Rules for reflecting across the axes and the line y=x — each one flips exactly one thing.
The Mnemonic
Three Reflection Lines, Three Simple Rules
Reflecting over the x-axis: (x, y) → (x, −y) — the x-coordinate stays the same, the y-coordinate flips sign. Reflecting over the y-axis: (x, y) → (−x, y) — the y-coordinate stays the same, the x-coordinate flips sign.
Reflecting over the line y = x: (x, y) → (y, x) — the two coordinates simply SWAP places, with no sign changes at all.
Reflecting over the x-axis flips y; reflecting over the y-axis flips x — the axis you reflect across stays fixed
💡 Memory Trick
"Rules for reflecting across the axes and y=x." Reflecting over the x-AXIS flips the Y (the OTHER letter) — and reflecting over the y-AXIS flips the X (again, the OTHER letter). Reflecting over y=x is different in kind: it swaps the coordinates entirely, since y=x is a diagonal line, not a straight axis.
Why It Works
Why Each Rule Flips What It Flips
Reflecting over the x-axis means the new point is exactly as far below the x-axis as the original point was above it (or vice versa) — same horizontal position, mirrored vertical position. Since the x-axis IS where y=0, mirroring the y-value around 0 is exactly what negating it does; the x-coordinate never changes because horizontal position isn't affected by a vertical mirror.
Reflecting over y=x works differently because y=x is a 45° diagonal line, not a horizontal or vertical axis. A point's mirror image across this specific diagonal is always found by literally swapping its coordinates — this can be shown by noting that the segment connecting a point and its y=x reflection is always perpendicular to y=x (slope of −1, the negative reciprocal of 1) and that the midpoint of that segment always lands exactly ON the line y=x.
Using It In A Proof
Applying Each Rule Correctly
Since all three rules look deceptively similar (flip one sign, or swap), being deliberate about which specific rule matches which reflection line prevents mixing them up.
1
Identify the specific line of reflection
Confirm whether the problem specifies the x-axis, y-axis, or the line y=x — each has a genuinely different rule.
2
Apply only that one rule
x-axis → flip y's sign only. y-axis → flip x's sign only. y=x → swap x and y, no sign changes.
3
Reflect each vertex separately for a full shape
When reflecting an entire polygon, apply the same single rule to every vertex individually, then reconnect them in the same order.
Full Worked Example
Reflecting a Triangle Over the Y-Axis
Given: Triangle with vertices A(2, 3), B(5, 1), C(2, −2). Find: the image after reflecting over the y-axis.
1
Confirm the correct rule
Reflecting over the y-axis: (x,y) → (−x, y).
2
Apply the rule to A
A(2,3) → A'(−2, 3).
3
Apply the rule to B and C
B(5,1) → B'(−5, 1). C(2,−2) → C'(−2, −2).
4
State the image
The reflected triangle has vertices A'(−2,3), B'(−5,1), C'(−2,−2).
Notice only the x-coordinates changed sign — every y-coordinate stayed exactly the same, confirming the y-axis reflection rule was applied consistently.
🎯 Quick Worked Example
Reflect the point (4, −7) over the line y = x.
1
Identify the rule. Reflecting over y=x: (x,y) → (y,x).
2
Apply the rule. Swap the coordinates: (4,−7) → (−7,4).
3
State the image. The reflected point is (−7, 4).
📌 Exam Application
Since all three rules involve either flipping a sign or swapping coordinates, exams often test all three in the same problem set specifically to check whether students have genuinely memorized which rule matches which line — practice all three enough that recalling the correct one becomes automatic rather than requiring re-derivation each time.
⚠️ Most Common Reflection Coordinate Rules Mistakes
Trap 1 — Flipping the wrong coordinate's sign: Reflecting over the x-axis flips Y, not x; reflecting over the y-axis flips X, not y — since the rule flips the OPPOSITE letter from the axis name, this is a very easy mix-up.
Trap 2 — Adding a sign flip to the y=x reflection: Reflecting over y=x is a pure swap with NO sign changes — accidentally adding a negative sign (writing (−y,−x) instead of (y,x)) confuses this rule with a rotation instead.
✓ Quick Self-Test
1) State the rule for reflecting over the x-axis. 2) State the rule for reflecting over the y-axis. 3) State the rule for reflecting over the line y=x. 4) Reflect the point (6, 2) over the x-axis. 5) Reflect the point (−3, 5) over the line y=x.