⚡ Grade 6 · Lesson 4 of 14
Mirror City: Reflecting Points on the Coordinate Plane
Every building in Mirror City has a twin across the river or across the ground — find it by flipping one coordinate's sign.
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The big idea: Reflecting a point across the x-axis or y-axis creates a mirror image the same distance from the axis, but on the opposite side. The reflection keeps one coordinate the same and simply flips the sign of the other, so you can find any mirror point without redrawing the whole picture.
A city built on two mirrors
Mirror City sits on a coordinate grid, with a river running along the x-axis and a tall glass wall running along the y-axis. Every building has a twin: one reflected across the river, one reflected across the wall. If you know a building's coordinates, you can find its reflection without drawing a thing.
A reflection flips a point to the opposite side of a line (the x-axis or y-axis) while keeping it exactly the same distance away.
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Reflecting flips one sign, not both
Reflecting across the x-axis keeps x the same and flips the sign of y. Reflecting across the y-axis keeps y the same and flips the sign of x. Only one coordinate changes at a time.
\[ (x, y) \;\xrightarrow{\text{reflect across } x\text{-axis}}\; (x, -y) \]
Flipping over the x-axis keeps the horizontal position and flips the vertical one.
\[ (x, y) \;\xrightarrow{\text{reflect across } y\text{-axis}}\; (-x, y) \]
Flipping over the y-axis keeps the vertical position and flips the horizontal one.
🎮 Mirror City Reflections LIVE
Reflect a point across the x- or y-axis and watch which coordinate flips its sign.
Distance to the mirror line never changes
A point and its reflection are always the same distance from the mirror line, just on opposite sides. A building 5 units north of the river reflects to one 5 units south of it — the distance (5) stays fixed, only the direction flips.
📝 Worked example: Reflect the point (3, 5) across the x-axis. What are the new coordinates?
- Reflecting across the x-axis keeps x the same and flips the sign of y.
- x stays 3.
- y flips from 5 to −5.
✓ The reflected point is <strong>(3, −5)</strong>.
📝 Worked example: Reflect the point (−4, 2) across the y-axis. What are the new coordinates?
- Reflecting across the y-axis keeps y the same and flips the sign of x.
- y stays 2.
- x flips from −4 to 4.
✓ The reflected point is <strong>(4, 2)</strong>.
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Flip only one coordinate at a time
A common mistake is flipping both coordinates' signs no matter which axis you're told to reflect across. Reflecting across the x-axis never changes x; reflecting across the y-axis never changes y. Flipping both signs describes a different move — a 180° rotation about the origin, not a reflection.
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Reflect twice, land back home
Reflect a point across the x-axis, then reflect the result across the x-axis again, and you're back where you started. Reflections are their own opposite: doing the same reflection twice always undoes it.
Check your understanding
1. Reflect (6, −2) across the x-axis. What is the new point?
Reflecting across the x-axis keeps x the same and flips the sign of y: −2 becomes 2.
2. Reflect (−3, 7) across the y-axis. What is the new point?
Reflecting across the y-axis keeps y the same and flips the sign of x: −3 becomes 3.
3. A point with y = 4 is reflected across the x-axis. What is the y-coordinate of its reflection?
Reflecting across the x-axis flips the sign of y: 4 becomes −4.
4. Which coordinate stays the same when you reflect a point across the y-axis?
Reflecting across the y-axis only flips the sign of x; y is unchanged.
5. If you reflect (2, 9) across the x-axis and then reflect the result across the x-axis again, where do you end up?
Reflecting across the same line twice undoes the first reflection, returning you to the original point.
✅ Key takeaways
- A reflection flips a point to the opposite side of a line, the same distance away.
- Reflecting across the x-axis keeps x the same and flips the sign of y: (x, y) → (x, −y).
- Reflecting across the y-axis keeps y the same and flips the sign of x: (x, y) → (−x, y).
- Only one coordinate changes sign at a time — flipping both describes a rotation, not a reflection.
- Reflecting the same point across the same line twice returns it to its original position.