Fixed points of a Mobius transformation

Find and classify the fixed points of

f(z)=\frac{3z+1}{z+3}

Under iteration there are fixed points where f(z_{0})=z_{0}.
z_{0}=\frac{3z_{0}+1}{z_{0}+3}
z_{0}^2+3z_{0}=3z_{0}+1
z_{0}^2=1
There are fixed points are 1 and -1. To classify them, we need to plug them into |f'(z)| to determine whether they are repulsive, neutral, attractive, or superattractive.
|f'(z)|=|\frac{3}{z+3} - \frac{3z +1}{(z+3)^2}| = |\frac{8}{(z+3)^2}|
|f'(1)|=|\frac{8}{4^2}| = \frac{1}{2}, since 0< \frac{1}{2} <1, the fixed point at 1 is attractive.
|f'(-1)|=|\frac{8}{2^2}| = 2, since 1<2, the fixed point at -1 is repulsive.