The general vertex form is
color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)
for a parabola with vertex at (color(red)(a),color(blue)(b))
Given:
color(white)("XXX")y=1/2x^2-1/6x+6/13
rArr
color(white)("XXX")y=1/2(x^2-1/3x)+6/13
color(white)("XXX")y=1/2(x^2-1/3x+(1/6)^2)+6/13-1/2*(1/6)^2
color(white)("XXX")y=1/2(x-1/6)^2+6/13-1/72
color(white)("XXX")y=1/2(x-1/6)^2+(6*72-1*13)/(13*72)
color(white)("XXX")y=color(green)(1/2)(x-color(red)(1/6))^2+color(blue)(409/936)
which is the vertex form with vertex at (color(red)(1/6),color(blue)(409/936))
The graph below of the original equation indicates that our answer is at least approximately correct.
graph{1/2x^2-1/6x+6/13 [-0.6244, 1.0606, -0.097, 0.7454]}
