Families of Functions and Curves

 
Objective: To provide a toolbox of aids for teaching students about families of functions. The toolbox includes  collections of animations that illustrate how functions change when certain parameters are varied. The animations are designed to be run on a variety of platforms and are grouped into galleries.

Level: Algebra and calculus courses in high school or college.

Prerequisites: Basic knowledge of a function and graphing equations. Users can choose function forms appropriate to the level of the material in their course.

Platform: A particular platform is not required. The animations in the collection can be viewed in a browser or other utility like Windows Media Player or Quicktime. We highly recommend that the animations be viewed with software that has a start/stop feature.

Instructor's Notes: (A successful web based assignment that uses some of the visualization tools in the gallery of demos is available as a pdf file by clicking here.)

As students progress in their study of algebra, the pervasive notion of a function is encountered with increasing regularity. Certainly a basic understanding of functions and the facility to manipulate them is requisite for true success in algebra and succeeding courses such as trigonometry and calculus. It has been said that 'Functions are simply the most important concept in mathematics.' 

We expect students to accumulate a catalog of functions and graphs that contains the basic examples that are studied as part of algebra, geometry, trigonometry, and calculus. We further expect students to be very familiar with members of the catalog. For example in an algebra or pre-calculus course students must be able to readily recall equations, domains, ranges, and basic graphs of linear, quadratic, and (basic) cubic equations. As they encounter geometry we expect them to add the conic sections to the catalog and then the 'circular' functions from trigonometry. Then we throw in logarithmic and exponential functions as these topics are discussed. By the time they reach calculus the catalog has grown to also include power functions, rational functions, polar curves, and some combinations of members of the catalog. We continue to expand the set of properties we expect them to quickly recall and use, such as, intercepts, increasing, decreasing, extrema, concavity, and possible asymptotes. 

Recent trends and standards have emphasized a balance in mathematics and certainly function concepts using the 'Rule of Four'; that is, topics should be presented numerically, graphically, symbolically, and verbally. In this collection of demos we emphasize the graphical aspect of functions and how the graph changes when we vary a parameter in the algebraic expression. We stress interrelationships using the question 'How does the graph f change as we vary a parameter.' Thus we link the symbolic expression to the graphical aspect by requiring a verbal description of a situation supplied in an animation. 

The study of a family of functions and curves of a particular type when the basic form is encountered can aid in the development of a student's catalog of curves, the recall of properties, familiarity with their graphs, and how we can use members of a family to model various behaviors. Using technology, calculators or computers, we have the opportunity to supply visual components to study families of curves. In addition a collection of animations can further compliment the visual tools available for instruction. With this in mind we have developed galleries of animations (see below) that can be used by instructors at various levels to enhance the idea of families of curves and graphs and how the members of the family change when certain parameters are varied. Instructors can choose those that are appropriate for their course and level. Students can also use the gallery for independent study or to complete class assignments on various families of functions. We illustrate several of the families from the gallery with the following three examples.

Example 1. The family of linear functions g(x) = mx + b contains two parameters m and b which control the slope and y-intercept of the graph of g(x) respectively. Very early in algebra the properties of slope are investigated and illustrated by various pictures. The following animation shows the behavior of this family as the (slope) parameter m is varied.

 

Example 2. The family of parabolas expressed in the form f(x) = a(x - h)2 + k has three parameters that control the shape and position of the graph. This form for the equation of a quadratic polynomial is often easier to use when determining graphic characteristics than the more familiar standard form f(x) = ax2 + bx + c. Here we illustrate the behavior of this family of functions as we vary the parameter h.

Example 3. One reason for studying families of functions is gain insight into the characteristics of curves for use in mathematical modeling. One important such model is the logistic-growth model; it is used to describe growth in which there is an upper limit. Members of the logistic family are often used to describe long-term population growth, the spread of disease (see the Logistic Curve Demo) , the spread of rumors, sales forecasts, and company growth. 

Galleries of Animations

Because of the variety of function and curve families we have developed galleries by grouping together related families, primarily in terms of the level in which they appear in courses from algebra through calculus. Click on a gallery:

 Polynomial & Rational Function Gallery

 Trigonometric Function Gallery

 Exponential & Logarithmic Function Gallery

 Conic Sections Gallery

 Polar Gallery

Each gallery has a collection of animations and associated software that can be downloaded for use in classes or projects. In addition, accompanying each gallery is a list of selected resources which focus on the particular type of function or curve represented in the gallery.

Software

In the galleries we indicate the software tool used to generate the animations and at times other software which has similar capabilities. We have grouped the set of programs in for each software platform into a zipped file which can be downloaded by clicking on the appropriate item.

Selected General Resources

Using your favorite search engine with appropriate key words for graphs, functions, etc. gives a plethora of sites which contain a wide variety of information. The focus of this demo is to provide toolboxes of animations of families of functions and curves that can be used by instructors and students to better understand the behavior of members of the family as parameters in expressions are varied. There are sites that are repositories for pictures and properties of functions and curves that have applets for displaying pictures of curves and some of these also allow variation of parameters. There are also numerous sites which provide general sketching applets. Following is a selected list of resources of both types which may provide materials to complement the animations available in the galleries of this demo.

Credits:  This demo was constructed by David R. Hill, Temple University for Demos with Positive Impact. Many of Geometer's Sketch Pad programs were developed by Mark Yates of the McCallie School. We also appreciate his generous contributions so the members of the project were able to learn construction techniques for Geometer's Sketch pad. The Excel worksheets were developed by members of the project who benefited from consultations with Deane Arganbright of University of Tennessee at Martin and with Walter Hunter of Montgomery County Community College  All software available in this demo is done so with the permission of the authors.

 


DRH          Last updated 9/15/2010 DRH

Since 8/29/03