The general vertex form of a quadratic equation is #y = a(x - h)^2 + k#,
where the vertex of the graph of the function is the point #(h, k)#. To convert this quadratic equation from standard form to vertex form, we will follow the process of completing the square. We will first isolate the #x#-terms on the right side of the equation, and then factor out the coefficient of the #x^2#-term.
#y = 1/2x^2 + 12x - 8#
#y + 8 = 1/2x^2 + 12x - 8 + 8#
#y + 8 = 1/2x^2 + 12x#
#y + 8 = 1/2(x^2 + 24x)#
The equation is now in the form necessary to complete the square. We will find the square of half the coefficient of the #x#-term to complete the square.
#(24/2)^2 = 12^2 = 144#
So, we will use #144# to complete the square.
#y + 8 + 72 = 1/2(x^2 + 24x + 144)#
Note that we added #72# to the left side of the equation because on the right side, the #144# is inside parenthesis to be multiplied by #1/2#, and #1/2(144) = 72#. Now factor the right side of the equation.
#y + 80 = 1/2(x + 12)^2#
Now, isolate the #y#-term on the left side of the equation to have the equation in vertex form.
#y + 80 - 80 = 1/2(x + 12)^2 - 80#
#y = 1/2(x + 12)^2 - 80#
