We report on the numerical simulation results of the propagation of Fourier-transformed super-Gaussian beams with cubic phase modulation, which we refer to as super-Airy beams. We show the effect of the truncation profile on the propagation of the "Airy" intensity distribution and propose methods to improve the non-diffracting propagation length. We show that super-Gaussian beams with cubic phase modulation would form, what we believe to be, truncated Airy beams after being optically Fourier-transformed. The results show that Airy beams with smaller curvature would preserve their non-diffracting characteristics for longer propagation distances than Airy beams with larger curvature. Airy beam curvature is a controllable parameter by the phase transformation step. The presented results suggest that Fourier-transforming super-Gaussian beams with cubic phase modulation would form Airy beams with up to > 350% increase in their non-diffracting propagation distance compared to Fourier-transformed Gaussian beams.