Question

1- How are translations represented as a function?


2- What is the relationship between a translation and a rigid motion?

Answer 1

Part two) I can help with - the relationship between the two, is that a rigid motion consists of both translations and rotations, which leaves the 'arrangement' so to speak unchanged.So there is a direct correlation between the two, as a rigid motion cannot exist without a translation.

Part One) I would say if you have a function f(x), translations in the y axis are usually denoted by something in the brackets with the x
So like y=(x+2) shifts the line y=x up the y axis by two
and y=x-2 moves the line y=x forward two along the x axis

Answer 2

Let me rephrase the top bit, I don't mean a rigid motion won't exist, I mean you won't get a rigid motion without a translation being involved