To determine the coordinates of point [tex]\( Q \)[/tex] before the reflection across the line [tex]\( x = -3 \)[/tex], we need to use the properties of reflections.
The reflection of a point [tex]\((x, y)\)[/tex] across a vertical line [tex]\( x = a \)[/tex] results in a new point whose coordinates are calculated as follows:
[tex]\[
(x', y') = (2a - x, y)
\][/tex]
Given the point [tex]\(Q'\)[/tex] is [tex]\((2, 4)\)[/tex] after the reflection and considering the vertical line [tex]\( x = -3 \)[/tex], we need to find the original coordinates [tex]\( Q \)[/tex].
Since [tex]\( Q' \)[/tex] is [tex]\((2, 4)\)[/tex] reflected across the line [tex]\( x = -3 \)[/tex], we use the formula for the reflection:
[tex]\[
Q' = (2a - x, y)
\][/tex]
Plugging in the values:
[tex]\[
Q' = (2 \cdot -3 - x, y) = (2, 4)
\][/tex]
We need to solve for [tex]\( x \)[/tex]:
[tex]\[
2(-3) - x = 2
\][/tex]
Simplifying this equation:
[tex]\[
-6 - x = 2
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-x = 2 + 6
\][/tex]
[tex]\[
-x = 8
\][/tex]
[tex]\[
x = -8
\][/tex]
Since the [tex]\( y \)[/tex]-coordinate does not change, [tex]\( y = 4 \)[/tex]. Therefore, the coordinates of point [tex]\( Q \)[/tex] are:
[tex]\(
(-8, 4)
\)[/tex]
Thus, the answer is [tex]\((D) (-8, 4)\)[/tex].
