the point z(4,-2) is rotated 180 about the origin. what is the image of z?
You need to determine the coordinates of symmetric point to origin, hence, first you need to determine the quadrant the point z is in.
Since x coordinate is positive and y coordinate is negative, the point z is in quadrant 4, hence, the symmetric point is in quadrant 2.
Since the image of the point z is in quadrant 2, the x coordinate is negative and y coordinate is positive such that:
Notice that x and y coordinates of image point z'(-4,2) have opposite values to the coordinates of original point z(4,-2).