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)
