Where in the world did the thunderstorm form and eventually become hurricane katrina

Hurricane Katrina formed over Southeastern Bahamas. 

At that time it was still a tropical storm - not yet as devastating. 
It then moved over Florida (around Florida is when it turned into a hurricane) and later into the Gulf of Mexico, where it gained strength and finally, it traveled towards the mainland of US. 

The whole existance of Katrina, from its beginning until its dissipation lasted 8 days.