Reflections flip a sign, a half-turn flips both, a slide adds a constant — geometry becomes arithmetic.
In geometry we flipped and slid whole figures by eye — fold a shape along a line, spin it about a pin, glide it across the page. On the coordinate plane those very same moves become clean arithmetic on the pair (x, y). Mirror a point across the x-axis and only its height changes sign; mirror it across the y-axis and only its across-ness flips; spin it a half-turn about the origin and both signs flip; slide it and you simply add to each coordinate. Four rules, each touching the coordinates in its own tidy way — and they are exactly the rules that will later move whole function graphs around.
You met these moves as pictures in reflection symmetry and translating figures. There you traced them with tracing paper. Here we give every point an address, and the moves turn into one-line rules you can carry out with a pencil and no picture at all. Each rule still does its picture — that is the test we hold ourselves to: the arithmetic and the figure must always agree.
Hold a mirror flat along the x-axis. Left and right are unchanged — the mirror runs that way — but up and down swap: whatever was two units above the axis lands two units below it. So the across-ness x is untouched and the height y changes sign:
reflect across the x-axis: (x, y) ↦ (x, −y)
Now stand the mirror up along the y-axis. This time up and down are unchanged and it is the across-ness that flips — a point three units to the right lands three units to the left:
reflect across the y-axis: (x, y) ↦ (−x, y)
The mirror line is the perpendicular bisector of every point-and-image segment: it sits halfway between P and its image P′, and meets that segment at a right angle. That is what "a reflection" means, and the sign rule simply records it in coordinates.
Reflect P(3, 2) across the x-axis: the x stays, the y flips sign → P′(3, −2) — the same 3 across, but 2 below.
Reflect the same P(3, 2) across the y-axis: the y stays, the x flips sign → P′(−3, 2) — still 2 up, but 3 to the left.
Pick a move for the little blue flag and watch where its image lands. For a slide, set how far right (a) and up (b) to go.
Now pin a point at the origin and spin the plane a half-turn — a 180° turn. Every point sweeps to the spot directly opposite through O, the same distance away on the far side. Both the across-ness and the height reverse, so both coordinates flip sign:
half-turn about the origin: (x, y) ↦ (−x, −y)
A figure that lands exactly on itself after this half-turn has point symmetry about O — the origin is its center. The point P, the origin O, and the image P′ always line up straight, with O the midpoint of the segment PP′: that is the whole meaning of "opposite, same distance."
A half-turn about O is exactly a reflection across one axis followed by a reflection across the other. Flip in the x-axis: (x, y) ↦ (x, −y). Then flip in the y-axis: (x, −y) ↦ (−x, −y). Both signs end up flipped — and the order does not matter here, since flipping the other axis first gives the very same (−x, −y).
Half-turn of P(3, 2) about O: flip both signs → P′(−3, −2). Check the picture — P sits up-and-right, P′ sits down-and-left, and the straight segment between them passes right through the origin.
You can explore this case with the same widget above — choose Half-turn about O and the dashed segment from a vertex through O to its image makes the "straight through the center" visible.
A translation is a slide: it moves every point the same distance in the same direction, with no flipping and no turning. Go a units right and b units up, and each point simply gains a on its x and b on its y:
translate by (a, b): (x, y) ↦ (x + a, y + b)
A negative a slides left; a negative b slides down. Because every point shifts by the same (a, b), the figure's shape, size, and facing are untouched — only its position changes. The little arrow from any point to its image is the same for all of them; that shared arrow is the translation.
Slide P(3, 2) by (a, b) = (−4, 1): add to each coordinate. New x = 3 + (−4) = −1; new y = 2 + 1 = 3. So P(3, 2) ↦ P′(−1, 3) — four steps left, one step up.
A translation never flips or turns a figure. If a letter R reads forward, its slid copy still reads forward; only reflections (and the half-turn) reverse it. If you find the copy facing the other way, you have reflected, not translated.
Real moves often come in pairs, and the trick is simple: apply the rules one at a time, in the stated order. Track the coordinates step by step, and never try to do both at once.
Take P(2, 3). First reflect it across the x-axis, then translate by (a, b) = (1, 2):
(2, 3) reflect in x→ (2, −3) add (1, 2)→ (3, −1)
Now swap the order — first slide, then reflect:
(2, 3) add (1, 2)→ (3, 5) reflect in x→ (3, −5)
Same point, same two moves — (3, −1) one way, (3, −5) the other. The results disagree, so for these two moves order matters. (Two flips about the two axes happened to agree in 20.2.2 — but you cannot count on it. Always work left to right through the recipe.)
The point is P(2, 3). The two moves are fixed: a reflection in the x-axis and a slide by (a, b) = (1, 2). Switch which one happens first and compare the landing spots.
Each move touches the address (x, y) in one tidy way. Learn the table and you can carry out any of these moves with a pencil, then trust the picture to match.
| Move | Rule on (x, y) | What changes | Example from (3, 2) |
|---|---|---|---|
| Reflect across the x-axis | (x, y) ↦ (x, −y) | y flips sign | (3, −2) |
| Reflect across the y-axis | (x, y) ↦ (−x, y) | x flips sign | (−3, 2) |
| Half-turn about O | (x, y) ↦ (−x, −y) | both flip sign | (−3, −2) |
| Translate by (a, b) | (x, y) ↦ (x + a, y + b) | add a to x, b to y | by (1, 2): (4, 4) |
Sign-flips give symmetry; adding a constant gives a slide. In the next lesson, Scaling from the Origin, we complete the set with the third operation — multiplying each coordinate — which stretches the whole plane from O. Together with the addresses from The Coordinate Plane, these rules are the grammar the function strand is written in.
Work each on paper first — name the rule, do the arithmetic, then open the answer.
Reflect (5, −2) across the x-axis.
Reflect in the x-axis ⇒ flip the y: (5, −2) ↦ (5, 2). The x stays 5; the −2 becomes +2.
Reflect the same point (5, −2) across the y-axis.
Reflect in the y-axis ⇒ flip the x: (5, −2) ↦ (−5, −2). The y stays −2; the 5 becomes −5.
Find the image of (4, 7) under a half-turn about O.
Half-turn ⇒ flip both signs: (4, 7) ↦ (−4, −7). The point lands diametrically opposite, the same distance through the origin.
Translate (−1, 2) by (a, b) = (3, −5).
Add to each coordinate: x = −1 + 3 = 2, y = 2 + (−5) = −3. So (−1, 2) ↦ (2, −3) — three right, five down.
A point and its image across the y-axis are (6, 1) and (−6, 1). What stayed the same, and why?
The y-coordinate (the height, 1) stayed the same; only the x flipped sign. A y-axis mirror runs straight up and down, so up-and-down distance is untouched — it only swaps left for right.
Reflect (3, 4) across the x-axis, then across the y-axis. Where does it land — and what single move is that?
Reflect in x: (3, 4) ↦ (3, −4). Then reflect in y: (3, −4) ↦ (−3, −4). Both signs ended up flipped, so the pair of flips is a single half-turn about O — exactly the rule (x, y) ↦ (−x, −y).
Six questions to lock it in. Tap the answer you think is right.
This lesson turns three rigid motions and a slide into arithmetic on an ordered pair, the move that lets coordinates carry the whole geometry of transformations into algebra. The single idea worth over-rehearsing is matching each operation to its job: flipping a sign is a reflection or half-turn (symmetry), while adding a constant is a translation (a slide). Students who can say which coordinate the move touches — "reflect in x flips the y" — rarely misremember the rules.
Three misconceptions to watch for. First, flipping the wrong coordinate: many learners reflexively negate the x for a reflection in the x-axis. Anchor it physically — a mirror lying along the x-axis leaves left-right alone, so only the height (y) can change. Second, believing a translation can turn or flip a figure; insist on the "same arrow for every point" picture, where a forward R stays forward. Third, assuming the order of two moves never matters; the worked example (flip-then-slide ≠ slide-then-flip) is the antidote — always track the coordinates left to right.
This material aligns with the US Common Core standards 8.G.A.1–3 (translations and reflections and their effect on coordinates), 6.NS.C.6b (signs and reflections across the axes), and G-CO.A / G-CO.B (rigid motions described in the coordinate plane). It builds directly on the picture-level work in reflection symmetry and translating figures, and sets up coordinate dilation in the next lesson.