All four transformations expressed as coordinate operations — a reference bridge into the Transformations section.
The Mnemonic
Every Transformation Is Just a Coordinate Rule
Every geometric transformation can be written as a simple rule for turning an original point (x, y) into a new point. This lesson serves as a reference summary before the full Transformations section covers each one in much greater depth: Reflections: over the x-axis (x,y)→(x,−y); over the y-axis (x,y)→(−x,y); over the line y=x (x,y)→(y,x). Rotations (counterclockwise about the origin): 90° → (x,y)→(−y,x); 180° → (x,y)→(−x,−y). Translations: shifting by (a,b) → (x,y)→(x+a, y+b). Dilations: scaling by factor k from the origin → (x,y)→(kx, ky).
Notice how each rule is really just applying earlier Coordinate Geometry ideas — a reflection over y=x literally swaps the coordinates (connecting to the symmetry of the Slope Formula itself), and a dilation directly matches the k-scaling behavior from Similar Figures applied coordinate-by-coordinate.
A quick reference table — each transformation expressed as one coordinate rule
💡 Memory Trick
"All four transformations expressed as coordinate operations." Reflections flip a sign or swap x and y. Rotations swap x and y AND flip a sign (in a specific pattern per angle). Translations simply ADD to each coordinate. Dilations simply MULTIPLY each coordinate by the same scale factor k.
Why It Works
A Preview, Not a Replacement
This summary intentionally doesn't derive WHY each rule works geometrically — that deeper derivation (using rotation angles, reflection line geometry, and so on) is exactly what the full Transformations section covers next. This lesson's purpose is narrower: to show that every transformation, no matter how it's later explained, always boils down to one clean coordinate rule you can apply directly to any point.
Having this reference table in Coordinate Geometry (before the dedicated Transformations lessons) means these rules are available immediately for any coordinate-based problem that needs them, without waiting for the deeper geometric explanations.
Using It In A Proof
Applying a Rule Directly to a Point
Using these rules is pure substitution — identify which transformation is being asked for, then apply its specific coordinate rule to each given point.
1
Identify the exact transformation
Determine which specific reflection, rotation, translation, or dilation the problem describes.
2
Look up (or recall) the matching coordinate rule
Match the transformation to its rule from the reference table above.
3
Apply the rule to each given point
Substitute each point's x and y values into the rule to get the new, transformed coordinates.
Full Worked Example
Applying Two Transformations in Sequence
Given: Point P(3, 5). Find: its final coordinates after first reflecting over the x-axis, then translating by (2, −4).
1
Apply the reflection first
Reflect over x-axis: (x,y)→(x,−y), so P(3,5) → (3, −5).
2
Apply the translation to the RESULT of step 1
Translate by (2,−4): (x,y)→(x+2, y−4), so (3,−5) → (3+2, −5−4).
3
Calculate the final coordinates
Final point: (5, −9).
Order matters when combining transformations — reflecting first and then translating can give a different final result than translating first and then reflecting, so always apply them in the exact sequence given.
🎯 Quick Worked Example
Point Q(−2, 6) is rotated 180° about the origin. Find its new coordinates.
1
Identify the rule. 180° rotation: (x,y) → (−x,−y).
2
Apply the rule. (−2, 6) → (−(−2), −6).
3
Calculate. Final point: (2, −6).
📌 Exam Application
Multi-step transformation problems (apply transformation A, then transformation B, to the same point) are common and specifically test whether the ORDER of operations is followed correctly — always apply each rule to the output of the previous step, never re-apply every rule to the original point independently.
⚠️ Most Common Coordinate Transformations Summary Mistakes
Trap 1 — Applying transformations out of order: When multiple transformations are chained, each one must be applied to the RESULT of the previous one, not to the original point each time — reversing the order can produce a genuinely different final coordinate.
Trap 2 — Mixing up similar-looking rules: Reflection over y=x, (x,y)→(y,x), and 90° rotation, (x,y)→(−y,x), look deceptively similar — always double check whether a sign flip is included before applying a rule.
✓ Quick Self-Test
1) State the coordinate rule for reflecting over the x-axis. 2) State the coordinate rule for a 90° counterclockwise rotation about the origin. 3) State the coordinate rule for a dilation by scale factor k. 4) Point R(4,−1) is translated by (−3,5) — find its new coordinates. 5) Why does the order matter when combining two transformations?