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.