To wrap a mattress when moving, you will need a mattress bag that is the correct size for your mattress. You can then follow these steps:
Lay the mattress bag flat on the ground.
Keep the mattress on top of the bag, the long edges of the mattress should aligned with the mattress bag.
Fold the short ends of the bag over the mattress and secure them with tape.
Roll the mattress up tightly, starting from the bottom.
Secure the rolled mattress with straps.