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 equation [tex]\( y = 6 \left( x - \frac{1}{2} \right) (x + 3) \)[/tex] in the [tex]\( xy \)[/tex]-plane, follow these steps:

1. Recall that the [tex]\( y \)[/tex]-intercept of a graph occurs where [tex]\( x = 0 \)[/tex].

2. Substitute [tex]\( x = 0 \)[/tex] into the given equation to determine the [tex]\( y \)[/tex]-value at this point.

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

3. Simplify the expression inside the parentheses first:

[tex]\[ 0 - \frac{1}{2} = -\frac{1}{2} \][/tex]
[tex]\[ 0 + 3 = 3 \][/tex]

4. Now, substitute these simplified values back into the equation:

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

5. Multiply the terms together:

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

6. Perform the multiplication step-by-step:

[tex]\[ 6 \cdot -\frac{1}{2} = -3 \][/tex]
[tex]\[ -3 \cdot 3 = -9 \][/tex]

7. Thus, the [tex]\( y \)[/tex]-intercept is:

[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]. Thus, the correct answer is:

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