Transformation Math

Guide to Math Transformation Types, Rules, Definitions, & Examples

transformation-mathWhat are Transformations in Math and Geometry?

Contents

Definition: A Transformation in Math is a process of moving an object (two-dimensional shape) from its original position to a new position.

The object in the original position (before transformation) is called the pre-image and the object in the new position (after transformation) is called the image.


Transformation Math Rules

Characteristics

Transformations are functions that take each point of an object in a plane as inputs and transforms as outputs (image of the original object) including translation, reflection, rotation, and dilation. Transformations are movements through space and it can be seen in many instances like diverse actions walking and running.

Uses

Transformations are useful in many real-world scenarios.

  • The movement of a robotic Mars Rover.
  • Strumming a guitar
  • Combined twist and turns of a roller-coaster.
  • The opening and closing of an artificial heart valve to control blood flow.

Importance

Transformations are quite important in many fields, such as the study of art, architecture, anthropology, and many more. Transformation sets the foundation for other areas of study, such as congruence and similarity, the verification of perpendicular segments, the derivation of the equation of a circle etc.


5 Types of Mathematical Transformations

Here are the 5 types of transformations in math uses in geometry.

  1. Line Reflections
  2. Point Reflections
  3. Rotations
  4. Dilations
  5. Translations

Transformation Math Examples

Line Reflections

What are line reflections?

Line reflection is folding or flipping an object over a mirror line (line of reflection). The original object is called the pre-image, and the reflection is called the image. An object and its reflection have the same size and shape, but the figure faces in opposite directions.

Reflections in the coordinate plane:

Reflection in the x-axis: Reflecting a point over the line x-axis, the x-coordinate remains the same, but the sign of y-coordinate is changed. The reflection of the point (x, y) over the line x-axis is the point (x, -y).

Example: When point Q with coordinates (2, 3) is reflecting over the x-axis line and mapped onto point Q’, the coordinates of Q’ are (2, -3).

Reflection in the y-axis:
Reflecting a point over the line y-axis, the y-coordinate remains the same, but the sign of x-coordinate is changed. The reflection of the point (x, y) over the line y-axis is the point (-x, y).

Example: When point Q with coordinates (4, 5) is reflecting over the y-axis line and mapped onto point Q’, the coordinates of Q’ are (-4, 5).

Reflection in the y = x: Reflecting a point over the line y = x, the x-coordinate and the y-coordinate change places. The reflection of the point (x, y) over the line y = x is the point (y, x).

Example: When point Q with coordinates (1, 3) is reflecting over the line y = x and mapped onto point Q’, the coordinates of Q’ are (3, 1).

Reflection in the y = -x: Reflecting a point over the line y = -x, the x-coordinate and the y coordinate change places and the signs are changed (negated).

The reflection of the point (x, y) over the line y = -x is the point (-y, -x).

Example: When point Q with coordinates (6, 1) is reflecting over line y = -x and mapped onto point Q’, the coordinates of Q’ are (-1, -6).


Point Reflections

What are point reflections?

A point reflection is a type of reflection that occurs when a figure is built around a single point called the point of reflection. In point reflection, a shape is reflected over a specific point, usually that point is the origin.

Reflection in the Origin:

Reflecting a point in the origin, both the x-coordinate and the y-coordinate are negated (their signs are changed).

In a Point reflection in the origin, the coordinate (x, y) changes to (-x, -y).

Example: When point P with coordinates (1, 2) is reflecting over the point of origin (0,0) and mapped onto point Q’, the coordinates of Q’ are (-1, -2).


Rotations

What are rotations?

A rotation turns an object about a fixed point called the center of the rotation. Rotation may be clockwise or counterclockwise. An object remains the same in shape and size after rotation, but the object may be turned in different directions.

Rotation of 90°

When a figure is rotated clockwise by 90°, each point of the figure has to be changed from (x, y) to (y, -x), it is similar to the rotation counterclockwise by 270° and when a figure is rotated counterclockwise by 90°, each point of the figure has to be changed from (x, y) to (-y, x), it is similar to the rotation clockwise by 270°.

Example: When point R with coordinates (3, 2) is rotated clockwise by 90° and mapped onto point R’, the coordinates of R’ are (2, -3).

Rotation of 180°

When a figure is rotated clockwise or counterclockwise by 180°, each point of the figure has to be changed from (x, y) to (-x, -y).

Example: When point R with coordinates (5, 1) is rotated clockwise by 180° and mapped onto point R’, the coordinates of R’ are (-5, -1).

Rotation of 270°

When a figure is rotated clockwise by 270°, each point of the figure has to be changed from (x, y) to (-y, x), it is similar to the rotation counterclockwise by 90° and when a figure is rotated counterclockwise by 270°, each point of the figure has to be changed from (x, y) to (y, -x), it is similar to the rotation clockwise by 90°.

Example: When point R with coordinates (2, 4) is rotated clockwise by 270° and mapped onto point R’, the coordinates of R’ are (-4, 2).


Dilations

What are Dilations?

Dilation is a transformation that changes the size of the image, but shape remains the same. A dilation that creates a larger image is called an enlargement. A dilation that creates a smaller image is called reduction. The scale factor measures how much the image is larger or smaller.
To dilate a figure in the coordinate plan multiply each coordinate by the scale factor.

Dilation of scale factor K:

The scale factor K determines whether the dilation is an enlargement or a reduction. If > 1, the dilation is an enlargement and if < 1, the dilation is a reduction.

The absolute value of the scale factor determines the size of the new image as compared to the size of the original image. When k is positive the new image and the original image are on the same side of the center, when k is negative they are on the opposite sides of the center.

Example: When point P with coordinates (2, 4) is dilated by the scale factor of k = 2 and mapped onto point P’, the coordinates of P’ are (4, 8).
When point P with coordinates (2, 4) is dilated by the scale factor of k = ½ and mapped onto point P’, the coordinates of P’ are (1, 2).


Translations

What are Translations?

Translation means moving. A translation is a type of transformation that moves a shape up, down, left or right without changing its appearance in any way. Translation moves every point of the shape in the same distance and in the same direction.

Translation of h, k:

Right h units: Add h to the x-coordinate: (x, y) → (x + h, y)

Left h units: Subtract h from the x-coordinate: (x, y) → (x – h, y)

Up K units: Add K to the y-coordinate: (x, y) → (x, y + k)

Down k units: Subtract h from the y-coordinate: (x, y) → (x, y- k)

Example: When point P with coordinates (5, 3) is translated 2 units right and mapped onto point P’, the coordinates of P’ are (7, 3).

When point P with coordinates (5, 3) is translated 2 units left and mapped onto point P’, the coordinates of P’ are (3, 3).

When point P with coordinates (5, 3) is translated 1 units up and mapped onto point P’, the coordinates of P’ are (5, 4).

When point P with coordinates (5, 3) is translated 1 units down and mapped onto point P’, the coordinates of P’ are (5, 2).