To find the reflection of point [tex]\( P = (-2, 7) \)[/tex] over the line [tex]\( y = 1 \)[/tex], follow these steps:
1. Identify the original coordinates: The coordinates of point [tex]\( P \)[/tex] are [tex]\( (-2, 7) \)[/tex].
2. Understand the line of reflection: The line of reflection is [tex]\( y = 1 \)[/tex].
3. Determine the reflection rule: When reflecting a point over the line [tex]\( y = c \)[/tex] (in this case, [tex]\( c = 1 \)[/tex]), the formula for the new coordinates is:
[tex]\[
(x, y) \to (x, 2c - y)
\][/tex]
Here, [tex]\( c = 1 \)[/tex].
4. Apply the reflection formula:
[tex]\[
x_{\text{reflected}} = x = -2
\][/tex]
[tex]\[
y_{\text{reflected}} = 2 \cdot 1 - 7 = 2 - 7 = -5
\][/tex]
5. Write the coordinates of the reflected point: The reflected point [tex]\( R_{y=1}(P) \)[/tex] is:
[tex]\[
(-2, -5)
\][/tex]
So, the reflection of [tex]\( P = (-2, 7) \)[/tex] over the line [tex]\( y = 1 \)[/tex] is [tex]\( R = (-2, -5) \)[/tex].
