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], you need to find the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
Here are the steps:
1. Start with the given equation:
[tex]\[
y = 6 \left( x - \frac{1}{2} \right) (x + 3)
\][/tex]
2. Substitute [tex]\( x = 0 \)[/tex] into the equation to find the [tex]\( y \)[/tex]-intercept:
[tex]\[
y = 6 \left( 0 - \frac{1}{2} \right) (0 + 3)
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
y = 6 \left( -\frac{1}{2} \right) (3)
\][/tex]
4. Perform the multiplications:
[tex]\[
y = 6 \cdot \left( -\frac{1}{2} \cdot 3 \right)
\][/tex]
[tex]\[
y = 6 \cdot \left( -\frac{3}{2} \right)
\][/tex]
5. Multiply the constants:
[tex]\[
y = 6 \cdot -\frac{3}{2} = -9
\][/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]\[
-9
\][/tex]
