Translation in Math

Translation in math involves moving shapes or graphs from one location to another. It is one of the easiest transformations and helps us understand how figures can slide on a plane without changing their size or shape. Learning translation makes it easier to study geometry, draw graphs, and even see how objects move in real life, like in games or animations. Translation helps us understand how figures move while keeping their properties intact, making it an essential concept in both geometry and algebra. In this guide, we will learn the definition of translation, its role in geometry, types of transformations, rules of translation, and its applications on the coordinate plane and graphs, supported by examples for better understanding.

 

Table of Contents

• Translation Math Definition
• Translation Geometry
• Types of Transformations in Math
• Translation on the Coordinate Plane
• Math Translation Rules
• Horizontal and Vertical Translations
• Graph Translation in Math
• Translation Function in Math
• Solved Examples
• Conclusion
• FAQs on Translation

 

Translation Math Definition  

In mathematics, translation is a type of transformation that moves a shape, graph, or object from one position to another on a plane without changing its size, shape, or orientation. It is simply sliding the figure in a specific direction by the same distance at every point, so only the location changes while the object itself remains unchanged.

 

Translation Geometry  

In translation geometry, geometric figures like triangles, rectangles, and circles are moved across the plane. The angles, sides, and size of the figure stay the same after translation. It is one of the most basic geometric transformations.  

In geometry, translation is a type of transformation that moves every point of a shape or figure the same distance in the same direction. It does this without changing the size, shape, or orientation.

 

Features of Translation in Geometry:

• It is a rigid transformation, meaning there is no distortion of the shape. 
• The figure is moved from one location to another.
• All points of the shape move uniformly.
• The shape remains congruent to its original position.

 

How It Works: 

If a shape is translated by a vector (a, b), then every point (x, y) on the shape moves to a new point (x + a, y + b).

Example: 
A triangle with vertices at A(1, 2), B(3, 2), and C(2, 4) is translated by (2, -1). 
New points will be: 
A' = (1 + 2, 2 - 1) = (3, 1) 
B' = (3 + 2, 2 - 1) = (5, 1) 
C' = (2 + 2, 4 - 1) = (4, 3)

Translation Vector: 
A vector like (a, b) shows you: 

• Move right if a > 0 or left if a < 0. 
• Move up if b > 0 or down if b < 0.

 

Types of Transformations in Math  

A transformation in math is an operation that changes the position, size, or shape of a figure. It helps us understand how shapes move or change on a coordinate plane.

There are four main types of transformations in math:

1. Translation (Sliding):

• Moves a shape from one place to another without rotating or flipping it.
• The shape keeps its size, shape, and orientation. 
• Example: Moving a triangle 3 units right and 2 units down. 

Rule: (x, y) → (x + a, y + b)

 

2. Reflection (Flipping):

• Flips a shape over a line, like a mirror image.
• Common lines of reflection are the x-axis, y-axis, or y = x.
• The shape's size and shape stay the same, but its orientation changes.

Example: Reflecting a point (3, 2) over the x-axis changes it to (3, -2).

 

3. Rotation (Turning):

• Turns a shape around a fixed point, usually the origin (0, 0).
• The shape rotates by a certain angle, like 90° or 180°, in a specific direction, either clockwise or counterclockwise.
• The size and shape remain the same.

Example: Rotating a point (2, 3) 90° counterclockwise around the origin changes it to (-3, 2).

 

4. Dilation (Resizing):

• Changes the size of a shape but keeps its proportions.
• A figure can be enlarged or reduced.
• It uses a scale factor to multiply the distances from a fixed point, known as the center of dilation.

Example: If the scale factor is 2, the point (3, 4) changes to (6, 8).

Translation is the simplest of all these types of transformations in math.  

 

Translation on Coordinate Plane  

Translation on the coordinate plane means moving a point or shape from one location to another using a set of coordinates. The movement is defined by a translation vector, which is a pair of numbers that shows how far to move horizontally and vertically.

Translation Rule:  
If a point is at (x, y) and it is translated by a vector (a, b), the new point becomes:  
(x + a, y + b)

• a = units moved horizontally (right if positive, left if negative)  
• b = units moved vertically (up if positive, down if negative)  

 

Example 1: Point Translation  
Translate the point A(2, 3) by (4, -2)  
New point A' = (2 + 4, 3 - 2) = (6, 1)  

 

Example 2: Shape Translation  
Translate a triangle with vertices:  
P(1, 2), Q(3, 2), R(2, 4) by the vector (2, 3)  

New vertices:  

P' = (1 + 2, 2 + 3) = (3, 5)  

Q' = (3 + 2, 2 + 3) = (5, 5)  

R' = (2 + 2, 4 + 3) = (4, 7)  

Translation on the coordinate plane helps students visualize how shapes move, which is important in geometry, graphing, and real-world applications like animations and game design.

 

Math Translation Rules  

Math translation rules define how to move shapes or points.  

General Translation Rule:
If a point is at (x, y) and you translate it by (a, b):

• New coordinates: (x + a, y + b)

• a = units moved horizontally

• b = units moved vertically

 

Translation Rules Table:

 

 

Horizontal and Vertical Translations  

In mathematics, a translation is a transformation that moves a figure or point from one position to another without changing its shape, size, or orientation.

There are two main types of translations:

  

Horizontal Translation:

A horizontal translation shifts a point or shape left or right on the coordinate plane.

Only the x-coordinate changes, while the y-coordinate stays the same.

Formula:

If a point (x, y) is moved h units horizontally, the new point is:

(x + h, y)

• If h is positive, the figure moves to the right.

• If h is negative, it moves to the left.

Example:

Translate the point (3, 5) horizontally by +4 units:

New point = (3 + 4, 5) = (7, 5)

 

Vertical Translation:

A vertical translation moves a point or shape up or down.

Only the y-coordinate changes, while the x-coordinate stays the same.

Formula:

If a point (x, y) is moved k units vertically, the new point is:

(x, y + k)

• If k is positive, the figure moves up.

• If k is negative, it moves down.

Example:

Translate the point (3, 5) vertically by -2 units:

New point = (3, 5 - 2) = (3, 3)

These are called horizontal and vertical translations and are used for translating graphs.  

 

Graph Translation in Math  

Graph translation in math involves shifting a function's graph either horizontally or vertically.  

Examples:  

• f(x) = x²  

• f(x - 2): shifts the graph 2 units to the right  

• f(x) + 3: shifts the graph 3 units up  

Graph translation in math helps us understand how changes in functions affect their graphs.  

 

Translation Function in Math  

A translation function in math is a type of transformation that moves every point of a shape or graph a fixed distance in a specific direction. It does not change the shape, size, or orientation; it only changes its position.

Translation Function for Points:  

If you have a point (x, y) and you want to translate it by a units horizontally and b units vertically, the translation function is:

T(x, y) → (x + a, y + b)

• If a > 0, move right.  
• If a < 0, move left.  
• If b > 0, move up.  
• If b < 0, move down.  

Example:  

Translate the point (2, 5) by a = 3 and b = -2:  

T(2, 5) → (2 + 3, 5 - 2) = (5, 3)

 

Solved Examples on Translation in Math

Example 1: Translation on Coordinate Plane (Point Translation)
Question: Translate the point A(3, 4) by the vector (2, -3). What is the new position of the point?
Solution:
Use the rule (x + a, y + b)
A = (3, 4)
Vector = (2, -3)
A' = (3 + 2, 4 - 3) = (5, 1)
Answer: A'(5, 1)

 

Example 2: Translation Geometry (Shape Translation)
Question: A triangle has vertices P(1, 2), Q(4, 2), and R(3, 5). Translate the triangle 3 units right and 2 units down.
Solution:
Translation vector = (3, -2)
P' = (1 + 3, 2 - 2) = (4, 0)
Q' = (4 + 3, 2 - 2) = (7, 0)
R' = (3 + 3, 5 - 2) = (6, 3)
Answer: P'(4, 0), Q'(7, 0), R'(6, 3)

 

Example 3: Graph Translation in Math (Function Shift)
Question: Given f(x) = x², translate the graph 2 units left and 4 units up.
Solution:
Left shift → x becomes (x + 2)
Upward shift → add 4
New function: f(x) = (x + 2)² + 4
Answer: f(x) = (x + 2)² + 4

 

Example 4: Translation Function in Math
Question: The graph of f(x) = |x| is translated 3 units right and 5 units down. What is the new function?
Solution:
Right shift → f(x - 3)
Down shift → subtract 5
New function: f(x) = |x - 3| - 5
Answer: f(x) = |x - 3| - 5

 

Example 5: Math Translation Rules with Quadrilateral
Question: Translate quadrilateral with vertices A(0, 0), B(2, 0), C(2, 3), D(0, 3) by (-1, -2).
Solution:
A' = (0 - 1, 0 - 2) = (-1, -2)
B' = (2 - 1, 0 - 2) = (1, -2)
C' = (2 - 1, 3 - 2) = (1, 1)
D' = (0 - 1, 3 - 2) = (-1, 1)
Answer: A'(-1, -2), B'(1, -2), C'(1, 1), D'(-1, 1)

 

Example 6: Identify Type of Translation
Question: If f(x) = x³ becomes f(x) = (x - 4)³ + 2, what type of translation is this?
Solution:
x - 4 → 4 units right
+2 → 2 units up
Answer: Horizontal translation 4 units right and vertical translation 2 units up

 

Conclusion  

Translation is an important concept in math that involves sliding objects or graphs across a plane. It aids in understanding geometry and graphing functions. With translation in math, students learn how to move shapes, follow math translation rules, and use translation functions to shift graphs.  

By mastering horizontal and vertical translations, graph translation in math, and translation on the coordinate plane, students gain a better understanding of how functions and shapes behave. Translation geometry is not only useful in math exams but also in real-world applications like design, animation, and navigation.  

 

Frequently Asked Questions on Translation in Maths

1. What is a translation in maths?

Answer: A translation in maths is a type of transformation that moves a shape or point from one position to another without rotating, resizing, or flipping it. It keeps the shape and size the same, only changing the position.

2. What are the 4 translations in math?

Answer: The four types of translations in math are:

• Translation (sliding)

• Rotation (turning)

• Reflection (flipping)

• Dilation (resizing)
  
  

3. What is an example of a translation?

Answer: Example: Moving a point A(2, 3) to the right by 4 units and up by 2 units. The new point becomes A'(6, 5). This is a translation by vector (4, 2).

4. What is the translation formula?

Answer: The translation formula is:
If a point is (x, y) and it is translated by (a, b), the new point is:
(x + a, y + b)

5. How to calculate a translation?

Answer: To calculate a translation:

• Determine how far to move horizontally (a) and vertically (b)

• Apply the formula: New point = (x + a, y + b)
  Example: Translate (3, 5) by (-2, 4) → (3 - 2, 5 + 4) = (1, 9)

 

Master translation in math with simple examples at Orchids International School.