The interaction between the femtosecond laser pulses and the gas plasma has been widely used to generate strong and broadband THz pulse radiation. A femtosecond laser pulse with slow turn-on, rapid turn-off shape was used to generate THz radiation by interacting with gas plasma. Based on the plasma current model, the properties of the THz generation from such scheme were investigated in detail. Because the electrons were accelerated to a fast velocity by such specially shaped laser pulses, their motions form a fast oscillation current, which emits electromagnetic wave with frequency in the THz region. The results show that such scheme can generate stronger and broader THz radiation than the normal two-color femtosecond laser scheme, although it loses some energy of the laser pulse. This proposal might offer a new way to develop plasma-based broadband THz radiation source.