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.

Answer 1

To complete a glide reflection for the point \( H(-2, -3) \) using the translation \( (x, y) \to (x - 2, y) \) and a reflection over the line \( y = -4 \), we can break it down into two steps: first, apply the translation, and then perform the reflection.

### Step 1: Translation
We apply the translation to \( H(-2, -3) \):
\[
\text{New point} = (x - 2, y) = (-2 - 2, -3) = (-4, -3).
\]
So, after translation, we have the point:
\[
H'(-4, -3).
\]

### Step 2: Reflection
Next, we reflect the point \( H'(-4, -3) \) over the line \( y = -4 \).

To reflect a point over a horizontal line \( y = c \), we use the formula:
\[
\text{Reflected point} = (x, 2c - y).
\]
Here, \( c = -4 \) and our \( y \) coordinate is \(-3\).

Calculating the new \( y \) coordinate:
\[
\text{New } y = 2(-4) - (-3) = -8 + 3 = -5.
\]

Thus, the reflected point is:
\[
H''(-4, -5).
\]

### Conclusion
The final result of the glide reflection of the point \( H(-2, -3) \) through the specified translation and reflection is:
\[
H''(-4, -5).
\]