Reflection Over the Axes

Reflection Over the Axes — Moosa Academy

Reflecting a point in the coordinate plane needs no drawing at all — just a sign change. Which coordinate flips depends on which axis you reflect over, and there is a simple way to remember which one it is.

Theorem The two rules

Over the x-axis:

(x,\, y) \rightarrow (x,\, -y)

Over the y-axis:

(x,\, y) \rightarrow (-x,\, y)

The coordinate that keeps its value is the one named after the axis. Reflecting over the x-axis leaves $x$ alone; reflecting over the y-axis leaves $y$ alone. The other coordinate simply changes sign.

Concept What the picture shows

The point and its image sit the same distance from the axis, on opposite sides. The dashed segment joining them crosses the axis at a right angle and is cut exactly in half.

That is the geometric meaning of the rule: the mirror line is the perpendicular bisector of the segment $AA'$ .

Example Reflecting two points

Reflect $A(4,\, 2)$ over the x-axis, and $B(-2,\, -3)$ over the y-axis.

A(4,\, 2) \rightarrow A'(4,\, -2)

— keep

x = 4

, flip

2

-2

B(-2,\, -3) \rightarrow B'(2,\, -3)

— keep

y = -3

, flip

-2

2

Flipping a negative gives a positive: $B$ was two units left of the y-axis, so its image is two units to the right.

Note Points that do not move

A point lying on the mirror line is its own image. Check it with the rule:

(5,\, 0) \rightarrow (5,\, -0) = (5,\, 0)

— on the x-axis, unchanged

(0,\, 7) \rightarrow (-0,\, 7) = (0,\, 7)

— on the y-axis, unchanged

These are the fixed points of the reflection. The origin is fixed by both, since it lies on both axes.

Note The four common reflections

Mirror	Rule	P(3, 4) becomes
x-axis	(x, y) → (x, −y)	(3, −4)
y-axis	(x, y) → (−x, y)	(−3, 4)
The line y = x	(x, y) → (y, x)	(4, 3)
The origin	(x, y) → (−x, −y)	(−3, −4)

The last row is not a reflection in a line but a half-turn about the origin. It happens to equal doing both axis reflections one after the other.

Summary

Over the x-axis: (x, y) → (x, −y).
Over the y-axis: (x, y) → (−x, y).
The coordinate named after the axis is the one that stays.
The mirror line is the perpendicular bisector of the segment joining a point to its image.
Any point on the mirror line is fixed.
Doing both reflections in turn gives (x, y) → (−x, −y), a half-turn about the origin.