Graphing the image of a complex number

[torquerotates]

Homework Statement

sketch y=2 under the image w=z^(2)

Homework Equations

z=x+iy

The Attempt at a Solution

z=x+(1)i=x+i {y=1 and x can be anything}
w=z^(2)=(x+i)^(2)=x^2+2xi-1

after regrouping, w=(x^2-1)+(2x)i and then I consider x^2-1 to be the real part and 2x to be the imaginary part. This is as far as I can get because I have no clue how to graph this.

 

[Dick]

I think you just confused because there are a couple of different meanings of x around. Try it this way (if you mean y=1 and not y=2 as you posted). A general point on the line y=1 is given (as you said) by t+i where t is any number. Squaring gives x=t^2-1 and y=2t splitting real and imaginary parts. Solve the y equation for t and substitute into the x equation to get an equation involving only x and y.

 

[torquerotates]

Ok so I just put x=t^2-1 and y=2t. From this I can conclude that t=y/2 and that x=(y^4)/4-1. This is just a parabola opening to the right facing in the x direction. Is this correct?

 

[Dick]

Sounds correct to me. You did mean x=y^2/4-1, right?

 

[torquerotates]

lol yeah.