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Topic: stop teaching shifting & stretching?
Replies: 14   Last Post: Oct 19, 1998 8:58 AM

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BDS McConnell

Posts: 4
Registered: 12/6/04
Re: stop teaching shifting & stretching?
Posted: Oct 19, 1998 8:50 AM
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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"

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 "self-similarity" of exponential functions: any scaled
version of the graph is congruent to the original graph.
(We studied self-similar Fractals in Pre-Calc, 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-
equals-shifting 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



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