jmc

Since the parabola has vertex $(4,2),$ the equation of the parabola is of the form $$y-2=k(x-4{)}^2.$$Since the parabola passes through $(2,0),$ we can plug in $x=2$ and $y=0,$ to get $$-2=4k,$$so $k=-\frac{1}{2}.$ Then $$y-2=-\frac{1}{2}(x-4{)}^2=-\frac{1}{2}{x}^2+4x-8,$$so the equation of the parabola is $\boxed{y=-\frac{1}{2}{x}^2+4x-6}.$