Sure, let's find the reflection of the point [tex]\( P = (-4, 3) \)[/tex] over the line [tex]\( y = 1 \)[/tex]. Here's a step-by-step solution:
1. Understand the Reflection Line:
- The given line of reflection is [tex]\( y = 1 \)[/tex].
2. Identify Coordinates of Point [tex]\( P \)[/tex]:
- Point [tex]\( P \)[/tex] has coordinates [tex]\( (x, y) = (-4, 3) \)[/tex].
3. Calculate the Reflected Point:
- To reflect a point over the line [tex]\( y = k \)[/tex]:
- The x-coordinate remains unchanged.
- The y-coordinate is transformed using the formula [tex]\( y' = 2k - y \)[/tex].
4. Apply the Formula:
- Here, [tex]\( k = 1 \)[/tex] and [tex]\( y = 3 \)[/tex].
- Compute the new y-coordinate: [tex]\( y' = 2 \cdot 1 - 3 = 2 - 3 = -1 \)[/tex].
5. Form the Reflected Point:
- The x-coordinate remains the same: [tex]\( x' = -4 \)[/tex].
- The new y-coordinate is: [tex]\( y' = -1 \)[/tex].
6. The Coordinates of the Reflected Point:
- Therefore, the reflected point [tex]\( R_{y=1}(P) \)[/tex] is [tex]\( (-4, -1) \)[/tex].
So, the reflection of [tex]\( P = (-4, 3) \)[/tex] over the line [tex]\( y = 1 \)[/tex] is [tex]\( R = (-4, -1) \)[/tex].
[tex]\[
R_{y=1}(P) = (-4, -1)
\][/tex]
Hence, the reflection [tex]\( R_{y=1}(P) \)[/tex] yields the coordinates [tex]\( (-4, -1) \)[/tex].
