Comparing graph and cograph (Version 2)

New Version 6 July 2011

This is a new version of a post originally created on 30 June, 2011. –Charles Wells

When you put the graph and cograph of a function with parameters side by side, interesting things may happen.  I have created the files CographExample.nb and CographExample.cdf to illustrate this.

The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish.  The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here.  The CDF Player allows you to change the parameters with the slidebars, so that you can experience the phenomena discusses in the example, but you cannot otherwise modify the file.

You cannot include a Mathematica computable document directly into a Word Press document, so here are screenshots of the cograph example with several different settings of its parameters.

Some Screenshots and sample questions follow.  These not in the original version.  I added these thanks to encouragement by Sam Alexander.

Questions for discussion

My idea is that students will manipulate the slider to see what happens and do some algebra on the relations between a, b and x to explain the phenomena that occur.

Questions about the graph (left figure).

G1. Prove that the two straight lines are parallel for any choice of a and b.

G2. When do the red and blue straight lines in the graph coincide? Answer: The lines coincide when $latex a=1 &fg=000000$ or $latex b=0 &fg=000000$.

• “When do the…” translates into “for what values of a and b do the…”. I predict that which way you ask the question will make a big difference for some students. To answer this question, you are not solving for x but for a and b. If you ask “When…” they have to discover this for themselves.
• The answer is disjunctive: This may be a new idea for some of the students.

G3. Find a formula in terms of a and b for the distance between the two straight lines in the graph. Answer: $latex |b-ab|$.

C1. When do the red and blue arrows in the cograph coincide? Answer: Same as answer to Question G1.

C2. In the cograph, for what aand b are the arrow targets for a given choice of x closer together than the arrow sources? Answer: $latex |a|>1$. b is irrelevant.

C3. Manipulate a and b. For some values the blue arrows all cross each other at the same point. (Same question for red arrows.) When does this happen? Answer: When a is negative.

• The abstract setting of the cograph is shifted by this question (and the next) as follows: The arrows originally provided a visual pointer from the input to the output of the function. All of a sudden we are treating them as mathematical objects(straight line segments in the plane).
• The abstraction is also broken in another way: The space between the source line and the target line is just a visual separation, but after this question both lines lie on the xy plane. The question turns a visual illustration into a mathematical object.

C4.  Describe the common point where all the red arrows cross when a is negative. (Same question for blue arrows.) Answer: The point is $latex \left(\frac{b}{1-a},\frac{3 a}{a-1}\right)&fg=000000$.

1 Comment

1. 1
Sam Alexander says:

What OS are you using? Apple? Linux? If you’re on Windows, taking screenshots is extremely easy, literally just push Alt+Prnt Scrn and then paste into any paint program. It’s saddening you won’t be including screenshots in future posts… the “Endograph and Cograph of real functions” post was beautiful.