Plot[
{V^2/144`,
Sequence[
Piecewise[{{V^2/344`, V < 12}}, I],
Piecewise[{{V^2/344`, V > 12}}, I]
]
},
{V, 8, 18},
PlotRange -> {0, 1.5},
PlotStyle -> AbsoluteThickness[3],
GridLines -> Automatic,
Epilog -> {Thick, Dashed, Black,
Line[{{{12, 0}, {12, 2}}, {{8, 1}, {18, 1}}}]}
] /. {sty__, x_Line, y_Line} :> {sty, y, Dashed, x}