Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)

297. Mirror Image on Y-Axis · Flip the x, keep the y

The segment PP’ is always perpendicular to the y-axis and bisected by it.

Reflecting in the y-axis sends (x, y) → (−x, y): the y-coordinate stays and the x-coordinate changes sign. The y-axis is the perpendicular bisector of the segment joining a point to its image.

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For subscribers · Selina ICSE: Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)

What this lesson covers

Try to break it

Drag P around. P’ always sits at (−x, y) — opposite x, same y. The line PP’ stays perfectly horizontal and the y-axis is its midpoint. Pull P onto the y-axis (x = 0) and P’ coincides with P; the y-axis is the fixed-point set for this reflection.

How you build it

Reflect a point across the y-axis.

Draw the segment AB.
Construct the perpendicular bisector of AB — the point where the bisector meets AB is O.
Mark point P.
With centre O, draw an arc of radius OP.
Mark point P′ where the arc cuts the bisector to the left of the y-axis.

The proof, step by step

Prove that the y-axis perpendicularly bisects the segment PP prime.

P is given at coordinates (x, y).
The Y-axis is the line x = 0.
The perpendicular distance from P to the Y-axis is x units.
Reflection preserves perpendicularity and distance.
Therefore, P’ lies on the perpendicular from P to the Y-axis, x units to the left.
Its coordinates are (-x, y).

Worked example

If point A(3, 5) is reflected in the Y-axis, what are the coordinates of its image A’?

Reflection in the Y-axis changes the sign of the x-coordinate while keeping the y-coordinate unchanged. Thus, A(3, 5) becomes A’(-3, 5).

A’(-3, 5) — correct
A’(3, -5)
A’(-3, -5)
A’(5, 3)