To determine the vertex of the quadratic equation [tex]\( y = -4x^2 - 16x - 12 \)[/tex], we use the formula for the vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex].
The vertex [tex]\((x, y)\)[/tex] of the parabola can be found using the following steps:
1. Find the x-coordinate of the vertex:
The x-coordinate of the vertex [tex]\( x_\text{vertex} \)[/tex] is given by
[tex]\[
x = \frac{-b}{2a}
\][/tex]
where [tex]\( a = -4 \)[/tex] and [tex]\( b = -16 \)[/tex].
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x = \frac{-(-16)}{2(-4)} = \frac{16}{-8} = -2
\][/tex]
2. Find the y-coordinate of the vertex:
Substitute [tex]\( x = -2 \)[/tex] back into the original equation to get [tex]\( y_\text{vertex} \)[/tex]:
[tex]\[
y = -4(-2)^2 - 16(-2) - 12
\][/tex]
Calculate each term step-by-step:
[tex]\[
-4(-2)^2 = -4 \times 4 = -16
\][/tex]
[tex]\[
-16(-2) = 32
\][/tex]
[tex]\[
-16 + 32 - 12 = 4
\][/tex]
Therefore, the coordinates of the vertex are [tex]\((-2, 4)\)[/tex].
The correct answer is:
[tex]\[
\boxed{(-2, 4)}
\][/tex]
