Vertical translation:

For \(\displaystyle{b}\ {>}\ {0}\),

The graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\ +\ {b}\) is the graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\) {shifted up b units}

The graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\ -\ {b}\) is the graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\ +\ {b}\) {shifted down b units}

Reflection:

Across the x-axis:

The graph of \(\displaystyle{y}=\ -{f{{\left({x}\right)}}}\) is the reflection of the graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\) across the x-axis

Across the y-axis:

The graph of \(\displaystyle{y}=\ {f{{\left(-{x}\right)}}}\) is the reflection of the graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\) across the x-axis

Vertically Stretching or Shrinking:

The graph of \(\displaystyle{y}=\ {a}{f{{\left(-{x}\right)}}}\) can be obtained from the graph of \(\displaystyle{y}={f{{\left({x}\right)}}}\) by

Stretching verttically for \(\displaystyle{m}{i}{d}{a}{m}{i}{d}\ {>}\ {1}\) or shrinking vertically for \(\displaystyle{a}\ {>}\ {m}{i}{d}{a}{m}{i}{d}\ {>}\ {1}\).

For \(\displaystyle{a}{<}{0}\), the graph is also reflected across the x-axis

. By the properties of transformation, the graph of \(\displaystyle{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}\) is,

The transformation of \(\displaystyle{y}={f{{\left({x}\right)}}}\) and shri

vertically by a factor of \(\displaystyle{\frac{{{1}}}{{{2}}}}\)

Then the graph is reflection about the x-axis and shifted up one unit.

From the given graphs, the graphs B satisfies the all condition's stated above.

Thus, the correct graph for the function \(\displaystyle{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}\) is B