Given:#color(white)(..)y=-3x^2+9x+1...........(1)#
Write as:#color(white)(..)y=-3(x^2color(green)(-3x))+1#
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Consider the RHS only
Write as: #-3(x-3/2)^2+1.............................(2)#
The #(-3/2)# comes from halving the coefficient of #x " in "color(green)( -3x)#
Expression (2) has an inherent error which we need to correct
#-3(x-3/2)^2#
#=-3( x^2 -3x+9/4)#
#= -3x^2+9x-27/4...................(3)#
Add the constant of +1 as shown in equation (1) giving
#= -3x^2+9x-27/4 + 1...................(3_a)#
When you compare #(3_a)# to (1) you see that the error introduced is #-27/4#
We correct for this by removing it from the vertex form equations using #color(blue)(+27/4)#
Thus the #underline(color(red)("incorrect"))# form of #y=-3(x-3/2)^2+1 color(blue)(" is adjusted by:")#
#y=-3(x-3/2)^2+1color(blue)(+27/4)#
Giving:
#y=-3(x-3/2)^2color(brown)(+31/4)#
