To find the coordinates of the image point [tex]\( C^{\prime} \)[/tex] when the point [tex]\( C(-4,-2) \)[/tex] is reflected across the [tex]\( y \)[/tex]-axis, follow these steps:
1. Understand reflection across the [tex]\( y \)[/tex]-axis:
- When a point is reflected across the [tex]\( y \)[/tex]-axis, the [tex]\( x \)[/tex]-coordinate of the point changes its sign, while the [tex]\( y \)[/tex]-coordinate remains the same.
2. Identify the coordinates of the original point [tex]\( C \)[/tex]:
- The original coordinates are [tex]\( (-4, -2) \)[/tex] where [tex]\( x = -4 \)[/tex] and [tex]\( y = -2 \)[/tex].
3. Apply the reflection rule:
- Reflecting across the [tex]\( y \)[/tex]-axis, the new [tex]\( x \)[/tex]-coordinate will be the opposite of the original [tex]\( x \)[/tex]-coordinate. Thus, the [tex]\( x \)[/tex]-coordinate changes from [tex]\(-4\)[/tex] to [tex]\( 4 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate remains unchanged, continuing to be [tex]\( -2 \)[/tex].
4. Form the coordinates of the reflected point [tex]\( C^{\prime} \)[/tex]:
- Combining the new [tex]\( x \)[/tex]-coordinate and the unchanged [tex]\( y \)[/tex]-coordinate, the coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (4, -2) \)[/tex].
Therefore, the coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( \boxed{(4, -2)} \)[/tex].
