Question

p (-2, 4)
y=1
if p = (-2, 4),
then find:
r_{y=1} (p)
(-2, ?)

Explanation:

Step1: Find the distance from P to \( y = 1 \)

The y - coordinate of \( P=(-2,4) \) is 4, and the line is \( y = 1 \). The distance \( d \) between the point \( (x,y) \) and the line \( y = k \) is \( |y - k| \). So \( d=|4 - 1|=3 \).

Step2: Find the y - coordinate of the reflection

The reflection of a point \( (x,y) \) over the line \( y = k \) has the same x - coordinate, and the y - coordinate \( y' \) satisfies \( k=\frac{y + y'}{2} \). We know \( x=-2 \), \( y = 4 \), \( k = 1 \). Substitute into the formula: \( 1=\frac{4 + y'}{2} \). Multiply both sides by 2: \( 2=4 + y' \). Then \( y'=2 - 4=-2 \).

Answer:

\(-2\)