Question

what would the following equation be in vertex form. y=-9x squared+72x+81.

I have tried for hours and i cannot figure it out

Answer 1

Bring constant to other side: y-81=9x^2+72x
Make constant in front of x^2 term one: (y-81)/9=x^2+8x
(y/9)-9=x^2+8x

complete the square...(x+a)^2= x^2+2abx+a^2.

x^2=x^2 so
8x=2ax= so a=4

4^2=a^2=16

The equation you want to create is (x+4)^2= x^2+8x+16

previously we had (y/9)-9=x^2+8x

so add 16 to make the equation desired
(y/9)-9=x^2+8x
+16 +16

(y/9)+7=x^2+8x+16
(y/9)+7=(x+4)^2

and now solve for y:

subtract 7:(y/9)=(x+4)^2-7

multiply by 9:y=[(x+4)^2-7]9
y=9(x+4)^2-7(9)
y=9(x+4)^2-63

Vertex form: a(x-h)^2+k=y where (h,k) is the coordinate of the vertex.

You have: 9(x+4)^2-63=y where (-4,-63) is the coordinate of the vertex.

:)