To solve this problem, we're going to reflect the given point [tex]\( P(2, -5) \)[/tex] across the [tex]\( y \)[/tex]-axis and find the coordinates of the reflected point [tex]\( P' \)[/tex]. Here is a detailed step-by-step solution:
### Step 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 sign while the [tex]\( y \)[/tex]-coordinate stays the same.
### Step 2: Original coordinates
We start with the given coordinates of point [tex]\( P \)[/tex]:
- [tex]\( P_x = 2 \)[/tex]
- [tex]\( P_y = -5 \)[/tex]
### Step 3: Reflect the point
To reflect [tex]\( P \)[/tex] across the [tex]\( y \)[/tex]-axis:
- Change the sign of the [tex]\( x \)[/tex]-coordinate: [tex]\( P_x \)[/tex] becomes [tex]\(-P_x \)[/tex]
- The [tex]\( y \)[/tex]-coordinate remains unchanged: [tex]\( P_y \)[/tex] stays as [tex]\( P_y \)[/tex]
### Step 4: Calculate the new coordinates
- The [tex]\( x \)[/tex]-coordinate of [tex]\( P' \)[/tex] will be [tex]\( -P_x = -2 \)[/tex]
- The [tex]\( y \)[/tex]-coordinate of [tex]\( P' \)[/tex] will be [tex]\( -5 \)[/tex]
So, the coordinates of the reflected point [tex]\( P' \)[/tex] are [tex]\((-2, -5)\)[/tex].
### Summary
Therefore, after reflecting the point [tex]\( P(2, -5) \)[/tex] in the [tex]\( y \)[/tex]-axis, the coordinates of [tex]\( P' \)[/tex] are [tex]\( (-2, -5) \)[/tex].
