To determine the image coordinates of point [tex]\(K\)[/tex] when reflected across the line [tex]\(y = -4\)[/tex], we need to follow a specific method for reflections. Let's go through the steps:
1. Identify the coordinates of point [tex]\(K\)[/tex]:
[tex]\(K(-4, -6)\)[/tex]
2. Understand the reflection concept:
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = c\)[/tex], the new y-coordinate, [tex]\(y'\)[/tex], of the reflected point can be determined using the formula:
[tex]\[
y' = 2c - y
\][/tex]
where [tex]\(c\)[/tex] is the y-coordinate of the line of reflection.
3. Plug in the known values:
- For the line [tex]\(y = -4\)[/tex], [tex]\(c = -4\)[/tex].
- The y-coordinate of [tex]\(K\)[/tex] is [tex]\(-6\)[/tex].
4. Calculate the new y-coordinate [tex]\(y'\)[/tex]:
[tex]\[
y' = 2(-4) - (-6) = -8 + 6 = -2
\][/tex]
5. The x-coordinate remains the same during a vertical reflection:
Therefore, the x-coordinate of [tex]\(K'\)[/tex] is still [tex]\(-4\)[/tex].
6. Combine the new coordinates:
Hence, the coordinates of [tex]\(K'\)[/tex] after reflection across the line [tex]\(y = -4\)[/tex] are:
[tex]\[
K'(-4, -2)
\][/tex]
7. Choose the correct multiple-choice answer:
The correct image coordinate for [tex]\(K'\)[/tex] is:
[tex]\[
K'(-4, -2)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{K^{\prime}(-4, -2)}
\][/tex]
