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How Math Transformations Work

In mathematics, transformations act to alter the shape of a graph by changing its height, width or direction. Transformations can occur singularly or as compositions of multiple transformations within a single equation. The four general types of transformations are translation, reflection, rotation and dilation. Understanding the rules of transformation allow you to more easily graph an equation by starting with the shape of a basic graph and shifting it appropriately.

Translation
- The concept of translation in mathematics involves shifting a graph vertically or horizontally on a coordinate plane. The translation of a graph is obtained by adding a value to one or both point coordinates of the (x, y) pair. Adding to the x-coordinate shifts the graph horizontally. For example, adding a -5 to the x-coordinate of the point (8,13) shifts the point 5 units to the left, resulting in the new point (3, 13). Similarly, adding to the y-coordinate shifts the graph vertically. For example, adding a 3 to the y-coordinate of the point (8, 13) shifts the point 3 units up, resulting in the new point (8, 16).
Reflection
- The reflection transformation occurs when a graph is flipped about the x or y axis. The reflection of a graph about the x-axis results in a change of sign for the y-coordinate of the coordinate pair (x, y). For example, the point (3, 4) reflected about the x-axis results in the new point (3, -4). Reflection about the y-axis results in a change of sign for the x-coordinate. For example, the point (3, 4) reflected about the y-axis results in the new point (-3, 4).
Rotation
- The rotating transformation of a graph involves rotating the graph around a central point a set number of degrees. The resulting point(s) on the graph are determined by the size of the angle the point is rotated through. Positive rotations move counterclockwise while negative rotations move clockwise. It is often necessary to use a compass to complete a rotation transformation. However, sometimes you can eyeball the graph and guess where the point will sit after rotation. For example, rotating the coordinate point (0, 1) 90 degrees counterclockwise results in the new point (0, -1).
Dilation
- A dilation transformation expands or shrinks the graph by specified factor. This factor, often called the scale factor, determines how much smaller or larger the graph becomes. Dilation is achieved by multiplying one or both coordinates in the coordinate pair (x, y). Dilating the x-coordinate alters the horizontal width of the graph. Dilating the y-coordinate alters the vertical height of the graph. For example, multiplying the x and y coordinates of the point (5, 7) by 2 results in a new point located at (5 * 2, 7 * 2) = (10, 14).

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