Discussion:  All Topics in Algebra II 
Topic:  Left Right Translation of Functions 
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Subject:  RE: Left Right Translation of Functions 
Author:  Craig 
Date:  Dec 12 2004 
> 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 if the person throwing the ball walks four feet to the right, and then
throws the ball (or, his twin is four feet to the right, throwing at the same
time)? If the original ball path is given by y = f(x), then the new ball's path
should be the same, somehow. If the shape is the same, then the "f" part is
correct, and if the height is the same, the "y" part is correct. Something has
to happen with the x and the 4. Y_new, when x = 4, should match Y_old when x =
0, four units less; so Y_new(x) = Y_old(x  4). In effect, f(x  4) takes f(x)
and moves it four units to the right.
Maybe a spreadsheet idea would help to clarify the situation, too... a column
for x, one for f(x), and one for g(x), which is supposed to be a copy of f(x)
moved four units to the right. You could give students a formula for f(x) [even
better, just a semirandom set of numbers], then make them figure out how to
graph g(x)... they'd have to realize they should use the fvalue from an
earlier row. If x is column A and f(x) is column B, then the entry for g(x) in
row n might look like =Bm, where m is some number less than n (if the
xincrement is 1, then m = n  4). Then students see g(n) = f(m) = f(n 
4).
Another approach may be too complicated for an Algebra 2 class... I believe I
saw this idea in the UCSMP "FST" textbook, intended to be the first of two
precalculus texts in the UCSMP series. Treat things parametrically! Think
(x1(t),y1(t)) and (x2(t),y2(t)), where the heights are exactly the same so y2(t)
= y1(t), and x2(t) = x1(t) + 4 (that is, x2 is 4 units to the right of x1). If
x1(t) = t, then x2(t) = t + 4, so t = x2(t)  4. Thus, y2(t) = y2(x2  4) =
y1(x2  4). This is difficult for precalculus students to get their minds
around, though. I think if you refer to the FST text you'll find some sort of
exploration with the parametric mode in a graphing calculator (or you can easily
adapt spreadsheets, similar to what I described above).
 
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