Transformations from the z-plane to the w-plane Quiz

1. Find the image of $z=1+2i$ under $w=z^2$.

• A. $-3+4i$
• B. $5+4i$
• C. $-3+2i$
• D. $1+4i$

2. The transformation $w=2z+3i$ maps the circle $|z|=1$ to which circle?

• A. $|w|=2$
• B. $|w-3i|=1$
• C. $|w-3i|=2$
• D. $|w+3i|=2$

3. Under $w=\frac1z$, the circle $|z|=4$ maps to:

• A. $|w|=4$
• B. $|w|=16$
• C. $|w|=\frac1{16}$
• D. $|w|=\frac14$

4. Under $w=\frac1z$, the line $\mathrm{Re}\,z=1$ maps to:

• A. a straight line through the origin
• B. a circle through the origin
• C. a circle not through the origin
• D. a straight line not through the origin

5. Under $w=z^2$, the half-line $\arg z=\frac{\pi}{3}$ maps to:

• A. $\arg w=\frac{2\pi}{3}$
• B. $\arg w=\frac{\pi}{6}$
• C. $\arg w=\frac{\pi}{3}$
• D. $\arg w=\frac{\pi^2}{9}$