I think you want to know how to determine the bifurcation at mu=0.75?
So f_mu has two fixed points. One at 0 and one at (-1+4mu)/(4mu). The single line up to mu=0.75 follows (-1+4mu)/(4mu) because it is a stable fixed point. Then at mu=0.75 we have that f_\mu'(-1+4mu)/(4mu)=-1 and for mu>0.75 we have that |f_\mu'(-1+4mu)/(4mu)|>1 so it becomes unstable.
If you investigate the map f_mu \circ f_mu you see that two fixed points are being "created" as we pass mu=0.75. For the original map f_mu the fixed points of f_mu \circ f_mu correspond to a period 2-orbit.
This transition is called a transcritical bifurcation because the stability of the fixed point at (-1+4mu)/(4mu) changes.