Graphing the image of a complex number

1. Jan 12, 2009

torquerotates

1. The problem statement, all variables and given/known data

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

2. Relevant equations
z=x+iy

3. 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.

2. Jan 12, 2009

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.

3. Jan 13, 2009

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?

4. Jan 13, 2009

Dick

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

5. Jan 13, 2009

lol yeah.