
Re: stop teaching shifting & stretching?
Posted:
Oct 19, 1998 8:50 AM


Hello ...
I haven't kept completely on top of the shifting/stretching controversy, but I'm in favor of teaching the topic in my classes because a proper understanding of shifts & stretches provides calculus students a painless introduction to the chain rule, as well as a convenient hook into the derivative of the exponential function.
For instance, then the following should be "obvious" when when one considers the effects of shifts and stretches on the slope of the tangent line to a function's graph ...
1) ( f(x) + d )' = f'(x) 2) ( a f(x) )' = a f'(x)
Likewise, these should be fairly straightforward, if perhaps a bit shy of "obvious" (horizontal transformations tend to be harder to grasp) ...
3) ( f( x + c ) )' = f'( x + c ) 4) ( f( b x ) )' = b f'( b x )
Notice that rules 3) & 4) alone combine into the rule
( f( b x + c ) )' = b f'( b x + c )
which is a simple case of the Chain Rule. Indeed, if you're really into using linearizations of functions to get at derivative rules and such, then this *is* the Chain Rule. (Take b x + c to be the linearization of the "inside" function.)
Notice also that 2) & 3) lead directly to the fact that the derivative of an exponential function f(x) = p^x (p constant) has the form f'(x) = P p^x (P constant). (Simply use the fact that f(x+c) = p^c f(x).)
I lead up to the calculus idea here by referring early on to the "selfsimilarity" of exponential functions: any scaled version of the graph is congruent to the original graph. (We studied selfsimilar Fractals in PreCalc, so the term gets their attention makes a connection between this idea and all those nifty jagged pictures. So, they don't tend to forget that exponential functions have this stretching equalsshifting property; when we get to doing slopes, they find out how useful that observation is.)
At any rate, the shifting and stretching (and let's not forget flipping over the line y=x) stuff doesn't just tell us how to graph functions conveniently. If it did, then using an automatic grapher would *would* make the material obsolete. In earlier courses, graphing savvy does *appear* to students as my goal when we discuss these things, but my real purpose is to instill in them a sense of what happens to the *function* (or relation). The graphs are merely visual fallout from the process I'm trying to get them to understand.
Regards,
Don

