Symmetry

108. Mirror on the Y-Axis · watch x flip sign while y stays put

P' is always the mirror image of P across the y-axis.

Reflection across the y-axis: point P(x, y) maps to P′(−x, y). The y-coordinate stays; the x-coordinate flips sign. The y-axis is the line of symmetry.

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Selina ICSE: Symmetry

What this lesson covers

Try to break it

Try moving P far up or down. Does the y-coordinate change? What about the distance to the y-axis?

How you build it

Reflect a point across the y-axis.

Pick the Point tool. Click on the y-axis to drop point A — anywhere on it. Use a grid intersection for a clean coordinate.
Click again on the y-axis, at a different spot, to drop point B. A and B together identify the mirror line.
Click anywhere to the right of the y-axis to drop point P — the point we will reflect.
Pick the Arc tool. Click A for the centre, then click P to set the radius. The arc passes through P and curves across the y-axis to the left. Every point on this arc is exactly the distance AP from A.
Stay on the Arc tool. Click B for the centre, then click P for the radius. This second arc also passes through P. To the left of the y-axis, the two arcs cross at exactly one point.
Switch to the Point tool. To the left of the y-axis, the two arcs cross at exactly one point. Click that intersection — this is P', the mirror image of P. Because AP' = AP and BP' = BP, both A and B are equidistant from P and P'. If P = (x, y), then P' = (-x, y).
Pick the Line tool. Click P, then P'. The line PP' crosses the y-axis at a right angle, and the y-axis splits it exactly in half. Both A and B being equidistant from the endpoints means the y-axis IS the perpendicular bisector of PP'.

The proof, step by step

Prove that reflection in the y-axis keeps the y-coordinate and negates the x-coordinate.

PP' is a horizontal line, so P and P' share the same y-coordinate.
The y-axis bisects PP' perpendicularly, so P' is as far left as P is right.
Therefore, if P is (x, y), its reflection P' must be (-x, y).

Worked example

Point A is located at (5, -3) on the coordinate plane. If A is reflected in the y-axis to form point A', what are the coordinates of A'?

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

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