**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) = ax^{2} + 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.