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Transformation 2: Dilation

3.9 Inverse Transformations

Note: In this note, we will only cover the inverse for translation(s) and dilation(s).

Inverse Transformations

  • A generic way of writing any combination of translation and dilations (which already includes reflections) is

y=A f(n(x+b))+c

where A,\ n,\ b,\ c \in R,\ n \neq 0.

In this transformation, we have (x,\ y) \rightarrow\left(\frac{x-b}{n},\ A y+c\right).

A simple idea is: Try to cancel out the effects on x. So x^{\prime}=\frac{x-b}{n}. As we originally have y=f(x), so f(n(x+b))=y if we adjust x to cancel n and b. So the new , here labelled as y^{\prime} to avoid confusion, is y^{\prime}=A y+c.

And finally (x,\ y) \rightarrow\left(x^{\prime},\ y^{\prime}\right) and sub x^{\prime},\ y^{\prime} to get the expression above. This is covered in 3.4 Combination of Transformations in more detail.

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3.2 Transformation 2: Dilation

Dilation

  • Dilation is a type of transformation. A translation is simply a ‘stretch’ or a ‘shrink’. The direction is important too, hence it can be a stretch along the x- and/or y-axis.
  • ‘A dilation of factor h from the x-axis and of factor k from the y-axis’, where h and k are positive numbers, is described by the rule

(x,\ y) \rightarrow(k x,\ h y)

Note: Notice the positions of and and their corresponding axis in the description.

  • If a function initially is y=f(x), the new function under the translation as described above is
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