This is a hyperbola with a horizontal and a vertical translation. Can you go from here?
Remember that absolute values gobble up all negatives and turn them positive. Can you go from here?
For the first one, the range will be and not as written by the OP above (I can't quote it easily because it's inside another quote). Prove It's explanation is good, so try to break it down and understand Prove It's posts. For the second one, think of the range of |x-3| and then think how dividing that by 2 affects it ("hint": in this case it doesn't). For the last one, think about the range of |x-6| and then the range of |x-6| + 4, etc.