# Algebraic Representation

## The Importance of Algebraic Representation in Mathematical Transformations

As with any busy individual, the prime minister of the United Kingdom cannot be in two places at once. When his presence is needed, he delegates the task to a representative. Similarly, in mathematics, algebraic expressions and equations use variables to stand in for actual numerical values, creating an algebraic representation.

Algebraic representation is the use of variables, numbers, and symbols to represent quantities in equations or expressions. It provides a concise and efficient way to describe mathematical relationships without the use of words. This form of representation can be applied to multiple areas, including geometrical transformations, formulae, and functions.

## Understanding Algebraic Representation

Geometrical transformations are changes that occur to a mathematical object, such as a shape, in terms of its position or size. These changes can involve translation, reflection, rotation, enlargement, or a combination of these. To gain a deeper understanding of transformations, check out our article on the subject.

When it comes to the algebraic representation of transformations, we work with geometric shapes that are being transformed along the x-axis and y-axis.

## Translation

Translation involves moving a shape in a specific direction, such as up, down, left, or right, or a combination of these. The shape remains unchanged in size and shape, but its position shifts. To learn more about translation, read our article on the topic.

To represent translation algebraically, we use coordinates to indicate the direction and distance the shape moves along the x-axis and y-axis. For example, if the shape is moving to the right on the x-axis, the coordinates are (x+1, y). Similarly, if it moves to the left, the coordinates are (x-1, y).

## Reflection

Reflection is the mirror image of a geometric shape when it is flipped across a line, known as the line of reflection or symmetry. Unlike translation, reflection changes the position of the shape, but it remains unchanged in size and shape. For instance, if we have a triangle with coordinates (3,5) and (7, 5), and we reflect it over the y-axis, the resulting coordinates are (-3, 5) and (-7, 5). You can see this in action in the image below.

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