Summary: We consider randomly flashing ratchets, where the potential acting can be switched to another at random time instants with Poisson statistics. Using coupled Fokker-Planck equations, we formulate explicit expressions of mean velocity, dispersion and quantities measuring thermodynamics. How potential landscapes and transitions affect the motility and energetics is exemplified by numerical calculations on random on-off ratchets. Randomly flashing ratchets with shifted sawtooth potentials are further discussed. We find that the dynamics and output power of such system present symmetry w.r.t. the shift between the two potentials \(\Delta_{\max} + \Delta_{\min}\), which is the sum of the shift between the two peaks \(( \Delta_{\max})\) and the shift between the two bottoms \(( \Delta_{\min})\). The mean velocity and output power both reach the optimal performance at \(\Delta_{\max} + \Delta_{\min} = 1\), provided that the asymmetry \(\alpha_i\) of potential \(U_i\) implies a positive flux respectively, i.e.\(, \alpha_i > 0.5\) for \(i = 1, 2\).