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Transformation 3: Reflections

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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3.3 Transformation 3: Reflection

Reflection

  • Reflection is a more familiar transformation to us: we usually see our reflections in a mirror.
  • In Cartesian plane, a line will be the ‘mirror’, and the function will be reflected along the ‘mirror’. Therefore, the line can be horizontal, vertical, or diagonal. Here, we will only consider horizontal and vertical reflection, as diagonal reflections are too complex for this course. In particular, we will only study reflections along the x-axis and y-axis.
  • A reflection along the x-axis gives the following transformation:

(x,\ y) \rightarrow(x,\ -y)

y=-f(x)

and a reflection along the y-axis gives the following transformation:

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