To determine in which quadrant point [tex]\( P' \)[/tex] is located after reflecting point [tex]\( P \)[/tex] across the [tex]\( x \)[/tex]-axis, we follow these steps:
1. Identify the coordinates of the original point [tex]\( P \)[/tex]: The coordinates of point [tex]\( P \)[/tex] are [tex]\( (-4, -7) \)[/tex].
2. Reflect [tex]\( P \)[/tex] across the [tex]\( x \)[/tex]-axis: Reflecting a point across the [tex]\( x \)[/tex]-axis changes the [tex]\( y \)[/tex]-coordinate to its negative while the [tex]\( x \)[/tex]-coordinate remains the same. Therefore, if [tex]\( P \)[/tex] has coordinates [tex]\( (x, y) \)[/tex], the reflected point [tex]\( P' \)[/tex] will have coordinates [tex]\( (x, -y) \)[/tex].
Applying this to our point [tex]\( P \)[/tex]:
[tex]\[
P = (-4, -7) \quad \text{reflects to} \quad P' = (-4, 7)
\][/tex]
3. Determine the quadrant in which [tex]\( P' \)[/tex] lies:
- Quadrant I: [tex]\( x > 0 \)[/tex] and [tex]\( y > 0 \)[/tex]
- Quadrant II: [tex]\( x < 0 \)[/tex] and [tex]\( y > 0 \)[/tex]
- Quadrant III: [tex]\( x < 0 \)[/tex] and [tex]\( y < 0 \)[/tex]
- Quadrant IV: [tex]\( x > 0 \)[/tex] and [tex]\( y < 0 \)[/tex]
Given the coordinates of [tex]\( P' \)[/tex]:
[tex]\[
P' = (-4, 7)
\][/tex]
We observe [tex]\( x = -4 \)[/tex] and [tex]\( y = 7 \)[/tex]. Since [tex]\( x < 0 \)[/tex] and [tex]\( y > 0 \)[/tex], [tex]\( P' \)[/tex] is located in Quadrant II.
Therefore, point [tex]\( P' \)[/tex] is located in Quadrant II.
