Question

Let w=f(z)=z^2+6+6i is a complex function. Then the image of the line x=5

 looks like what?

Answer 1

$w=u+iv=f(z)=(5+iy)^2+6+6i=31-y^2 +i(10y+6)$

$v=10y+6$ , hence $y=(v-6)/10$

$u=31-y^2=31-(v-6)^2/100$

Therefore the line $x=5$ transforms under the function $f(z)$ into a parabola with the vertex at (31,6) and the axis $v=6$.