Yes, that is a painted turtle.
It looks pretty small, please tell me it's in an aquarium with a filter and clean water, basking spots, and regulated temps on either end of the tank, and proper lighting?
At that size/age it should be eating things like worms, small small crayfish, feeder fish (minnows and guppys), aquatic plants like anacharis, and small turtle pellets.
Painted turtles need UV light and heat lamps, and plenty of places to get out of the water if they choose, but still plenty of swimming room at same time. Even at that size the tank should be at least 30 gallons (about 1/3 full of water) to allow it room to grow and for temp regulation. By the time its full grown it should have at least 75 gallons for a tank size to allow it space enough to get exercise.