A linearly polarized laser pulse has been employed as a wiggler in a free-electron laser (FEL) in the presence of a plasma background for generating short wavelength radiation down to the extreme ultraviolet ray and X-ray spectral regions. Introducing plasma background in the FEL interaction region would lessen the beam energy requirement and also enhance both the beam current and the electron-bunching process. This configuration affords the possibility of scaling the device to more compact FELs and would have a higher tunability by changing the plasma density and the temperature of the electron beam. Electron trajectories have been analyzed using single-particle dynamics. The effect of plasma density on electron orbits has been investigated. A polynomial dispersion relation considering longitudinal thermal motion has been derived, by employing perturbation analysis. Numerical studies indicate that by increasing plasma density, the growth rate for groups I and II decreases, while the growth rate for group III increases. In addition, the effect of beam temperature and cyclotron frequency on the growth rate has been discussed. It has been found that by increasing the thermal velocity of the electron beam, the growth rate for groups I and III trivially decreases, while it increases for group II orbits. Besides, an increase in cyclotron frequency cause growth enhancement for group I orbits, while it present a growth decrement for group II and III orbits.