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Math · Geometry & Spatial Reasoning
Coordinate Transformations: Translations, Reflections & Rotations
Build on ordered pairs by translating, reflecting and rotating points and simple figures on the coordinate plane, then describe each transformation with precise coordinate rules.
The idea
A transformation moves a point or figure according to a rule. A translation slides, a reflection flips across a line, and a rotation turns around a centre.
Rigid transformations preserve lengths and angle sizes. The figure changes position or orientation, but it does not stretch or shrink.
Coordinate rules make each move testable: translations add the same amount to coordinates, axis reflections change selected signs, and 90° rotations around the origin swap coordinates in a predictable way.
Worked examples
Example 1 · Translate a point
Translate (−2, 3) by 5 units right and 4 units down.
- Add 5 to x: −2+5=3.
- Subtract 4 from y: 3−4=−1.
- Write the new ordered pair.
(3, −1).
Example 2 · Reflect across the y-axis
Reflect (4, −6) across the y-axis.
- A y-axis reflection keeps y unchanged.
- Change the sign of x: 4→−4.
- Write the image point.
(−4, −6).
Example 3 · Reflect across the x-axis
Reflect (−5, 2) across the x-axis.
- An x-axis reflection keeps x unchanged.
- Change the sign of y: 2→−2.
(−5, −2).
Example 4 · Rotate 90° counterclockwise
Rotate (3, 1) 90° counterclockwise about the origin.
- Use the rule (x,y)→(−y,x).
- Swap the coordinates and negate the new x.
- (3,1)→(−1,3).
(−1, 3).
Example 5 · Identify a transformation
Point A=(2,−4) moves to A′=(−2,−4). What transformation is shown?
- The y-coordinate stayed the same.
- The x-coordinate changed sign.
- That is the rule for reflecting across the y-axis.
Reflection across the y-axis.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Across the y-axis only x changes sign; across the x-axis only y changes sign.
Why this fails: Every point in the figure must receive the same translation vector.
Why this fails: Clockwise: (x,y)→(y,−x). Counterclockwise: (x,y)→(−y,x).
Why this fails: Translations, reflections and rotations preserve distances.
Why this fails: These rules assume the origin unless another centre is stated.
Foundation
Foundation practice
Apply one-step translations and axis reflections.
1. Translate (1,2) 3 units right.
2. Translate (−4,5) 2 units left.
3. Translate (3,−1) 4 units up.
4. Translate (−2,6) 5 units down.
5. Reflect (5,2) across the y-axis.
6. Reflect (−3,7) across the x-axis.
7. Reflect (0,4) across the y-axis.
8. Reflect (6,0) across the x-axis.
9. What coordinate rule reflects across the y-axis?
10. What coordinate rule reflects across the x-axis?
Core
Core practice
Use two-coordinate translations and 90°/180° rotations about the origin.
11. Translate (2,−3) by (4,5).
12. Translate (−5,1) by (−2,−6).
13. Rotate (2,5) 90° counterclockwise.
14. Rotate (2,5) 90° clockwise.
15. Rotate (−4,3) 180° about the origin.
16. Rotate (0,6) 90° clockwise.
17. (3,−2) becomes (3,2). Name the transformation.
18. (−1,4) becomes (4,1). Name the transformation.
19. (2,3) becomes (−3,2). Name the transformation.
20. After a translation, a segment was 7 units long. How long is its image?
Reasoning
Reasoning practice
Work backward, combine moves and reason about preserved properties.
21. A point translated 4 right and 3 down ends at (8,1). Where did it start?
22. After reflection across the y-axis, the image is (6,−2). What was the original?
23. Reflect (3,4) across the y-axis, then translate 2 right.
24. Translate (−2,1) 5 up, then reflect across the x-axis.
25. Rotate (1,3) 90° clockwise twice.
26. Which changes under a reflection: side length, angle size, orientation or area?
27. A translation maps every point (x,y) to (x−3,y+2). Describe the move.
28. A point follows (x,y)→(−x,−y). What rigid transformation is this?
29. Which point stays fixed under every rotation about the origin?
30. Why is (x,y)→(2x,2y) not a rigid transformation?
Stretch
Stretch practice
Transform figures and connect coordinates to geometric structure.
31. Triangle A(1,1), B(4,1), C(1,3) translates by (−2,4). Find A′.
32. Using the same translation, find B′.
33. Using the same translation, find C′.
34. The original triangle has side lengths 3, 2 and √13. What happens to its perimeter after the translation?
35. Reflect rectangle vertices (1,1),(5,1),(5,3),(1,3) across the y-axis. Give the image of (5,3).
36. Rotate (−2,5) 90° counterclockwise.
37. Rotate the result of p36 another 90° counterclockwise.
38. Reflect (4,−2) across the x-axis, then across the y-axis.
39. A student says rotating (3,2) 90° counterclockwise gives (2,−3). What did the student do?
40. Why can you compare an original figure and its rotated image using corresponding side lengths?
Answer key
- 1. (4, 2)
Add 3 to x; y stays the same. - 2. (−6, 5)
Subtract 2 from x. - 3. (3, 3)
Add 4 to y. - 4. (−2, 1)
Subtract 5 from y. - 5. (−5, 2)
A y-axis reflection changes the sign of x. - 6. (−3, −7)
An x-axis reflection changes the sign of y. - 7. (0, 4)
A point on the y-axis stays fixed. - 8. (6, 0)
A point on the x-axis stays fixed. - 9. (x,y)→(−x,y)
Only x changes sign. - 10. (x,y)→(x,−y)
Only y changes sign. - 11. (6, 2)
Add 4 to x and 5 to y. - 12. (−7, −5)
Add −2 to x and −6 to y. - 13. (−5, 2)
Use (x,y)→(−y,x). - 14. (5, −2)
Use (x,y)→(y,−x). - 15. (4, −3)
A 180° turn changes both signs. - 16. (6, 0)
(0,6)→(6,0). - 17. Reflection across the x-axis.
x stayed the same; y changed sign. - 18. 90° clockwise rotation about the origin.
The rule is (x,y)→(y,−x). - 19. 90° counterclockwise rotation about the origin.
The rule is (x,y)→(−y,x). - 20. 7 units.
Translations preserve distance. - 21. (4, 4)
Undo by moving 4 left and 3 up. - 22. (−6, −2)
Reflecting again across the same axis returns the original. - 23. (−1, 4)
(3,4)→(−3,4)→(−1,4). - 24. (−2, −6)
First (−2,6), then change y to −6. - 25. (−1, −3)
Two 90° clockwise turns equal a 180° rotation. - 26. Orientation.
A reflection reverses orientation but preserves lengths, angles and area. - 27. 3 units left and 2 units up.
Subtracting from x moves left; adding to y moves up. - 28. A 180° rotation about the origin.
Both coordinate signs change. - 29. (0, 0)
The centre of rotation does not move. - 30. It changes distances by scaling the figure.
Rigid motions preserve distance. - 31. (−1, 5)
Apply the same vector to every vertex. - 32. (2, 5)
(4,1)+(−2,4)=(2,5). - 33. (−1, 7)
(1,3)+(−2,4)=(−1,7). - 34. It stays the same.
Translations preserve every side length. - 35. (−5, 3)
Negate x only. - 36. (−5, −2)
Use (x,y)→(−y,x). - 37. (2, −5)
A second quarter-turn gives the 180° image. - 38. (−4, 2)
The two reflections change both signs, equivalent to a 180° rotation. - 39. Used the clockwise rule.
Counterclockwise should give (−2,3). - 40. Rotation preserves distance, so corresponding side lengths are equal.
Rigid transformations preserve size and shape.