1.5 Reflections — Transformations

Standards: G-CO.A.4 · G-CO.A.5 · G-CO.B.6

Given a point or figure and a mirror line, you will produce the reflected image and verify the rule by applying the same reflection twice.

Definitions and rules

Reflection over the axes and diagonals

over the x-axis: (x, y) → ( x, −y) over the y-axis: (x, y) → (−x, y) over y = x: (x, y) → ( y, x) over y = −x: (x, y) → (−y, −x)

Reflecting over an axis flips the sign of the other coordinate.

Why it has to be this: the mirror line stays put, so whatever measures distance from it is what changes sign.

How to check any reflection in five seconds

Do it twice. A reflection undoes itself — reflect and reflect again and you are back where you started, every time, for every mirror line.

So if your rule applied twice does not return the original point, the rule is wrong.

Here is the result that ties the day together. Reflect a point over the x-axis, then reflect that over the y-axis:

Take (5, 2).

over x-axis: (5, 2) → (5, −2) over y-axis: (5, −2) → (−5, −2) and R180(5, 2) = (−5, −2)

What that means

Two reflections in perpendicular mirror lines produce exactly a half turn. The flips compose into a rotation.

That is worth knowing for its own sake, and it is the first hint of what composition can do — two of one kind of motion adding up to a different kind entirely.

Worked examples

Example 1 — a whole figure over the y-axis

Reflect triangle A(2, 1), B(5, 1), C(2, 4) over the y-axis.

Suggested work

Givens

A(2, 1) B(5, 1) C(2, 4) mirror line: the y-axis

Tools

over the y-axis: (x, y) → (−x, y)

Primitives

the rule is fixed for the whole figure; only the inputs change

Computation

A(2, 1) → (−(2), 1) = A′(−2, 1) B(5, 1) → (−(5), 1) = B′(−5, 1) C(2, 4) → (−(2), 4) = C′(−2, 4)

Notice: every y stayed put. Only distance from the mirror changed.

Example 2 — a point reflected over y = −x

Reflect P(−3, 5) over y = −x.

Suggested work

Givens

P(−3, 5) mirror line: y = −x

Tools

over y = −x: (x, y) → (−y, −x)

Primitives

x = −3 y = 5

Computation

P(−3, 5) → (−(5), −(−3)) = P′(−5, 3)

Notice: The coordinates exchange roles and both signs change.

Why the rule can be checked quickly

Reflect the image with the same rule: (−5, 3) → (−3, 5). A reflection must undo itself.

Example 3 — a point reflected over y = x

Reflect Q(2, 6) over y = x.

Suggested work

Givens

Q(2, 6) mirror line: y = x

Tools

over y = x: (x, y) → (y, x)

Primitives

x = 2 y = 6

Computation

Q(2, 6) → Q′(6, 2)

Why the coordinates swap

The line y = x exchanges horizontal and vertical distance. Points on the mirror line already have equal coordinates and remain fixed.

Common traps

The sign that catches everyone

Reflecting over the x-axis changes the sign of y. It feels backwards, and it is the single most common error on this topic. The mirror line does not move, so the coordinate that measures distance from it is the one that flips.

Practice

Reflect S(−6, 3) over the x-axis.
Answer — tap to show
S′(−6, −3)
Suggested work — tap to show
Givens
S(−6, 3) mirror line: the x-axis
Tools
over the x-axis: (x, y) → (x, −y)
Primitives
x = −6 y = 3
Computation
= (−6, −(3)) = (−6, −3) x did not move. Only y changed sign.
Reflect A(4, 2) over the y-axis.
Answer — tap to show
A′(−4, 2).
Reflect V(2, 7) over the line y = x.
Answer — tap to show
V′(7, 2)
Suggested work — tap to show
Givens
V(2, 7) mirror line: y = x
Tools
over y = x: (x, y) → (y, x)
Primitives
x = 2 y = 7
Computation
= (7, 2) Check by doing it twice: (7, 2) → (2, 7) ✓ back to the start
A(2, −5) maps to A′(5, −2), and B(−1, 3) maps to B′(−3, 1). Identify the reflection line and state whether the image is congruent to the preimage.
Answer — tap to show
The reflection line is y = −x. The image is congruent to the preimage because a reflection is a rigid motion.
Suggested work — tap to show
Givens
A(2, −5) → A′(5, −2) B(−1, 3) → B′(−3, 1)
Tools
over y = −x: (x, y) → (−y, −x)
Primitives
Test the same candidate rule on both pairs.
Computation
A: (2, −5) → (−(−5), −(2)) = (5, −2) B: (−1, 3) → (−(3), −(−1)) = (−3, 1) Both pairs match the y = −x reflection rule. A reflection preserves congruence.
A classmate reflects W(4, −1) over the x-axis and writes W′(−4, −1). Find their mistake.
Answer — tap to show
They are wrong. The correct image is W′(4, 1).
Suggested work — tap to show
Givens
W(4, −1) mirror line: the x-axis Classmate wrote W′(−4, −1)
Tools
over the x-axis: (x, y) → (x, −y)
Primitives
x = 4 y = −1
Computation
= (4, −(−1)) = (4, 1) The mistake: they flipped x instead of y — that is the rule for the y-axis, not the x-axis.