To find the image coordinates of point [tex]\( D \)[/tex] after reflecting it across the line [tex]\( y = 4 \)[/tex], we need to follow a systematic approach:
### Step-by-Step Solution:
1. Identify the Original Coordinates of [tex]\( D \)[/tex]:
[tex]\( D \)[/tex] has the coordinates [tex]\( (-1, -4) \)[/tex].
2. Understand the Line of Reflection:
The line of reflection given is [tex]\( y = 4 \)[/tex].
3. Determine the Distance Between Point [tex]\( D \)[/tex] and the Reflection Line:
To reflect a point across a horizontal line, we measure the vertical distance from the point to the line of reflection.
- The y-coordinate of [tex]\( D \)[/tex] is [tex]\( -4 \)[/tex].
- The line [tex]\( y = 4 \)[/tex] is 4 units above the x-axis.
- Distance from [tex]\( D \)[/tex] to the line: [tex]\( 4 - (-4) = 4 + 4 = 8 \)[/tex] units.
4. Reflect the Point Across the Line:
To find the y-coordinate of the reflected point [tex]\( D' \)[/tex], we move this distance upwards from the line of reflection.
- Since [tex]\( D \)[/tex] is 8 units below the reflection line and reflection means equal distance on the opposite side, we add 8 units above the reflection line.
- Calculating the new y-coordinate: [tex]\( 4 (line) + 8 = 12 \)[/tex].
5. Retain the x-coordinate:
The x-coordinate does not change because the reflection is happening across a horizontal line. Therefore, [tex]\( x_{D'} = -1 \)[/tex].
6. Write the Coordinates of the Reflected Point [tex]\( D' \)[/tex]:
Thus, the coordinates of [tex]\( D' \)[/tex] are [tex]\( (-1, 12) \)[/tex].
### Conclusion:
The image coordinates of [tex]\( D' \)[/tex] after reflecting [tex]\( D(-1, -4) \)[/tex] across the line [tex]\( y = 4 \)[/tex] are [tex]\( \mathbf{(-1, 12)} \)[/tex].
Therefore, the correct answer is:
[tex]\[ D^{\prime}(-1, 12) \][/tex]
