image of a circle

Hint:

Do this intuitively and visually.

Think of this mapping as a function from R^2 to R^2, and think of w and z as vectors in R^2.

Ask yourself what the image of a particular point on the circle will be under this mapping. Then ask yourself what the image of the entire circle will look like. Then find an equation that describes that image. Try to visualise what the addition of 1/2 and the multiplication by 1/2 are doing to points on your circle. How are those operations changing what your circle looks like? What do picture you end up with?

And don't forget that your circle's image is technically in the uv-plane, not the xy-plane...

...so the equation of your image should be written in terms u and v, instead of x and y.

Oh, and one more thing...

It will be easier to do this problem intuitively and visually if you first rewrite your mapping as:

w = 1/2 * z + 1/4

(In any case, my approach only works here because this mapping is a very simple one. Normally, you'd have to solve for x and y in terms of u and v and substitute what you got back into your original equation to get the answer.)