To find the [tex]\( y \)[/tex]-intercept of the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex] in the [tex]\( xy \)[/tex]-plane, follow these steps:
1. Recall that the [tex]\( y \)[/tex]-intercept of a graph occurs where [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the given equation to determine the [tex]\( y \)[/tex]-value at this point.
[tex]\[ y = 6 \left( 0 - \frac{1}{2} \right) (0 + 3) \][/tex]
3. Simplify the expression inside the parentheses first:
[tex]\[ 0 - \frac{1}{2} = -\frac{1}{2} \][/tex]
[tex]\[ 0 + 3 = 3 \][/tex]
4. Now, substitute these simplified values back into the equation:
[tex]\[ y = 6 \left( -\frac{1}{2} \right) (3) \][/tex]
5. Multiply the terms together:
[tex]\[ y = 6 \cdot \left( -\frac{1}{2} \right) \cdot 3 \][/tex]
6. Perform the multiplication step-by-step:
[tex]\[ 6 \cdot -\frac{1}{2} = -3 \][/tex]
[tex]\[ -3 \cdot 3 = -9 \][/tex]
7. Thus, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ y = -9 \][/tex]
Therefore, 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]. Thus, the correct answer is:
[tex]\[ \boxed{-9} \][/tex]
