To determine the [tex]$y$[/tex]-intercept of the graph of the equation [tex]\( y = 6\left(x-\frac{1}{2}\right)(x+3) \)[/tex], we need to find the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = 6\left(x-\frac{1}{2}\right)(x+3) \)[/tex]:
[tex]\[
y = 6 \left(0 - \frac{1}{2}\right) (0 + 3)
\][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[
y = 6 \left(-\frac{1}{2}\right) (3)
\][/tex]
3. Calculate the product:
[tex]\[
y = 6 \left(-\frac{1}{2} \cdot 3\right)
\][/tex]
[tex]\[
y = 6 \left(-\frac{3}{2}\right)
\][/tex]
4. Multiply the constants:
[tex]\[
y = -9
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of the graph is [tex]\(-9\)[/tex].
The correct answer is [tex]\(\boxed{-9}\)[/tex].
