To find the value of [tex]\( d \)[/tex] in the given quadratic equation [tex]\( y = a(x-2)(x+4) \)[/tex], we'll follow these steps:
1. Identify the properties of the parabola:
The given equation [tex]\( y = a(x-2)(x+4) \)[/tex] can be expanded into standard quadratic form, but it's also beneficial to recognize the roots and symmetry properties directly from the factored form.
2. Determine the vertex [tex]\( x \)[/tex]-coordinate:
The roots of the quadratic equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = -4 \)[/tex]. The vertex [tex]\( x \)[/tex]-coordinate is the midpoint of the roots.
[tex]\[
c = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1
\][/tex]
3. Find the vertex [tex]\( y \)[/tex]-coordinate [tex]\( d \)[/tex]:
To find [tex]\( d \)[/tex], substitute [tex]\( x = c = -1 \)[/tex] back into the quadratic equation [tex]\( y = a(x-2)(x+4) \)[/tex].
[tex]\[
d = a(-1 - 2)(-1 + 4)
\][/tex]
Simplify inside the parentheses:
[tex]\[
d = a(-3)(3)
\][/tex]
Multiplying the values:
[tex]\[
d = a \cdot (-3) \cdot 3 \implies d = a \cdot (-9) \implies d = -9a
\][/tex]
Thus, the value of [tex]\( d \)[/tex] is equal to [tex]\( -9a \)[/tex], and the correct answer is:
A) [tex]\( -9a \)[/tex]
