To solve the problem of reflecting point [tex]\( A \)[/tex] with coordinates [tex]\((-4, 3)\)[/tex] in the line [tex]\( y = 2 \)[/tex], let's proceed with the following steps:
1. Identify the horizontal line of reflection [tex]\( y = 2 \)[/tex]:
The line [tex]\( y = 2 \)[/tex] is a horizontal line, which means it runs parallel to the x-axis at [tex]\( y = 2 \)[/tex].
2. Determine the vertical distance from point [tex]\( A \)[/tex] to the line [tex]\( y = 2 \)[/tex]:
Since point [tex]\( A \)[/tex] has coordinates [tex]\((-4, 3)\)[/tex] and the line of reflection is at [tex]\( y = 2 \)[/tex], we calculate the vertical distance as follows:
[tex]\[
\text{Distance to line} = 3 - 2 = 1
\][/tex]
3. Reflect point [tex]\( A \)[/tex] over the line [tex]\( y = 2 \)[/tex]:
To reflect point [tex]\( A \)[/tex], we need to take the distance to the line and subtract it twice from the original y-coordinate of point [tex]\( A \)[/tex]. The reflection formula in this context is:
[tex]\[
\text{Reflected y-coordinate} = 3 - 2 \times 1 = 3 - 2 = 1
\][/tex]
4. Determine the coordinates of the reflected point:
The x-coordinate remains unchanged during a reflection over a horizontal line. Therefore, the new coordinates of the reflected point are:
[tex]\[
(\text{Reflected x-coordinate}, \text{Reflected y-coordinate}) = (-4, 1)
\][/tex]
Thus, the coordinates of the image of point [tex]\( A \)[/tex] after reflection in the line [tex]\( y = 2 \)[/tex] are:
[tex]\[(-4, 1)\][/tex]
