[e-mailed and posted to the ap-calc mailing list]:
On Mon, 12 Oct 1998 13:54:27 -0400, "William J. Larson" <Bill_Larson@compuserve.com> wrote about stop teaching shifting & stretching?:
:Since my calculus students will have graphers, is it the :consensus that I should stop teaching my precalculus :students such tricks to aide graphing as shifting & stretching :functions and symmetry? I'd love to stop teaching this, because it :would free time for other "essential" topics, which I do not :get to.
I would certainly not concur with that.
Students should learn to understand function behavior and how changing one quantity affects the overall behavior of the function.
For instance, if I take a given function y = f(x) and change it to produce a new function y = 2f(x) the students should CERTAINLY understand that this has the effect of doubling all the function values, or y-coordinates, which is WHY the graph appears to stretch.
Perhaps, since they have graphers, you could ask them to explain, in terms of x and f(x), why the graph appears the "stretch".
Understanding interactions between quantities in equations and being able to express these understandings is important in mathematics, I believe.
Symmetry, I believe, is an extremely important topic. I just took Math 317 at Cal Poly Pomona this summer (Fourier Analysis and Laplace Transforms) and we repeatedly remarked on the symmetry of functions in that course. I spend time on that topic as part of my PreCalc review at the beginning of my PreCalc course. I want my students to be able to use symmetry as an aide to save them time and work on Calculus problems.
Out of curiosity, what are the other "essential" topics that you do not get to that you believe are more important? With the advent of calculators such as the TI-89, I'd think we could rationalize away almost our whole K-12 curriculum, if we so chose to.