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

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.
 

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.

• 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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