Question

What are the coordinates of Y′ after a reflection across y=1 ? (1 point) Responses Y′(−2,3) upper Y prime left parenthesis negative 2 comma 3 right parenthesis Y′(−2,−1) upper Y prime left parenthesis negative 2 comma negative 1 right parenthesis Y′(4,5) upper Y prime left parenthesis 4 comma 5 right parenthesis Y′(4,−1)

Answer 1

To find the coordinates of a point \( Y \) after it is reflected across the line \( y = 1 \), we can follow these steps:

1. **Identify the Original Point**: Let's denote the original point \( Y \) as \( (x, y) \).
2. **Calculate the Distance to the Line**: The distance from the point \( Y \) to the line \( y = 1 \) is given by \( d = y - 1 \).
3. **Determine the Reflective Point**: The reflected point \( Y' \) will be the same distance below the line as the original point is above it, thus you subtract \( d \) from \( 1 \):
\[
Y' = (x, 1 - d) = (x, 1 - (y - 1)) = (x, 2 - y).
\]

Now, let's examine the choices to see which point they represent after reflection across the line \( y = 1 \):

1. \( Y'(-2, 3) \):
- Reflection: \( y = 2 - 3 = -1 \). So \( Y'(-2, -1) \).
2. \( Y'(-2, -1) \):
- Reflection: \( y = 2 - (-1) = 3 \). So this is not the final answer.
3. \( Y'(4, 5) \):
- Reflection: \( y = 2 - 5 = -3 \). So \( Y'(4, -3) \) which is not listed.
4. \( Y'(4, -1) \):
- Reflection: \( y = 2 - (-1) = 3 \). So this is not the final answer.

Only the calculated point \( Y'(-2, -1) \) appears directly in your choices as the reflection of \( (-2, 3) \) over the line \( y = 1 \).

Therefore, the correct answer is:
**Y′(−2,−1)**.