Symmetry (including Reflection and Rotation)
185. Mirror Image on the Axis · how reflection flips the y-coordinate
P' is always the exact reflection of P across the x-axis.
Reflection across the x-axis: P(x, y) ↦ P′(x, −y). The x-coordinate stays; y flips sign. Distance from the x-axis is preserved.
What this lesson covers
Try to break it
Drag P around. P' always sits exactly opposite across the x-axis — same x-coordinate, opposite y-coordinate. The line PP' always crosses the x-axis at 90°. Try to drag P so PP' isn't perpendicular to the axis; you can't.
How you build it
Reflect a point across the x-axis.
- Draw line AB along the x-axis — click two points on the horizontal axis.
- Place point P anywhere above the x-axis.
- Drop a perpendicular from P to line AB, marking the foot as A.
- Extend PA to P' such that PA equals AP'. This is the reflection.
The proof, step by step
Prove that P prime is the reflection of P across the x-axis.
- PA is drawn perpendicular to the x-axis.
- By construction, the length PA equals the length AP'.
- Therefore, P' is the reflection of P across the x-axis.
- Since A has coordinates (x, 0), the y-coordinate of P' must be -y, giving P'(x, -y).
Worked example
If point A(3, 5) is reflected in the x-axis, what are the coordinates of its image?
Reflection in the x-axis keeps the x-coordinate unchanged and changes the sign of the y-coordinate. So, (3, 5) becomes (3, -5).
- (3, 5)
- (-3, 5)
- (3, -5) — correct
- (-3, -5)