Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)
299. The Unmoved Point · discovering invariant points under reflection
P stays exactly where it is when reflected across the x-axis.
A point is invariant under a transformation if it maps to itself. Under reflection in a line, exactly the points on the mirror line are invariant — so a point on the x-axis stays put when reflected in the x-axis.
What this lesson covers
Try to break it
Drag P up and away from the x-axis — P' appears below, mirroring. Slide P back toward the axis; P and P' converge. The moment P sits exactly on the x-axis (y = 0), P' coincides with P — that's the invariant. Every point on the mirror line is unmoved by the reflection.
How you build it
Mark an invariant point.
- Click on the x-axis to place invariant point P.
The proof, step by step
Prove that a point on the x-axis is unchanged by reflection in the x-axis.
- An invariant point maps to itself under a transformation.
- Reflection in the x-axis transforms any point (x, y) to (x, -y).
- For the point to be invariant, its image must equal the original: (x, -y) = (x, y).
- Comparing coordinates gives -y = y, which simplifies to 2y = 0, so y = 0.
- Thus, any point with y-coordinate 0 (i.e., lying on the x-axis) is invariant under reflection in the x-axis.
Worked example
A point P(a, b) is invariant under reflection in the x-axis. What must be the value of b?
For a point to be invariant under reflection in the x-axis, its image must coincide with the original point. Reflection in the x-axis maps (a, b) to (a, -b). Equating coordinates gives b = -b, which implies 2b = 0, so b = 0.
- a
- 0 — correct
- 1
- b