From this image find the rotation angle and the expansion factor.
$z\in\Bbb C$
I am unsure how to find a way to accurately plot the image. I understand that the image should expand because the vector falls outside of the circle $|z+1|\lt\frac{1}{2}$, but i am unsure of how to find the factor of expansion or angle of rotation.
So $f(i) = i^2 + 2i = -1+2i$ This means the vector $0 + i = (0,1)$ with angle $\frac{\pi}{2}$ and magnitude $1$ was sent to the vector $-1+2i = (-1,2)$ with angle equal to $\arctan(\frac{2}{-1}) = \arctan(-2)$ and magnitude $\sqrt{(-1)^2+2^2} = \sqrt{5}$. To me it sounds like the angle has "rotated" by $\arctan(-2) - \frac{\pi}{2}$ and the magnitude has "expanded by a factor" of $\sqrt{5}/1 = \sqrt{5}$ if that is what you are asking.
By the way, it's a little weird (but not wrong) to say `vector,' because f(z) is not a linear transformation on the real two-plane (not linear with respect to R and certainly not linear with respect to C).