To complete a glide reflection for the point \( H(-2, -3) \) using the translation \( (x, y) \rightarrow (x - 2, y) \) and a reflection over the line \( y = -4 \), we'll follow these steps:
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Apply the translation: The translation \( (x, y) \rightarrow (x - 2, y) \) transforms point H(-2, -3) as follows:
\[ H' = (-2 - 2, -3) = (-4, -3) \]
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Reflect over the line \( y = -4 \): To reflect the point \( H'(-4, -3) \) over the line \( y = -4 \), we need to find the distance from \( H' \) to the line \( y = -4 \).
- The y-coordinate of \( H' \) is \(-3\).
- The line \( y = -4 \) is below \( H' \) (since \(-3 > -4\)).
- The distance from \( H' \) to the line \( y = -4 \) is:
\[ \text{Distance} = -3 - (-4) = -3 + 4 = 1 \]
This means the reflected point \( H'' \) will be 1 unit below the line \( y = -4 \):
\[ H'' = (-4, -4 - 1) = (-4, -5) \]
Hence, the point \( H'' \) after performing the glide reflection is:
\[ \boxed{(-4, -5)} \]