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 Discussion: All Topics in Algebra II Topic: Left Right Translation of Functions

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 Subject: RE: Left Right Translation of Functions Author: Mathman Date: Nov 29 2004
On Nov 29 2004, The Non-Whistler wrote:
> What I try do is to ask "What is the solution to f(x)=0. For
> example, why is f(x)=(x-2)^2 a shift to the RIGHT of 2 in the x-
> direction of f(x)=x^2. Well, solving (x-2)^2=0 gives x=2, so it has
> to be to the right. Support this with the graphs.

The support might be done by calculating and plotting a few points with the
following argument, plotting the graphs together on the same axes:

Calculate a few suitable points and graph y = x^2

Now consider y = (x-2)^2.

The "new" y value is calculated as follows:

If x = 2, the y-value is what it would have been when x = (2-2)...that is,
0, [and is now 2 units to the right at x = 2.]

If x = 3, the y-value is what it would have been when x =(3-2) ...that is,
+1, [and is now 2 units to the right at x = 3.]

If x = ....and so on.

Doing that sufficient times with a sufficient number of graphs should make the
result obvious enough to declare a pattern exists.

However, I think she was looking for "real world" physical setup that the kids
could relate to by throwing a ball, or whatever, and that's not always possible.
I think that re-arranging where possible [even here if allowing for the
positive results] might be as revealing.  That is:  y = (x-2)^2, so x-2 =
sqrt(y), x = sqrt(y) + 2, and the 2 is added to increase the value of x.

This raises the point over which students get confused when doing "word
problems": In an expression A = B + C, which is the larger? A, or B?  in the
above example, they might have trouble seeing that although the quantity C is
added to the right hand side, it is the left hand quantity that has been
increased.  It is such issues as that which must *first* be dealt with and
overcome in their studies.  Examining older texts [late 1800s, early 1900s] and
those of today, one can see readily that such important basic ideas are now
glossed over, rather than being an intricate part of the study.  So students now
study "more", but in much less detail.  Their confusion has good solid
ground.

David.

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