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 start by understanding that the [tex]\( y \)[/tex]-intercept occurs where the graph intersects the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex] because the [tex]\( y \)[/tex]-axis corresponds to [tex]\( x = 0 \)[/tex] in the coordinate plane.
Therefore, to find the [tex]\( y \)[/tex]-intercept, we substitute [tex]\( x = 0 \)[/tex] into the given equation:
[tex]\[ y = 6 \left( 0 - \frac{1}{2} \right) (0 + 3) \][/tex]
Now, let's simplify it step by step:
[tex]\[ y = 6 \left( -\frac{1}{2} \right) (3) \][/tex]
First, calculate [tex]\( 0 - \frac{1}{2} \)[/tex]:
[tex]\[ 0 - \frac{1}{2} = -\frac{1}{2} \][/tex]
Then, calculate [tex]\( 0 + 3 \)[/tex]:
[tex]\[ 0 + 3 = 3 \][/tex]
Next, multiply [tex]\(-\frac{1}{2}\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[ -\frac{1}{2} \times 3 = -\frac{3}{2} \][/tex]
Finally, multiply [tex]\(6\)[/tex] by [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ 6 \times -\frac{3}{2} = -9 \][/tex]
Thus, 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]\[ \boxed{-9} \][/tex]
