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Topic:  Left Right Translation of Functions 
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Subject:  RE: Left Right Translation of Functions 
Author:  Mathman 
Date:  Nov 28 2004 
They see the problem with a quantity added. That's why they see the vertical
shift clearly?
1. Move the ladder?
2. Rearrange the formula using simple examples to have "x = ", and the
"subtracted" quantity becomes "added" in *precisely* the same sense that it was
in the ydirection.
It is not always simple to rearrange, so there has to be some wisdom in what
you pick as examples. Then they *must* abtract to generalise to all functions,
even those not yet seen, having seen some convincing simpler examples. That's
part and parcel of the nature of the study.
Also, this is the computer age, and I'm told you must use it as often as
possible, so have them graph a few on the computer, changing the translation
parameters. They can then see that the same pattern occurs no matter the
function.
You didn't say what is the "real world" example you mentioned. However, do be
careful. If students are capable of handling transformations in algebra they
should be able to abstract. Otherwise they should stick to more basic practical
mathematics for "real world" application. The truth is that it becomes more
abstract as the study advances, and what you might call "real world"
requirements for their understanding will fail as it becomes more complex.
Consider how a knowledge of Complex Numbers and Vectors [and phase shifts] adds
to the number of electronic toys now available. You don't get more "real world"
than that, or atomic energy, or physical organic chemistry, or rates of chemical
reaction [really applied in the world of the chemical industry, not just theory]
using systems of differential equations.
David.
> Most students have no problems understanding a real world
> application that shows a vertical shift in a function. For example,
> if students are asked to think of a graph of the path of a ball
> ball that is thrown from a person holding it at waist level, and
> then another graph that shows the same throw from a person that is
> standing on a ladder throwing it from waist level, they can easily
> see that the vertical translation makes sense. What is so much
> harder to explain is the horizontal shift. I have only seen it
> explained through a real world example in one text. All the
> students "know" that you shift it to the left/right according to the
> number in the parenthesis, but I don't think they understand why it
> is the opposite of the number, or how it would relate to a real
> world situation. Does anyone have a good way to explain this?
 
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