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: \[ H''(-4, -5). \]
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