Discussion:  All Topics in Algebra II 
Topic:  Left Right Translation of Functions 
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Subject:  RE: Left Right Translation of Functions 
Author:  Mike Shepperd 
Date:  Dec 4 2004 
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((xh)/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
For example:
y=4(x5)^2+3
(y3)/4=(x5)^2
Start with graph of y=x^2
Dilate factor 4 in y direction (stretch)
Translate distance 5 in x direction
Translate distance 3 in y direction
For example:
y=4sin(2x5)+3
(y3)/4=sin((x2.5)/(1/2))
Start with graph of y=sin(x)
Dilate factor 1/2 in x direction (squash)
Dilate factor 4 in y direction (stretch)
Translate distance 2.5 in x direction
Translate distance 3 in y direction
Note:
amplitude = 4
period = 2pi x 1/2 = pi
A negative value of a involves a reflection in the x axis.
A negative value of b involves a reflection in the y axis.
This approach is supported by two TI83 programs GRAPHFN and TRIG that can be
downloaded at:
http://www.teachers.ash.org.au/mikemath/ti83.html
On Nov 27 2004, Susan wrote:
> 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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