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Topic:  Left Right Translation of Functions 
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Subject:  RE: Left Right Translation of Functions 
Author:  Alan Cooper 
Date:  Nov 29 2004 
> . . . What is so much harder to explain is the horizontal shift. . .
> . . . . Does anyone have a good way to explain this?
It's not exactly an applied example, but I find this helps:
If you stand facing the graph of y=f(x) at some position on the xaxis then in
order to plot the corresponding point on the graph of y=f(x+a) you need to plot
a yvalue equal to the height of the f graph at position x+a (ie if a>0 that
is a units to the right of where you are standing). So reach over to the point
at position x+a on the f graph and ask, "If I have to reach to the right to find
the point on the f graph, how am I going to move it when I pull it in front of
me?"
This leads to the summary "When we reach to the right we pull to the left, and
we reach left to pull right". The same idea can also be used for scaling 
"Reach in to pull out" and "Reach out to pull in".
Of course you don't want to be addressing the board for too long, so in order to
avoid too much fancy choreography it might work better to have a student do the
standing and reaching.

For applied examples, perhaps time dependency ones work best:
eg if r=f(t) is the graph of rainfall at time t with t=0 being midnight, then if
I get up at 8am, the amount of rain I see T hours after I get up will be f(T+8).
Using just one copy of the graph and labelling the time axis with both t and T
values (eg one above and one below) may help to sort things out and show that
from the point of view of the T graph the y axis (T=0) is at t=8 and as the
yaxis moves to the right the graph appears to move to the left (maybe hold up a
window frame or transparent set of axes and ask students to say what appears to
be happening as the window is moved to the right). But I know from experience
that it's a good idea to be a bit less spontaneous than usual with this kind of
example  there's a reason that kids find it confusing!
Alan
P.S. A major source of confusion in many situations is the (bad) habit of using
the same name for a function and its dependent variable. eg if you start talking
about r(t) and r(T) in the above example then we'll all get confused.
 
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