To find the reflection of the point [tex]\( P = (-1, 5) \)[/tex] across the vertical line [tex]\( x = 1 \)[/tex], we'll follow these steps:
1. Identify the line of reflection: The given vertical line of reflection is [tex]\( x = 1 \)[/tex].
2. Determine the x-coordinate of the reflected point: For reflection across a vertical line [tex]\( x = c \)[/tex], the x-coordinate of the reflected point [tex]\( R \)[/tex] can be found using the formula:
[tex]\[
x_{\text{reflected}} = 2c - x_P
\][/tex]
Here, [tex]\( c = 1 \)[/tex] and [tex]\( x_P = -1 \)[/tex]. Substituting these values into the formula gives:
[tex]\[
x_{\text{reflected}} = 2 \cdot 1 - (-1) = 2 + 1 = 3
\][/tex]
3. Determine the y-coordinate of the reflected point: The y-coordinate remains unchanged during reflection across a vertical line, so:
[tex]\[
y_{\text{reflected}} = y_P = 5
\][/tex]
4. Combine the coordinates: Hence, the coordinates of the reflected point [tex]\( R \)[/tex] are:
[tex]\(
R_{x=1}(P) = (3, 5)
\)[/tex]
So, the reflection of the point [tex]\( P = (-1, 5) \)[/tex] across the line [tex]\( x = 1 \)[/tex] is [tex]\( (3, 5) \)[/tex].
