Slide, flip, turn and resize — each one is a small rule on the coordinates.
Start with the point A(2, 3)
Reflect it over the x-axis.
Reflecting over the x-axis negates y only: (2, 3) → (2, −3). B reflects over the y-axis instead. The axis named is the one that stays put — points slide across it, so the other coordinate is the one that flips.
(x, y) → (x + a, y + b)
4 right, 2 down
a = 4, b = −2
(2 + 4, 3 − 2)
add both
= (6, 1)
the image
Right and up are positive; left and down are negative. Size and shape never change — a translation only moves the figure.
Only the image of one vertex is given? Read the move off that vertex. If P(4, −2) lands on P′(1, 3), the move is 3 left and 5 up: (x − 3, y + 5). Every other vertex gets the same move, so Q(0, 1) goes to (−3, 6).
| Mirror | Rule | (2, 3) becomes |
|---|---|---|
| x-axis | (x, −y) | (2, −3) |
| y-axis | (−x, y) | (−2, 3) |
| the line y = x | (y, x) | (3, 2) |
| the origin | (−x, −y) | (−2, −3) |
| the line y = 1 | (x, 2 − y) | (2, −1) |
| the line x = −1 | (−2 − x, y) | (−4, 3) |
Reflecting over the x-axis changes y. The axis you flip over is the one that stays put.
For a mirror that is not an axis, count the gap. (2, 3) is 2 above the line y = 1, so its image is 2 below it, at y = −1. In general, over y = c the new y is 2c − y; over x = c the new x is 2c − x.
Over the x-axis, the x-coordinate is unchanged. Writing (−2, 3) there is flipping over the wrong axis.
| Turn | Rule | (2, 3) becomes |
|---|---|---|
| 90° counterclockwise | (−y, x) | (−3, 2) |
| 180° | (−x, −y) | (−2, −3) |
| 270° counterclockwise | (y, −x) | (3, −2) |
| 360° | (x, y) | back to start |
Clockwise 90° is the same as counterclockwise 270° — read the direction carefully.
How far did the point travel? (2, 3) turns 90° counterclockwise to (−3, 2). The gap between a point and its image is the distance formula: √(2 − (−3))² + (3 − 2)² = √25 + 1 = √26. Do not confuse it with √13, each point's distance from the origin.
(x, y) → (kx, ky)
k = 2 on (2, 3)
scale factor 2
= (4, 6)
twice as far from the origin
enlarges
shrinks
Dilation is the only one of the four that changes size. Lengths scale by k and areas by k².
| Transformation | Size | Shape | Orientation |
|---|---|---|---|
| translation | same | same | same |
| rotation | same | same | turned |
| reflection | same | same | mirrored |
| dilation | changed | same | same |
The first three are rigid motions — the image is congruent to the original. A dilation gives a similar figure instead.
Order matters when you combine them. Reflecting then translating rarely gives the same point as translating then reflecting.
Clockwise. Rotate (5, −2) 90° clockwise about the origin.
90° clockwise = 270° counterclockwise
use that row of the table
(x, y) → (y, −x)
the y comes first
(5, −2) → (−2, −5)
quadrant IV turns into quadrant III ✓
What is different: the table is written counterclockwise, so a clockwise turn borrows the 270° rule. Using (−y, x) turns the wrong way and lands on (2, 5).
The other diagonal. Reflect (−3, 4) over the line y = −x.
(x, y) → (−y, −x)
swap, and change both signs
(−3, 4) → (−4, 3)
the image
midpoint (−3.5, 3.5) lies on y = −x ✓
the mirror sits halfway
What is different: over y = x we only swap. Over y = −x we swap and negate both, which here leaves the point in the same quadrant.
A centre away from the origin. Dilate (3, 1) by a scale factor of 3 about the centre (1, 2).
from the centre: (3 − 1, 1 − 2) = (2, −1)
measure from the centre, not from (0, 0)
× 3: (6, −3)
stretch that arrow
image = (1 + 6, 2 − 3) = (7, −1)
add it back on to the centre
What is different: multiplying the coordinates directly only works when the centre is the origin.
3 × (3, 1) = (9, 3) is the dilation about (0, 0). With centre (1, 2), subtract the centre, multiply, then add the centre back.
The point (a, b) is translated 2 left and 5 up, then reflected over the y-axis. It lands on (3, 1). Find (a, b).
undo the reflection: (3, 1) → (−3, 1)
last move first; a reflection undoes itself
undo the translation: (−3 + 2, 1 − 5)
2 right and 5 down — the opposite move
(a, b) = (−1, −4)
check: (−1, −4) → (−3, 1) → (3, 1) ✓
Undoing starts from the last step. Translations flip direction, a 90° counterclockwise turn is undone by a 90° clockwise turn, and reflections and 180° turns undo themselves. Running the chain forwards from the answer is the slip the wrong choices are built on.
Two moves can also equal one: over the x-axis then over the y-axis sends (x, y) → (x, −y) → (−x, −y), the same as a 180° turn.
Expect a rule read off a grid (90° turns, y = x), a mirror line that is not an axis, a dilation asking for perimeter (× k) or area (× k²), or a chain to undo. The fastest check is to push your answer forwards through the moves and see that it lands on the given image.
Start with B(−4, 1)
Rotate it 180° about the origin.
A 180° turn about the origin negates both coordinates: (−4, 1) → (4, −1). A only swaps the coordinates — that is a reflection in y = x, not a turn. A half turn never swaps them. C and D negate just one coordinate: those are the reflections in the x-axis and the y-axis.
