Help understanding textbook on Landau Ginzburg theory

[MisterX]

Main Question or Discussion Point

This question is about the Landau Ginzburg of phase transitions which seem to take this classical field theory form.
I don't understand the meaning of the 2nd to last equation
$$b(T) = b'(T -T_c) $$
does that mean b(T) has be redefined in the previous two equations relative to the original definition of $b(T)$. Does the prime $\prime$ indicate a derivative? Here is the entire case, with $A$ being the free energy of some scalar field m(r), which may be interpreted as magnetization. T is temperature.

 

[UVCatastrophe]

It's a coordinate change; both $b$ and $b'$ are functions. Suppose $f(x) = (x - a)^2$. What is $f(x-a)$ ? Does $f(x) = f(x-a)$ ? If not, can you find $g(x)$ such that $f(x-a) = g(x)$?