#color(blue)("Method:")#
First simplify the equation so that it is in standard form of:
#color(white)("xxxxxxxxxxx)y=ax^2+bx+c#
Change this into the form:
#color(white)("xxxxxxxxxxx)y=a(x^2+b/ax)+c# This is NOT vertex form
Apply # -1/2xxb/a= x_("vertex")#
Substitute #x_("vertex")# back into the standard form to determine
#y_("vertex")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:#color(white)(.....) y=3(x-3)^2-x^2+12x-15#
#color(blue)("Step 1")#
#y=3(x^2-6x+9)-x^2+12x-15#
#y=3x^2-18x+27-x^2+12x-15#
#y=2x^2-6x+12# ...........................................(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#
Write as: #y=2(x^2-3x)+12#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
#color(green)(x_("vertex") = (-1/2)xx(-3)=+3/2)#.........................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4")#
Substitute value at (2) into equation (1) giving:
#y_("vertex")=2(3/2)^2-6(3/2)+12#
#y_("vertex")=18/4-18/2+12#
#y_("vertex")=18/4-36/4+12#
#color(green)(y_("vertex")=-9/2+12=15/2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#"Vertex "-> (x,y)->(3/2,15/2)->(1 1/2, 7 1/2)#
