To rewrite the equation [tex]\(y = 9x^2 + 9x - 1\)[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic equation is given by [tex]\( y = a(x-h)^2 + k \)[/tex].
Let's follow these steps to convert [tex]\(y = 9x^2 + 9x - 1\)[/tex] into vertex form:
1. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[
y = 9(x^2 + x) - 1
\][/tex]
2. Complete the square inside the parentheses. To do this, we look at the coefficient of [tex]\(x\)[/tex], take half of it, square it, and add and subtract this square inside the parentheses.
The coefficient of [tex]\(x\)[/tex] inside the parentheses is 1. Half of 1 is [tex]\(\frac{1}{2}\)[/tex], and squaring it gives [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex].
Add and subtract [tex]\(\frac{1}{4}\)[/tex] inside the parentheses:
[tex]\[
y = 9(x^2 + x + \frac{1}{4} - \frac{1}{4}) - 1
\][/tex]
This can be rearranged as:
[tex]\[
y = 9\left((x + \frac{1}{2})^2 - \frac{1}{4}\right) - 1
\][/tex]
3. Distribute the factor 9 and simplify:
[tex]\[
y = 9(x + \frac{1}{2})^2 - 9 \cdot \frac{1}{4} - 1
\][/tex]
[tex]\[
y = 9(x + \frac{1}{2})^2 - \frac{9}{4} - 1
\][/tex]
[tex]\[
y = 9(x + \frac{1}{2})^2 - \frac{9}{4} - \frac{4}{4}
\][/tex]
[tex]\[
y = 9(x + \frac{1}{2})^2 - \frac{13}{4}
\][/tex]
Therefore, the vertex form of the equation [tex]\(y = 9x^2 + 9x - 1\)[/tex] is:
[tex]\[
y = 9\left(x + \frac{1}{2}\right)^2 - \frac{13}{4}
\][/tex]
Thus, the correct choice is:
[tex]\[y=9\left(x+\frac{1}{2}\right)^2-\frac{13}{4}\][/tex]
