The Shape Translation Calculator calculates where a shape’s points end up after sliding it a set distance horizontally and vertically.
Every point moves by the same vector (a, b) using the rule (x, y) → (x + a, y + b). Enter the points, how far to move right (+) or left (−), and how far to move up (+) or down (−); the calculator lists the image points and graphs both shapes.
Shape Translation Calculator
Translate (slide) a shape by a vector — move each point left/right and up/down — and graph the result.
Formula
(x, y) → (x + a, y + b)
Where:
- (x, y) = original point
- a = horizontal move (positive = right, negative = left)
- b = vertical move (positive = up, negative = down)
The translation vector is often written as ⟨a, b⟩ or as a column vector.
How the Calculation Works
Step 1: Add a to every x-coordinate.
Step 2: Add b to every y-coordinate.
Step 3: Join the new points in the same order. The shape keeps its size, angles and orientation.
Variables Explained
| Input | Meaning | Example |
|---|---|---|
| Points | One x, y pair per line | 2, 3 |
| Move right/left (a) | Horizontal shift | 4 or −3 |
| Move up/down (b) | Vertical shift | −2 or 5 |
Example Calculation
Given: A(1, 1), B(5, 1), C(2, 4); translate 4 right and 2 down, vector ⟨4, −2⟩
Rule: (x, y) → (x + 4, y − 2)
Result: A′(5, −1), B′(9, −1), C′(6, 2)
Finding the vector from a point and its image
If P(2, 3) moves to P′(−1, 7), then a = −1 − 2 = −3 and b = 7 − 3 = 4, so the vector is ⟨−3, 4⟩.
Shape Properties
A translation is a rigid transformation: the image is congruent to the original and faces the same way. Every point moves the same distance, √(a² + b²), in the same direction.
Dimensions Explained
Writing translation rules
“3 units left and 5 units up” becomes (x, y) → (x − 3, y + 5). Combining two translations just adds their vectors.
Frequently Asked Questions
What is the rule for a translation?
(x, y) → (x + a, y + b), where a and b are the horizontal and vertical moves.
How do you translate a shape 3 units left?
Subtract 3 from every x-coordinate: (x, y) → (x − 3, y).
How do you find the translation vector?
Subtract the original coordinates from the image coordinates of any one point.
Does translation change the shape’s size or orientation?
No. Only its position changes.
What is the difference between a translation and a rotation?
A translation slides a shape without turning it; a rotation turns it around a center point.