Question 748054
This  solution will be taken part of the way only:


{{{f(x)=(1/2)x^2+4x+19/3}}}, complete the square,..
{{{f(x)=(1/2)*(x^2+8x+38/3)}}}
The term necessary will be {{{(8/2)^2=16}}}
{{{f(x)=(1/2)(x^2+8x+16+38/3-16)}}}
{{{f(x)=(1/2)((x+4)^2+(38-3*16)/3)}}}
{{{f(x)=(1/2)((x+4)^2+2*(19-3*8)/3)}}}
{{{f(x)=(1/2)((x+4)^2+2*6/3)}}}
{{{f(x)=(1/2)((x+4)^2+2*2)}}}
{{{highlight(f(x)=(1/2)(x+4)^2+2)}}}


Now from the standard form of this function, the vertex, symmetry axis, and knowing that the vertex is either min or max can be read directly from the equation.  


{{{graph(300,300,-10,12,-10,12,(1/2)(x+4)^2+2)}}}