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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: Mike Shepperd
Date: Dec 4 2004
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

For example:
y=4(x-5)^2+3
(y-3)/4=(x-5)^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(2x-5)+3
(y-3)/4=sin((x-2.5)/(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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