How to graph this complex equation

Question

Trying to graph $Im((z + 4 - 4i)^2) = 1$ on the complex plane.

I'm having trouble interpreting this equation to sketch it by hand. Fully expanding this is not helpful (or practical) when trying to sketch this by hand.

I understand how to graph complex equations such as $|z|^2 = 25$ (a circle of radius 5) and $|z - 2 - 2i| = 3$ (circle of radius 3 with centre $2 + 2i$). You can substitute $z = x + iy$ for all of these types of problems; but, I generally understand their geometric form due to the meaning of the modulus and so on.

As for this hyperbola; I'm not sure of the approaches / steps required to sketch this by hand.

Answer 1

As you mention, we just plug in $z=x+iy$ to get $(z+4-4i)^2 = \{(x+4)+(y-4)i\}^2$.

The imaginary part is $2(x+4)(y-4)$ so we have

$$2(x+4)(y-4)=1$$

Your plot should look like a hyperbola.

Answer 2

The substiution $z=x+iy$ works just as well in this example. Do the algebra ... we get $2(x+4)(y-4)=1$. So the solution is a rectangular hyperbola with asymptotes at $x=-4$ and $y=4$.