Question

Complete a glide reflection to the point H(-2, -3) with the translation (x, y) → (x – 2, y) and a reflection over y = -4.



Find the point H”.

Answer 1

To perform the glide reflection to the point \( H(-2, -3) \) using the translation \( (x, y) \rightarrow (x - 2, y) \) and a reflection over the line \( y = -4 \), we will follow these steps:

1. **Apply the Translation:**
The translation \( (x, y) \rightarrow (x - 2, y) \) moves the point \( H(-2, -3) \).

\[
H' = (-2 - 2, -3) = (-4, -3)
\]

2. **Reflect over the line \( y = -4 \):**
To reflect the point \( H'(-4, -3) \) over the line \( y = -4 \), we first determine the distance from the point \( H' \) to the line \( y = -4 \).

The y-coordinate of point \( H' \) is \(-3\), and the line \( y = -4 \) is below this point. The distance from \( y = -3 \) to \( y = -4 \) is:

\[
-3 - (-4) = 1
\]

Since the reflection will move the point down by twice this distance (1 unit down to the line and another unit after crossing the line), the new y-coordinate will be:

\[
-4 - 1 = -5
\]

The x-coordinate remains the same during reflection, so:

\[
H'' = (-4, -5)
\]

Thus, the final point after completing the glide reflection is:

\[
\boxed{(-4, -5)}
\]