You've got the circuit for the pot to Arduino connection just as I set it up. I took this from a YouTube video series but mine goes to a Nano instead of a Uno. Other than that it's identical.
So as I understand it, the pot acts as a voltage divider to feed the analog input between 0 to 5 volts. But it's a ratio kinda thing so it could produce the short range of values at the end points? Anyway, that's all there is to the circuit.