Gif Animation             Mov Animation Vary b: Gif Animation              Mov Animation Reciprocal of x2 function GSP also an EX file Vary a: Gif Animation              Mov Animation Vary b: Gif Animation              Mov Animation Reciprocal of x3 function GSP also an EX file Vary a: Gif Animation              Mov Animation Vary b: Gif Animation               Mov Animation Zipped downloads ==> Download (all) Animation files (large file) Download (12) Excel files Download (6) Java applets

Other Selected Resources:

• A demo that includes quadratics developed from the "locus of points" definitions of the conic sections that can be implemented on a whiteboard (or blackboard) is available at  www.mathdemos.org. Click on Go to Demo List, then click on Physical Demos, and finally click on 'Constructing the Conic Sections on a Whiteboard'

• Another demo at www.mathdemos.org shows in an  immediate way that parabolas (quadratics) arise as graphs of natural phenomena. A ball is given a push up an inclined plane. As the ball moves up and then back down the inclined plane its distance from the starting point at the base is recorded by a motion sensor. The time vs. distance data is simultaneously projected on a computer monitor. When the ball returns to the starting point the data points are connected and the resulting graph is parabolic. If we would continuously record the data the resulting curve would indeed be a parabola. At  www.mathdemos.org. click on Go to Demo List, then click on Physical Demos, and finally click on 'Parabola!'

• There are a wide variety of applets for graphing polynomials available; we mention just a few here.
graphing cubics ==>

• A very nice applet for displaying cubics to help users to relate the coefficients in a cubic polynomial to the shape of its graph is available at http://thorin.adnc.com/~topquark/math/poly3nt.html A picture of the applet shows basic directions and its style.

• The US Air Force Acadamy has a nice collection of Mathematica notebooks on mathematics topics including polynomials and rational functions; http://www.usafa.af.mil/dfms/mma/precalc.htm

• NCTM as part of their Illuminations collection has a interactive math applet combines the features of a
spreadsheet and a graphing calculator. It can be used as a general tool or customized for specific purposes. For example,
this tool can be used to investigate polynomials, rational functions, and much more. The applet can be downloaded from http://illuminations.nctm.org/mathlets/grapher/index.html . Be sure to read their guidelines on the use and distribution of the resource.

• The Function Institute at http://id.mind.net/~zona/mmts/functionInstitute/functionInstitute.html has resources for polynomials, rational functions and more.

• The Math Forum at http://mathforum.org/mathtools/cell.html?rc=support&new_rc=tool and the Math Archives at http://archives.math.utk.edu/ have a wide variety of resources for functions of they type displayed in this gallery.

Credits:

• The Geometer's Sketchpad routines were constructed by Mark Yates of The McCallie School and are
used with his permission.

• The Excel files were written by Xia Zhao, a student at Temple University, and David R. Hill, Temple University.

• Special thanks to Deane Arganbright of University of Tennessee at Martin and to Walter Hunter of Montgomery County Community College for their assistance with the development of the Excel files that accompany this gallery. The book "Mathematical Modeling with Excel", by Erich Neuwrith and Deane Arganbright (Brooks-Cole) provides a rich source of information for developing Excel routines for mathematical instruction.

• This gallery was developed by David R. Hill, Temple University.

8/10/03DRH          Last updated  2/11/2005

As of 8/28/03