In thins activity, you can study how an image [math]w[/math] (on the right widow) of a complex number [math]z[/math] (on the left window) is constructed by a function[br][math]f:\mathbb{C}\longrightarrow\mathbb{C}[/math][br][math]z\mapsto w=f\left(z\right)=z^2-z[/math][br]You can change this function by double clicking on image point [math]w_i[/math] and modify the formula [math]z_i^2-z_i[/math][br]where [math]i[/math] vary from 1 to 5:[br][math]z_1=i[/math] [br][math]z_2=1+i[/math] [br][math]z_3=-2-5i[/math][br]you can move them to any other positions and see the result on the right window.[br]

The number [math]z_4=x_0+iy[/math] describe a point on the line segment [math]\left(x=x_0;-2\le y\le2\right)[/math] and it's image[br][math]w_4=z_4^2-z_4=x_0^2-x_0-y^2+iy\left(2x_0-1\right)=a-y^2+iby[/math][br]where [math]a=x_0^2-x_0[/math] and [math]b=2x_0-1[/math] are constants who depend on the value of [math]x_0[/math] given by the slider.[br]The image as you see it is a portion of parabola[br][math]u=a-\left(\frac{v}{b}\right)^2[/math][br]on the plane [math]w=u+iv[/math] with [math]-2\left|b\right|\le v\le2\left|b\right|[/math] corresponding to [math]-2\le y\le2[/math].[br]As conclusion, the image of a line (or a portion of it) is a parabola (or a portion of it).[br][br]To well illustrate this problem, we added another example:[br]The image of a square is a complicated form who we can't describe it by a simple equation. To find this image, we used the tracing technique. You can move the brown point [math]z_5[/math] on the square and see the resulting form of the structured set of points [math]w_5[/math].[br]