The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Some background

• Generally, I have advocated using all sorts of images and metaphors to enable people to think about particular mathematical objects more easily.
• In previous posts I have illustrated many ways (some old, some new, many recently using Mathematica CDF files) that you can provide such images and metaphors, to help university math majors get over the abstraction cliff.
• When you have to prove something you find yourself throwing out the images and metaphors (usually a bit at a time rather than all at once) to get down to the rigorous view of math [1], [2], [3], to the point where you think of all the mathematical objects you are dealing with as unchanging and inert (not reacting to anything else).  In other words, dead.
• The simple example of a family of functions in this post is intended to give people a way of thinking about getting into the rigorous view of the family.  So this post uses image-and-metaphor technology to illustrate a way of thinking about one of the basic proof techniques in math (representing the object in rigor mortis so you can dissect it).  I suppose this is meta-math-ed.  But I don’t want to think about that too much…
• This example also illustrates the difference between parameters and variables. The bottom line is that the difference is entirely in how we think about them. I will write more about that later.

 A family of functions

This graph shows individual members of the family of functions \( y=a\sin\,x\) for various values of . Let’s look at some of the ways you can think about this.

• Each choice of  “shows the function for that value of the parameter “.  But really, it shows the graph of the function, in fact only the part between and .
• You can also think of it as showing the function changing shape as changes over time (as you slide the controller back and forth).

Well, you can graph something changing over time by introducing another axis for time.  When you graph vertical motion of a particle over time you use a two-dimensional picture, one axis representing time and the other the height of the particle. Our representation of the function  is a two-dimensional object (using its graph) so we represent the function in 3-space, as in this picture, where the slider not only shows the current (graph of the) function for parameter value but also locates it over on the axis.

The picture below shows the surface given by as a function of both variables and . Note that this graph is static: it does not change over time (no slide bar!). This is the family of functions represented as a rigorous (dead!) mathematical object.

If you click the “Show Curves” button, you will see a selection of the curves in middle diagram above drawn as functions of for certain values of . Each blue curve is thus a sine wave of amplitude . Pushing that button illustrates the process going on in your mind when you concentrate on one aspect of the surface, namely its cross-sections in the direction.

Reference [4] gives the code for the diagrams in this post, as well as a couple of others that may add more insight to the idea. Reference [5] gives similar constructions for a different family of functions.

References

1. Rigorous view in abstractmath.org 
2. Representations II: Dry Bones (post)
3. Representations III: Rigor and Rigor Mortis (post)
4. FamiliesFrozen.nb.
5. AnotherFamiliesFrozen.nb (Mathematica file showing another family of functions)