Question

What are the coordinates of Y′ after a reflection across y=1?

(1 point)
Responses

Y′(4,−1)
upper Y prime left parenthesis 4 comma negative 1 right parenthesis

Y′(4,5)
upper Y prime left parenthesis 4 comma 5 right parenthesis

Y′(−2,−1)
upper Y prime left parenthesis negative 2 comma negative 1 right parenthesis

Y′(−2,3)

Answer 1

To find the coordinates of point Y′ after reflecting across the line \( y = 1 \), we first need the coordinates of point Y.

Let’s assume the coordinates of point Y are \((x, y)\). When reflecting a point across a horizontal line like \( y = k \), the new y-coordinate can be calculated as follows:

1. Find the distance of the original point from the line. This is given by \( d = y - k \).
2. The new y-coordinate after reflection will be \( k - d \).

In this case, we reflect across \( y = 1 \):

1. The distance from point Y to the line \( y = 1 \) is \( d = y - 1 \).
2. The new y-coordinate \( y' \) after reflection will be:

\[
y' = 1 - (y - 1) = 2 - y
\]

Now we can apply this transformation for the y-coordinates of the given responses, assuming the coordinate of Y was (4, -1):

1. Using \( y = -1 \):

\[
y' = 2 - (-1) = 2 + 1 = 3
\]

So, the reflected point Y′ becomes \( Y′(4, 3) \).

Using the options given:

- \( Y′(4, -1) \) - incorrect,
- \( Y′(4, 5) \) - incorrect,
- \( Y′(−2,−1) \) - incorrect,
- \( Y′(−2, 3) \) - incorrect because it has a wrong x-coordinate.

None of the given options correctly describe the new coordinates of the point after reflection based on the assumption that the original coordinates of Y are (4, -1).

To summarize, if Y is at (4, -1), the reflection across \( y = 1 \) yields \( Y′(4, 3) \), but this exact result is not listed in the responses.