In the [tex]\(xy\)[/tex]-plane, what is the [tex]\(y\)[/tex]-intercept of the graph of the equation [tex]\(y=6\left(x-\frac{1}{2}\right)(x+3)\)[/tex]?

A. [tex]\(-9\)[/tex]
B. [tex]\(-\frac{1}{2}\)[/tex]
C. 3
D. 9



Answer :

To find the [tex]$y$[/tex]-intercept of the graph of the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex], we need to determine the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is equal to 0. This is because the [tex]$y$[/tex]-intercept occurs where the graph intersects the [tex]$y$[/tex]-axis, and the [tex]$y$[/tex]-intercept is always at a point where [tex]$x = 0$[/tex].

Here are the steps to solve for the [tex]$y$[/tex]-intercept:

1. Substitute [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex].
2. Calculate the resulting value of [tex]\( y \)[/tex].

Following these steps:

[tex]\[ y = 6 \left( 0 - \frac{1}{2} \right) (0 + 3) \][/tex]

First, simplify inside the parentheses:

[tex]\[ y = 6 \left( -\frac{1}{2} \right) (3) \][/tex]

Next, multiply the terms together:

[tex]\[ y = 6 \cdot -\frac{1}{2} \cdot 3 \][/tex]

[tex]\[ y = -3 \cdot 3 \][/tex]

Finally:

[tex]\[ y = -9 \][/tex]

So, 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].

Therefore, the correct answer is:

[tex]\[ \boxed{-9} \][/tex]