Image of x+y=1 under f(z) = z^2

[complexnumber]

For f(z) = z^2 find the image of x + y = 1

f(z) = z^2 = (x + iy)^2 = x^2 + 2ixy - y^2

u(x,y) = x^2 - y^2
v(x,y) = 2xy

 

[Mark44]

Here's how I would approach this problem. Every point on the line has coordinates (in the complex plane) of (x, 1 - x). A point on this line can be represented as z = x + i(1 - x). What does your function f do to these complex numbers?