# Translation and Compressions Applet

This applet is designed to develop intuition about translation, expansions and contractions of functions.  In particular it graphs a function named Func, and a second function ShiftedFunc defined by

ShiftedFunc(x)=c*Func(a*x+b))+d.

The values of a, b, c, and d are controlled by sliders.  The definition of Func(x) and the view window can be changed with the control panel.

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Min X = Max X = Min Y = Max Y =

It is worthwhile to note the transformation caused by changing the prameters one at a time:

• Setting (a, b, c, d) = (1, 0, 1, 0) gives the original function.
• Setting (a, b, c, d) = (1, h, 1, 0) gives ShiftedFunc(x) = Func(x+h) a horizontal translation left by h.
• Setting (a, b, c, d) = (1, 0, 1, k) gives ShiftedFunc(x) = Func(x)+k a vertical translation up by k.
• Setting (a, b, c, d) = (m, 0, 1, 0) gives ShiftedFunc(x) = Func(m*x) a horizontal contraction by m.
• Setting (a, b, c, d) = (1, 0, n, 0) gives ShiftedFunc(x) = n*Func(x) a vertical expansion by n.
• Setting (a, b, c, d) = (-1, 0, 1, 0) gives ShiftedFunc(x) = Func(-x) a reflection across the y axis.
• Setting (a, b, c, d) = (1, 0, -1, 0) gives ShiftedFunc(x) = -Func(x) a reflection across thex axis.
It is also wothwhile to note functions that show off transformations clearly:
• Func(x)=abs(x) has a single corner.  Translations are easy to recognize.  However a horizontal contraction by m is indistinguishable from a vertical expansion by 1/m.
• Func(x)=x^2 has a single vertex.  Translations are easy to recognize.  However a horizontal contraction by m is indistinguishable from a vertical expansion by 1/m^2.
• Func(x)=sqrt(1-x^2) graphs as a half cicle.  Almost all transformations are easy to recognize.  The only problem is that a reflection across the y axis gives back the same graph.

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