We are starting with the parent function \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}\)

STEP 1: Vertically compress the graph by a factor of \(\displaystyle\frac{{1}}{{4}}\), to get \(\displaystyle{y}={\left(\frac{{1}}{{4}}\right)}{x}^{{2}}\)

STEP 2: Reflect the graph across x-axis, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{x}^{{2}}\)

STEP 3: Shift the graph by 2 units to the left, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{\left({x}+{2}\right)}^{{2}}\)

STEP 4: Shift the graph by 2 units downwards, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{\left({\left({x}+{2}\right)}^{{2}}\right)}-{2}\) which is the required function g(x)

STEP 1: Vertically compress the graph by a factor of \(\displaystyle\frac{{1}}{{4}}\), to get \(\displaystyle{y}={\left(\frac{{1}}{{4}}\right)}{x}^{{2}}\)

STEP 2: Reflect the graph across x-axis, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{x}^{{2}}\)

STEP 3: Shift the graph by 2 units to the left, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{\left({x}+{2}\right)}^{{2}}\)

STEP 4: Shift the graph by 2 units downwards, to get \(\displaystyle{y}=-{\left(\frac{{1}}{{4}}\right)}{\left({\left({x}+{2}\right)}^{{2}}\right)}-{2}\) which is the required function g(x)