We present here quantitative diagnostic tools for cool giants that employ low-resolution near-infrared spectroscopy in the K band for stellar population studies. In this study, a total of 260 cool giants (177 stars observed with X-shooter and 83 stars observed with NIFS) are used covering a wider metallicity range than in earlier works. We measure equivalent widths of some of the selected important K-band spectral features like NaI, FeI, and ^12^CO after degrading the spectral resolution (R~1200) to investigate the spectral behaviour with fundamental parameters (e.g. effective temperature and metallicity). We derive empirical relations to measure effective temperature using the ^12^CO first-overtone band at 2.29 and 2.32{mu}m and show a detailed quantitative metallicity dependence of these correlations. We find that the empirical relations based on solar-neighborhood stars can incorporate large uncertainty in evaluating T_eff_ for metal-poor or metal-rich stars. Furthermore, we explore all the spectral lines to establish the empirical relation with metallicity and find that the quadratic fit of the combination of NaI and ^12^CO at 2.29{mu}m lines yields a reliable empirical relation at [Fe/H]~<-0.4dex, while a linear fit of any line offers a good metallicity scale for stars having [Fe/H]>~0.0dex.