The scaling of the axial potential profile measured in the Tandem Mirror Experiment-Upgrade (TMX-U) is reported. These measurements were done with the end walls at ground potential. As predicted theoretically, the results suggest that a deep thermal barrier is achieved when the mirror confined electron fraction in the end cells exceeds 90% of the total density and when there is sufficient pumping of cold, potentially confined ions. In operation with shallow thermal barriers, a potential maximum near the inner turning point of the sloshing ions is frequently observed. The central cell confining potential associated with this local maximum is large enough to have a significant effect on the ion axial confinement time. Measurements of the total ion confining potential were difficult. There is evidence of classical scaling of the parallel particle confinement with the magnitude of the ion confining potential. However, in some cases, long axial confinement times are accompanied by enhanced ambipolar radial losses. These losses are interpreted as occurring in the population of passing ions. Finally, it is observed that the overall machine potential is determined by the plasma densities, the thermal barrier depth and the electron cyclotron heating power, consistent with theoretical predictions based only on collisional and radiofrequency diffusion of the electrons.