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”.

1 answer

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:

  1. Apply the translation: The translation \( (x, y) \rightarrow (x - 2, y) \) transforms point H(-2, -3) as follows:

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

  2. Reflect over the line \( y = -4 \): To reflect the point \( H'(-4, -3) \) over the line \( y = -4 \), we 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:

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