  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 Na I, Fe I, and
$^{12}$CO after degrading the spectral resolution (R $\sim$ 1200) to
investigate the spectral behavior 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 $\mu$m
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 Na I and $^{12}$CO at 2.29 $\mu$m
lines yields a reliable empirical relation at [$Fe/H$] $\leq$ $-$0.4 dex, while
a linear fit of any line offers a good metallicity scale for stars having
[$Fe/H$] $\geq$ 0.0 dex.
