To find 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 determine the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is equal to 0. This is because the [tex]$y$[/tex]-intercept occurs where the graph intersects the [tex]$y$[/tex]-axis, and the [tex]$y$[/tex]-intercept is always at a point where [tex]$x = 0$[/tex].
Here are the steps to solve for the [tex]$y$[/tex]-intercept:
1. Substitute [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex].
2. Calculate the resulting value of [tex]\( y \)[/tex].
Following these steps:
[tex]\[
y = 6 \left( 0 - \frac{1}{2} \right) (0 + 3)
\][/tex]
First, simplify inside the parentheses:
[tex]\[
y = 6 \left( -\frac{1}{2} \right) (3)
\][/tex]
Next, multiply the terms together:
[tex]\[
y = 6 \cdot -\frac{1}{2} \cdot 3
\][/tex]
[tex]\[
y = -3 \cdot 3
\][/tex]
Finally:
[tex]\[
y = -9
\][/tex]
So, the [tex]$y$[/tex]-intercept of the graph of the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex] is [tex]\( -9 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{-9}
\][/tex]
