Represent the following curves in the z-plane in the form z(t)

Question

Represent the following curve in the z-plane in the form z=z(t).

$$y = {x^2}\quad 1 \le x \le 3 $$

I know the answer is $$z(t) = t + i{t^2}\quad 1 \le t \le 3 $$ however I don't have any idea how to arrive at that answer.

Answer 1

z=z(t)= x(t) + i y(t) Now if you set x = t then as y=x² , we get y=t² . So we write z=t+it²