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Topic: Left Right Translation of Functions

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Subject:   RE: Left Right Translation of Functions
Author: Mike Shepperd
Date: Dec 6 2004
In teaching high school Mathematics, I am happy to plead guilty to the frequent
use of a "why through pattern" approach, especially when linked to the use of

By omitting the word "symmetry" in my original email, I may have failed to
establish clearly a "why argument". The symmetry of the form (y - k)/a =
f((x-h)/b) demonstrates that the dilation factor a and translation distance k
in the y direction have their counterparts b and h in the x direction.

In my experience, students have little trouble in understanding the
transformations in the y direction but they often need help with the
transformations in the x direction.

For Windows users, download Winplot if you have not already discovered this
great freeware from Rick Parris. Enter the equation y=asin((x-h)/b)+k and
create sliders for the values of a,b,h,k. Enjoy!

On Dec  5 2004, Mathman wrote:
> On Dec  4 2004, Mike Shepperd wrote:
> There is no difference in
> the treatment of translations (and
> dilations) between those in
> the x direction and those in the y
> direction. To see this
> clearly, consider the equation of a function
> in the form:
(y -
> k)/a = f((x-h)/b)

The graph of this function
> is the graph of
> y=f(x) subject to the following transformations:
> dilation factor
> b in x direction
  dilation factor a in y direction
> translation
> distance h in x direction
  translation distance k in y
> direction

Respectfully, Mike ... Doesn't that just state *what*
> happens, with following samples, but not *why*?  That is, it states
> what each parameter does, but gives no reason.  It is true that, as
> yo usuggest, a study of graphs reveals the patterns, but the
> students need to see it in the algebra.  Since the problem the
> students are having is knowing why something is moved in a positive
> direction when a number is negative, isn't it more informative to
> show that y-a = A becomes y = A+a, and the move is in the positive
> direction? Likewise, if y = x-d, then x=y+d, and the move is again
> in the positive direction.  Alternately, y+a = A results in y=a-a,
> and a move in the negative direction ,and the same for x.  Functions
> unresolvable for "x" do need further explanation, but the idea can
> still be shown.


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