To determine the [tex]\( y \)[/tex]-intercept 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]\( x \)[/tex] is equal to 0.
Here's the step-by-step process:
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = 6\left(0 - \frac{1}{2}\right)(0 + 3)
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
y = 6\left(-\frac{1}{2}\right)(3)
\][/tex]
3. Multiply the values:
[tex]\[
y = 6 \cdot -\frac{1}{2} \cdot 3
\][/tex]
4. Calculate the product step-by-step:
- First, multiply [tex]\(-\frac{1}{2} \cdot 3\)[/tex]:
[tex]\[
-\frac{1}{2} \cdot 3 = -\frac{3}{2}
\][/tex]
- Next, multiply [tex]\( 6 \cdot -\frac{3}{2} \)[/tex]:
[tex]\[
6 \cdot -\frac{3}{2} = 6 \cdot -1.5 = -9
\][/tex]
Thus, when [tex]\( x = 0 \)[/tex], [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].
The correct answer is [tex]\( -9 \)[/tex].
