CuTe is a two-dimensional (2D) layered material, yet forming a quasi-one-dimensional (quasi-1D) charge-density-wave (CDW) along the a-axis in the ab-plane at high transition temperature $T_{CDW}=335$ K. However, the anisotropic properties of CuTe remain to be explored. Here we performed combined transport, polarized infrared reflectivity, and ultrafast pump-probe spectroscopy to investigate the underlying CDW physics of CuTe. Polarized optical measurement clearly revealed that an energy gap gradually forms along the a-axis upon cooling, while optical evidence of gap signature is absent along the b-axis, suggesting pronounced electronic anisotropy in this quasi-2D material. Time-resolved optical reflectivity measurement revealed that the amplitude and relaxation time of photo-excited quasiparticles change dramatically across the CDW phase transition. Taking fast Fourier transformation of the oscillation signals arising from collective excitations, we identify the 1.65-THz mode as the CDW amplitude mode, whose energy softens gradually at elevated temperatures. Consequently, we provide further evidence for the formation of completely anisotropic CDW order in CuTe, which is quite rare in quasi-2D materials.