Same Math › Geometry · grades 9–10 › Transformations with coordinate rules
What are coordinate rules for transformations?
Every slide, flip, turn or resize of a shape on a grid has a rule that says what happens to each point. A translation 3 right and 2 down is (x, y) → (x + 3, y − 2). Reflecting over the x-axis is (x, y) → (x, −y). Rotating 90° counterclockwise is (x, y) → (−y, x). Kids apply the rule to each corner and label the new points A′, B′, C′.
- The original shape, labeled A, B, C.
- The image, labeled A′ (A prime) and so on.
- Apply the rule to every corner, then connect.
Why schools teach it
Moves you can compute, not just draw
Eighth graders describe transformations with coordinates (8.G.A.3), and geometry students define them precisely and use them for congruence (HSG.CO.A.5).
Rotations are hard to draw by eye. The rule makes them mechanical: swap the numbers, then fix the signs.
At the kitchen table
What to say, and what to skip
Say this
What's the rule, written as (x, y) → ?
Apply it to A first. Where does A′ go?
Did the shape keep its size?
Everything but a dilation does.Is it counterclockwise or clockwise?
Most rules are written counterclockwise.
Skip this
- Mixing up reflecting over the x-axis and the y-axis.Over the x-axis, y changes sign.
- Forgetting to swap for 90° turns.(x, y) → (−y, x) swaps the numbers.
- Drawing the new shape before computing points.Compute, plot, then connect.
See it in action
Rotating A(2, 5) by 90°, 180° and 270° counterclockwise
90°: (x, y) → (−y, x)
A′(−5, 2)180°: (x, y) → (−x, −y)
A′(−2, −5)270°: (x, y) → (y, −x)
A′(5, −2)
A 270° counterclockwise turn is the same as 90° clockwise.
Free practice sheet
Make a coordinate transformations sheet
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Parents also ask
What's a rigid motion?
A move that keeps size and shape: translations, reflections and rotations. Dilations change size.
What is a dilation?
A resize from a center point. With scale factor 2 from the origin: (x, y) → (2x, 2y).
What does the prime mark mean?
A′ is “A prime,” the new location of A. A″ is after a second move.
Do rotations always turn around the origin?
In most Algebra and Geometry problems, yes, unless another center is named.