As the book says, an affine coordinate function of a line is $A\rightarrow \mathbb{R}$ and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is implicitly taken when defining the affine coordinate function.
Since for each of the 2 affine lines, the affine coordinate function is simply a number that is multiplied to the basis, what i think is that $f(\alpha )=r\alpha $. But this seems not to be correct. Why ?
The book is :Paul Bamberg, Shlomo Sternberg - A Course in Mathematics for Students of Physics Vol.1 . Pag 47 es 1.2
Affine lines do not have an "origin". That's pretty much what makes an affine line different from a coordinate axis.
When you pick an affine coordinate function, you are assigning coordinates to points on the affine line in a uniform way. Whichever point on the affine line happens to get the zero coordinate is chosen to the be origin relative to that choice of coordinate function, but it is not an intrinsic property of the affine line itself.
So there is no general requirement when one affine line is mapped onto to another affine line that some "origin" on the first affine line will map to an "origin" on the other affine line.
Therefore the most general form of relation from one affine coordinate function to another will have both the "scale" coefficient $r$ and the "shift translation" term $s$:
$$ F(\alpha) = r\alpha + s $$