To rewrite the equation [tex]\( y = 9x^2 + 9x - 1 \)[/tex] in vertex form, we need to express it in the form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's begin by comparing the given quadratic equation to the standard form [tex]\( y = ax^2 + bx + c \)[/tex]:
- Here, [tex]\( a = 9 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -1 \)[/tex].
Next, we calculate the vertex [tex]\((h, k)\)[/tex] using the vertex formula:
1. The x-coordinate of the vertex [tex]\((h)\)[/tex] is given by:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
h = -\frac{9}{2 \cdot 9} = -\frac{9}{18} = -\frac{1}{2}
\][/tex]
2. The y-coordinate of the vertex [tex]\((k)\)[/tex] is found by substituting [tex]\( h \)[/tex] back into the equation:
[tex]\[
k = c - \frac{b^2}{4a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
k = -1 - \frac{9^2}{4 \cdot 9} = -1 - \frac{81}{36} = -1 - \frac{9}{4} = -\frac{4}{4} - \frac{9}{4} = -\frac{13}{4}
\][/tex]
So, the vertex is [tex]\( \left( -\frac{1}{2}, -\frac{13}{4} \right) \)[/tex].
Now, substitute [tex]\( a = 9 \)[/tex], [tex]\( h = -\frac{1}{2} \)[/tex], and [tex]\( k = -\frac{13}{4} \)[/tex] into the vertex form equation:
[tex]\[
y = 9 \left( x + \frac{1}{2} \right)^2 - \frac{13}{4}
\][/tex]
Comparing this result with the given options:
- [tex]\( y = 9 \left( x + \frac{1}{2} \right)^2 - \frac{13}{4} \)[/tex]
- [tex]\( y = 9 \left( x + \frac{1}{2} \right)^2 - 1 \)[/tex]
- [tex]\( y = 9 \left( x + \frac{1}{2} \right)^2 + \frac{5}{4} \)[/tex]
- [tex]\( y = 9 \left( x + \frac{1}{2} \right)^2 - \frac{5}{4} \)[/tex]
The correct equation that represents [tex]\( y = 9x^2 + 9x - 1 \)[/tex] in vertex form is:
[tex]\[
\boxed{y = 9 \left( x + \frac{1}{2} \right)^2 - \frac{13}{4}}
\][/tex]
