Y=9x^2+9x-1 written in vertex form

Y=9x^2+9x-1 is to be re-written in "vertex form," i. e., y-k = a (x-h) ^2, where a is a constant coefficient and (h, k) is the vertex.

Y=9x^2+9x-1 can be re-written as y - 1 = 9 (x^2 + x).

We can complete the square of x^2 + x:

x^2 + 2 (1/2) + (1/2) ^2 - (1/2) ^2

and this can be simplified as follows: (x+1/2) ^2 - 1/4

Go back to y - 1 = 9 (x^2 + x) and subst. (x+1/2) ^2 - 1/4 for (x^2 + x):

y - 1 = 9[ (x+1/2) ^2 - 1/4]

Then the desired equation in vertex form is y-1 = 9[ (x+1/2) ^2 - (-1/4) ]

or y - 1 = 9 [ (x+1/2) ^2 ] + 9/4, or y - 1 - (9/4) = 9 (x+1/2) ^2, or,

finally, y - (13/4) = 9 (x+1/2) ^2. The vertex is at (1/2, 13/4).