Translation coordinates are written using a notation that shows the original point and where it moves after the translation. The standard way to write a translation mapping a point (x, y) is ( 𝑥 , 𝑦 ) → ( 𝑥 + 𝑎 , 𝑦 + 𝑏 ).
In this notation, (x, y) represents the original coordinates of a point, and (x + a, y + b) represents the new coordinates of the point after the translation.
Understanding the Translation Notation
The notation ( 𝑥 , 𝑦 ) → ( 𝑥 + 𝑎 , 𝑦 + 𝑏 ) clearly indicates how a point is transformed on a coordinate plane.
- (x, y): The initial position of any point before the translation occurs.
- →: This arrow symbol means "maps to" or "becomes".
- (x + a, y + b): The final position of the point after the translation.
Here, a represents the horizontal displacement (how many units the point moves left or right), and b represents the vertical displacement (how many units the point moves up or down).
- If
ais positive, the movement is to the right. - If
ais negative, the movement is to the left. - If
bis positive, the movement is upwards. - If
bis negative, the movement is downwards.
As the reference states, in general, if a translation has a horizontal displacement of a units and a vertical displacement of b units, then the point (x, y) is mapped to (x + a, y + b). This is written concisely as ( 𝑥 , 𝑦 ) → ( 𝑥 + 𝑎 , 𝑦 + 𝑏 ). This mapping preserves the lengths of line segments, meaning the size and shape of figures do not change under translation.
Examples of Writing Translation Coordinates
Let's look at a few examples to solidify this concept.
Example 1: Simple Translation
Suppose we want to translate a point 3 units to the right and 2 units up.
- Horizontal displacement
a= 3 - Vertical displacement
b= 2
The translation rule is written as:
( 𝑥 , 𝑦 ) → ( 𝑥 + 3 , 𝑦 + 2 )
If we apply this rule to a specific point, say (1, 5), its new coordinates would be:
(1, 5) → (1 + 3, 5 + 2) = (4, 7)
Example 2: Translation with Negative Displacements
Suppose we want to translate a point 4 units to the left and 1 unit down.
- Horizontal displacement
a= -4 - Vertical displacement
b= -1
The translation rule is written as:
( 𝑥 , 𝑦 ) → ( 𝑥 + (-4) , 𝑦 + (-1) ) which simplifies to ( 𝑥 , 𝑦 ) → ( 𝑥 - 4 , 𝑦 - 1 )
Applying this rule to the point (6, 0):
(6, 0) → (6 - 4, 0 - 1) = (2, -1)
Table of Translation Examples
| Original Point (x, y) | Horizontal Shift (a) | Vertical Shift (b) | Translation Rule | New Point (x+a, y+b) |
|---|---|---|---|---|
| (2, 3) | +5 (Right) | +1 (Up) | (x, y) → (x + 5, y + 1) | (7, 4) |
| (-1, 4) | -2 (Left) | +3 (Up) | (x, y) → (x - 2, y + 3) | (-3, 7) |
| (0, 0) | +7 (Right) | -4 (Down) | (x, y) → (x + 7, y - 4) | (7, -4) |
| (5, -2) | -3 (Left) | -1 (Down) | (x, y) → (x - 3, y - 1) | (2, -3) |
This standard notation ( 𝑥 , 𝑦 ) → ( 𝑥 + 𝑎 , 𝑦 + 𝑏 ) is the primary way to write translation coordinates and rules in coordinate geometry, clearly showing the relationship between a point's initial and final position after a shift.
