To find the [tex]\( y \)[/tex]-intercept of the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) ( x + 3 ) \)[/tex], we set [tex]\( x = 0 \)[/tex] because the [tex]\( y \)[/tex]-intercept is the point where the graph of the equation crosses the [tex]\( y \)[/tex]-axis.
Let's substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 6 \left( 0 - \frac{1}{2} \right) ( 0 + 3 ) \][/tex]
Now simplify the expression inside the parentheses:
[tex]\[ = 6 \left( -\frac{1}{2} \right) ( 3 ) \][/tex]
Multiply the constants:
[tex]\[ = 6 \left( -\frac{1}{2} \times 3 \right) \][/tex]
[tex]\[ = 6 \left( -\frac{3}{2} \right) \][/tex]
Finally, multiply:
[tex]\[ = 6 \times -\frac{3}{2} \][/tex]
[tex]\[ = -18 / 2 \][/tex]
[tex]\[ = -9 \][/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].
