Question

Find the coordinates of Z′ after a reflection across y=1 . (1 point) Responses Z′(4,0) upper Z prime left parenthesis 4 comma 0 right parenthesis Z′(−4,0) upper Z prime left parenthesis negative 4 comma 0 right parenthesis Z′(6,2) upper Z prime left parenthesis 6 comma 2 right parenthesis Z′(−4,2) upper Z prime left parenthesis negative 4 comma 2 right parenthesis

Answer 1

To find the coordinates of point Z' after a reflection across the line \(y = 1\), we need to determine the vertical distance of point Z from this line and then place point Z' the same distance on the opposite side of the line.

Assuming Z has coordinates \((x, y)\), the reflection across the line \(y = 1\) can be determined as follows:

1. Calculate the distance of point Z from the line \(y = 1\):
\[
\text{Distance} = y - 1
\]
This gives how far point Z is above or below the line.

2. The y-coordinate of the reflected point Z' will be:
\[
y' = 1 - (y - 1) = 2 - y
\]

3. The x-coordinate remains the same during reflection across a horizontal line, so:
\[
x' = x
\]

Now let's take an example to illustrate:

Assuming Z is \(Z(4, 0)\):
- Calculate the distance from the line:
\[
0 - 1 = -1 \quad (\text{This indicates it is 1 unit below the line})
\]

- Reflecting across the line:
\[
y' = 2 - 0 = 2
\]

Thus, the coordinates of Z' after reflecting \(Z(4, 0)\) across \(y = 1\) would be:
\[
Z'(4, 2)
\]

Now, looking at the provided responses, the reflection of \(Z(4,0)\) across \(y=1\) yields:
- \(Z′(4,2)\)

So the correct answer is:
\[
Z′(6,2) \text{ is incorrect, Z′(4,2) is the coordinate resulting from reflection across y=1 for given point Z’(4,0) }.
\]