Transmission jumping hard is the malfunction causing check engine light to come on,most likely its the tcc solonoid causing this or a possibility of causing this.
The check engine light (service engine soon) comes on and stays on when a problem is detected by the self diagnosis system of your vehicle. Generally, the problem is in the emissions or something...
That means that the computer has stored a "fault code". You'll need to get a code scanner to find out which component is having problems. You can usually borrow, rent or buy them at most auto parts...
You can take your car auto parts place like autozone, hilow, etc, they should have a portable computer to attached to your car and it will give them a code for the reason of the check engine light is...