Coordinate Rules for Transformations
Every slide, flip and turn on the coordinate plane has a short rule that tells you exactly where each point goes. Learn to read the rules, find them, and chain them together.
Know these first
- Rotations 8.G.1
This unlocks
What you'll be able to do
- Apply a coordinate rule to points and figures.
- Name the transformation a rule describes.
- Find the rule from a figure and its image.
- Combine two rules into one sequence and find where a point ends up.
The big idea
A rule like (x, y) → (−y, x) is a recipe: take each point, build the new point from its old coordinates.
A few rules come up again and again. Adding to x or y is a translation. Changing a sign is a reflection over an axis. Swapping x and y is a reflection over y = x. Swapping AND changing one sign is a 90° rotation.
To find a rule, test two points - one point can fit more than one rule. To combine rules, apply the first, then apply the second to the result.
Words you need
- Coordinate rule
- A recipe like (x, y) → (x + 2, −y) that says where every point goes.
- Composition
- Doing one transformation, then another, to the result.
- Image
- Where a point lands, written with a prime: A′.
Common rules
| Rule | Transformation |
|---|---|
| (x, y) → (x + a, y + b) | Translation |
| (x, y) → (x, −y) | Reflection over the x-axis |
| (x, y) → (−x, y) | Reflection over the y-axis |
| (x, y) → (y, x) | Reflection over y = x |
| (x, y) → (−y, x) | Rotation 90° counterclockwise |
| (x, y) → (−x, −y) | Rotation 180° |
| (x, y) → (y, −x) | Rotation 90° clockwise |
Worked examples
Example 1
Apply the rule (x, y) → (−y, x) to the point (2, 5).
- The new x is the old y with its sign changed: −5.
- The new y is the old x: 2.
- The image is (−5, 2) - a 90° counterclockwise turn.
Answer: (−5, 2)
Example 2
Point (3, 1) maps to (−1, 3), and point (0, 2) maps to (−2, 0). Find the rule.
- In both, the new x is the old y with its sign changed, and the new y is the old x.
- The rule is (x, y) → (−y, x), a 90° counterclockwise rotation.
Answer: (x, y) → (−y, x)
Example 3
Apply (x, y) → (x, −y), then (x, y) → (x + 2, y), to the point (1, 3).
- First rule: (1, 3) → (1, −3).
- Second rule on the result: (1, −3) → (3, −3).
Answer: (3, −3)
Common mistakes
Testing a rule with only one point. One point can fit several rules. Check a second point before you decide.
Applying combined rules in the wrong order. Do the first rule, then apply the second rule to the NEW point. Order can change the answer.
Mixing up (−y, x) and (y, −x). (−y, x) turns counterclockwise; (y, −x) turns clockwise. Test with (1, 0).
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Apply the rule (x, y) → (−y, x) to the point (4, 1).
- Apply the rule (x, y) → (−x, −y) to the point (−3, 4).
- A board game flips pieces with the rule (x, y) → (y, x). A piece is at (1, 6). Where does it go? Word problem
- A robot follows the rule (x, y) → (x + 2, y − 5). It starts at (−1, −1). Where does it end? Word problem
- A sticker at (4, −2) is moved by the rule (x, y) → (−x, y). Where does it end up? Word problem
Tier 2 Intermediate One more step to think through
- Point (2, 1) maps to (1, −2), and point (0, 3) maps to (3, 0). Find the rule.
- Point (4, 1) maps to (4, −1), and point (−2, 3) maps to (−2, −3). Find the rule.
- Apply the rule (x, y) → (y, −x) to triangle ABC with A(1, 2), B(3, 2) and C(1, 4).
- A designer applies the rule (x, y) → (−y, x) to the corners (2, 0) and (2, 3). Where do they go? Word problem
- A pattern maps the point (5, −2) to (−5, 2) and the point (1, 3) to (−1, −3). What rule is it using? Word problem
- The rule (x, y) → (y, x) is applied to (3, 7), and then applied again to the result. Where does the point end up? Word problem
- A tile maps (1, 4) to (−4, 1) and (2, 0) to (0, 2). What rule is it using? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Apply (x, y) → (−x, y), then (x, y) → (−y, x), to the point (2, 3).
- Point (1, 2) maps to (−2, −1), and point (3, 0) maps to (0, −3). Find the rule.
- An animation applies (x, y) → (−y, x) and then (x, y) → (x + 1, y + 1) to the point (2, 4). Where does it end up? Word problem
- A shape's points (2, 1) and (4, 1) land at (−1, 2) and (−1, 4). What rule maps the shape? Word problem
- The rule (x, y) → (−y, x) is applied three times in a row to the point (1, 2). Where does it end up? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Point (1, 1) maps to (−1, −3), and point (2, 3) maps to (0, −5). Find two rules that, applied in order, do this.
- Which rule reverses a figure's orientation: (x, y) → (−y, x) or (x, y) → (y, x)? Word problem
- A game applies the rule (x, y) → (−x, y + 3) to a triangle with corners (1, 0), (3, 0), (1, 2). Where do the corners go? Word problem
Answers
Tier 1 Foundations
- (−1, 4). Apply (x, y) → (−y, x) to each point.
- (3, −4). Apply (x, y) → (−x, −y) to each point.
- (6, 1). Apply (x, y) → (y, x) to each point.
- (1, −6). Apply (x, y) → (x + 2, y − 5) to each point.
- (−4, −2). Apply (x, y) → (−x, y) to each point.
Tier 2 Intermediate
- (x, y) → (y, −x). Apply the rule to every original point; each one lands on its image point.
- (x, y) → (x, −y). Apply the rule to every original point; each one lands on its image point.
- (2, −1), (2, −3), (4, −1). Apply (x, y) → (y, −x) to each point.
- (0, 2), (−3, 2). Apply (x, y) → (−y, x) to each point.
- (x, y) → (−x, −y). Apply the rule to every original point; each one lands on its image point.
- (3, 7). Apply (x, y) → (y, x); then (x, y) → (y, x) to each point.
- (x, y) → (−y, x). Apply the rule to every original point; each one lands on its image point.
Tier 3 Advanced
- (−3, −2). Apply (x, y) → (−x, y); then (x, y) → (−y, x) to each point.
- (x, y) → (−y, −x). Apply the rule to every original point; each one lands on its image point.
- (−3, 3). Apply (x, y) → (−y, x); then (x, y) → (x + 1, y + 1) to each point.
- (x, y) → (−y, x). Apply the rule to every original point; each one lands on its image point.
- (2, −1). Apply (x, y) → (−y, x); then (x, y) → (−y, x); then (x, y) → (−y, x) to each point.
Tier 4 Challenge
- (x, y) → (x, −y) then (x, y) → (x − 2, y − 2). Apply the rule to every original point; each one lands on its image point.
- (x, y) → (y, x). That rule is a reflection over y = x, and reflections reverse orientation. (x, y) → (−y, x) is a rotation, which keeps it.
- (−1, 3), (−3, 3), (−1, 5). Apply (x, y) → (−x, y); then (x, y) → (x, y + 3) to each point.
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