X-rays are widely used in the field of optical imaging. A novel type of X-ray source is the relativistic nonlinear Thomson scattering (RNTS). Within the framework of RNTS, we investigated in detail the effects of different intensities of linearly polarized tightly focused lasers on the maximum radiation power, optimal position of an electron (initial position of the electron that yields the maximum radiation power in the entire space), as well as the motion and spatial radiation features of the electron at the optimal position. Our results reveal, for the first time, that the optimal position and maximum radiation power of the electron exhibit significant linear and exponential dependences, respectively, on the laser amplitude. After interacting with the laser, the initially stationary electron at the optimal position first undergoes an oscillatory motion and then moves linearly. The entire trajectory of the electron is asymmetric. As the laser intensity increases, the spatial and angular distributions of the radiation become smaller, and the collimation of radiation increases. In the direction of maximum power radiation, the azimuth angle φ remains at 0°, whereas the corresponding polar angle θ decreases from 41° to 20°, indicating that the spatial radiation approaches the z-axis. The time when the maximum radiation power is obtained is approximately 50.7 fs. Additionally, the asymmetry and micro-double-peak structure of the temporal spectra, as well as the modulation characteristics of the frequency spectra, were further investigated in the direction of the maximum power radiation. These findings will assist researchers in achieving high-power and ultrashort X-rays in experiments, thereby increasing the resolution and imaging speed of optical imaging.