How do i show that this function preserves angle? There is no function given along which this is mapped. I know that its derivative is $f'(z)=2z+1$ so it is conformal for all points except 1 and $z_0$ not equal to zero. But how do i show this preserves angle?
An analytic function is conformal precisely where its derivative is non-$0$. Thus, conformality holds everywhere except at $z=-\frac12$.