Question

Find the coordinates of Z′
after a reflection across y=1
.

(1 point)
Responses

Z′(−4,0)
upper Z prime left parenthesis negative 4 comma 0 right parenthesis

Z′(4,0)
upper Z prime left parenthesis 4 comma 0 right parenthesis

Z′(−4,2)
upper Z prime left parenthesis negative 4 comma 2 right parenthesis

Z′(6,2)

Answer 1

To find the coordinates of the point \( Z' \) after reflecting a point \( Z \) across the line \( y = 1 \), we first need to understand the reflection process.

1. **Identify the starting point \( Z \)**: The original coordinates are not provided, so we'll let \( Z(x, y) \) denote the original point.
2. **Calculate the distance from \( Z \) to the line \( y = 1 \)**:
- The formula for the reflection of a point \( (x, y) \) over the line \( y = k \) is:
\[
Z'(x, 2k - y)
\]
- Here, \( k = 1 \).

3. **Reflection Formula**: Therefore, after reflecting across the line \( y = 1 \), the new coordinates \( Z' \) become:
\[
Z'(x, 2(1) - y) = Z'(x, 2 - y)
\]

Now, we can deduce the new \( y \)-coordinate based on the possible responses given. The resulting \( y \)-coordinate after reflection must be computed from \( y \).

To determine which choices from the options you provided could be valid \( Z' \) coordinate pairs, consider that:

- If the original point \( Z \) is located above the line \( y=1 \) (i.e., \( y > 1 \)), the reflected point \( Z' \) will fall below the line.
- If the original point \( Z \) is below the line \( y=1 \) (i.e., \( y < 1 \)), the reflected point \( Z' \) will end up above the line.

Given the options:
1. \( Z'(-4, 0) \)
2. \( Z'(4, 0) \)
3. \( Z'(-4, 2) \)
4. \( Z'(6, 2) \)

The coordinates \( y = 0 \) from options 1 and 2 indicate that the reflection must originate from a point with a \( y \) coordinate greater than 1 (since it reflects down to \( y = 0 \)).

The coordinates \( y = 2 \) in options 3 and 4 suggest that the reflection must originate from a point with a \( y \) coordinate below 1 (since it reflects down to \( y = 2\)).

- Reflections of \( y=2 \) over \( y=1 \) yield \( y=0 \) (this is not correct for options 3 and 4).
- Reflections of \( y=0 \) over \( y=1 \) would yield \( y=2 \) (correct for options 1 and 2).

Since both \( Z'(-4, 0) \) and \( Z'(4, 0) \) meet the logic of reflecting down from \( y > 1 \), let’s summarize.
To identify the valid reflection, we confirm:
- Choose options \( Z'(-4, 0) \) or \( Z'(4, 0) \) as both candidates are valid based on our analysis.

However, since the specific original point is not given, from the options provided, we conclude they both are suitable outputs reflecting across \( y=1 \):

So the answer must be **Either**:
- \( Z'(-4, 0) \) or \( Z'(4, 0) \).