As long as you follow the basic rules of mathematics you can change any equation into other forms.
Note that the (x-h)^2 part occurs when completing the square so lets have a play with that approach and see what we get.
Starting point:
Given:" "y=-3x^2color(white)(.)+6x" "+10
Write as: " "y=-3(x^2+6/(-3)x)+10+k
At this stage k=0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 1")" - halve the "6/(-3)x
"y=-3(x^2+6/(-3xx2)x)+10+k
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 2")" - remove the "x" from "6/(-3xx2)x
"y=-3(x^2+6/(-3xx2))+10+k
but 6/(-3xx2)=-1 giving:
"y=-3(x^2-1)+10+k
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 3") - move the exponent (power) outside the brackets
"y=-3(x-1)^2+10+k
Note that -3(-1)^2+k=0 so k=+3-> positive
"y=-3(x-1)^2+13
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 4") - move the 13 to the other side
y-13=-3(x-1)^2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 5") - divide both sides by -3
-1/3(y-13)=(x-1)^2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 6") - forcing the derived equation into the required format.
Target format: " "(x-h)^2=4p(y-k)
We have got the (x-h)^2 part ->(x-1)^2
We have got the (y-k) part ->(y-13)
Set -1/3=4p
Divide both sides by 4 =>p=-1/(3xx4)=-1/12
Thus we have:" "(x-1)^2=4(-1/12)(y-13)
![Tony B]()
