We report an asymmetric behavior of optical pulses during their propagation through a time-varying linear optical medium. The refractive index of the medium is considered to be varying with time and complex, such that a sufficient amount of gain and loss is present to realize their effect on pulse propagation. We have exploited the universal formula for optical fields in time-varying media. Numerically simulated results reveal that pulses undergo opposite temporal shifts around their initial center position during their bi-directional propagation through the medium along with corresponding spectral shifts. Moreover, the peak power and accumulated chirp (time derivative of accumulated phase) of the output pulse in both propagation directions are also opposite in nature, irrespective of their initial state. Numerically simulated behavior of the pulses agrees well with the analytical solutions. © 2018 Optical Society of America.