To determine the y-intercept of the graph 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 set to 0. The y-intercept occurs where the graph intersects the y-axis, which is when [tex]\( x = 0 \)[/tex].
Let's substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 6 \left( 0 - \frac{1}{2} \right)(0 + 3) \][/tex]
Now, simplify inside the parentheses first:
[tex]\[ y = 6 \left( -\frac{1}{2} \right) (3) \][/tex]
Next, multiply the values:
[tex]\[ y = 6 \times \left( -\frac{1}{2} \right) \times 3 \][/tex]
First, multiply [tex]\( 6 \times -\frac{1}{2} \)[/tex]:
[tex]\[ 6 \times -\frac{1}{2} = -3 \][/tex]
Now, multiply that result by 3:
[tex]\[ -3 \times 3 = -9 \][/tex]
Therefore, the y-intercept of the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right)(x + 3) \)[/tex] is [tex]\( y = -9 \)[/tex].
So the correct answer is:
[tex]\[ -9 \][/tex]
