Geometric Transformations
Move figures on the coordinate plane with precision: translate, reflect, rotate, and dilate — and describe exactly which transformation you used.
1 What is a transformation?
A transformation is a rule that moves every point of a figure to a new position. The original figure is the preimage; the result is the image, usually labeled with primes: P becomes P′.
Rigid motions preserve size and shape — the image is congruent to the preimage. A dilation changes size by a scale factor k: the image is similar, not congruent (unless k = 1).
2 The mapping rules
Every transformation on the coordinate plane is a rule of the form (x, y) → (new x, new y). Memorize these nine — they cover nearly every problem you will meet.
Reflect over x-axis: (x, y) → (x, −y) · over y-axis: (x, y) → (−x, y)
Reflect over y = x: (x, y) → (y, x) · over origin: (x, y) → (−x, −y)
Rotate 90° clockwise: (x, y) → (y, −x) · 90° counterclockwise: (x, y) → (−y, x)
Rotate 180°: (x, y) → (−x, −y) · Dilate by k: (x, y) → (kx, ky)
Notice: a 180° rotation gives the same coordinates as a reflection over the origin. A dilation with k between 0 and 1 shrinks the figure; k > 1 enlarges it; negative k also flips it.
3 Worked examples
Each example applies one rule directly to a point. Apply the rule, then check the image.
Reflect P(3, −2) over the x-axis.
Rule: (x, y) → (x, −y). So (3, −2) → (3, 2). The x-coordinate stays; the y-coordinate changes sign.
Rotate P(3, −2) 90° clockwise about the origin.
Rule: (x, y) → (y, −x). So (3, −2) → (−2, −3). The coordinates trade places and the new y takes the sign.
Translate P(3, −2) by the vector ⟨4, 1⟩.
Rule: (x, y) → (x + 4, y + 1). So (3, −2) → (7, −1). Slide right 4, up 1.
Dilate P(−2, 5) by scale factor k = 3 centered at the origin.
Rule: (x, y) → (3x, 3y). So (−2, 5) → (−6, 15). Multiply every coordinate by 3.
Reflect P(4, 1) over the line y = x.
Rule: (x, y) → (y, x). So (4, 1) → (1, 4). The coordinates trade places — nothing else changes.
Rotate P(−3, 4) 180° about the origin.
Rule: (x, y) → (−x, −y). So (−3, 4) → (3, −4). Both signs flip — the point lands in the opposite quadrant.
P(2, 1) maps to P′(5, 4) and Q(−1, 3) maps to Q′(2, 6). Name the transformation.
Check the shift: 5 − 2 = 3 and 4 − 1 = 3; 2 − (−1) = 3 and 6 − 3 = 3. Every point shifts by ⟨3, 3⟩, so this is a translation. One point pair is never enough — always check a second pair.
4 Common mistakes
The three errors that cost the most points on transformation problems.
Wrong: reflecting (4, 1) over y = x to get (−4, −1) or (4, 1) unchanged.
Right: the coordinates trade places and nothing else: (4, 1) → (1, 4).
Wrong: dilating triangle ABC by k = 2 but doubling only A and B.
Right: multiply every coordinate of every vertex by k —
(x, y) → (2x, 2y) for A, B, and C alike.
Wrong: rotating (3, −2) 90° clockwise to get (2, 3).
Right: clockwise is (x, y) → (y, −x), giving (−2, −3);
counterclockwise is (x, y) → (−y, x), giving (2, 3). Say the direction out loud before you write.
5 Key vocabulary
Words to know
- Preimage — the original figure before the transformation.
- Image — the resulting figure, labeled with primes (A′, B′, C′).
- Rigid motion — a transformation preserving size and shape: translation, reflection, rotation.
- Scale factor (k) — the multiplier in a dilation; k > 1 enlarges, 0 < k < 1 shrinks.
- Center of dilation — the fixed point distances are measured from (usually the origin).
- Orientation — the order of vertices around a figure; reflections reverse it, rotations preserve it.
6 Quick check
Try these before moving on — click to reveal each answer.
7 Next steps
Now drill the skill with endless randomized problems.