In mathematics, transformations of a parent cosine function can be used to create a new cosine function. In order to change the parent cosine function to the cosine function below, the following transformations are required:

## Amplitude

The amplitude is the height of a cosine graph and refers to the maximum absolute value of the graph’s y-coordinate. This value can be modified by multiplying the parent cosine function’s amplitude by a scalar value. For example, if the parent cosine function’s amplitude is 5, multiplying it by 2 will make the new cosine graph’s amplitude 10.

## Period

The period of a cosine graph refers to the length of a single repetition of the graph. This value can be modified by multiplying the parent cosine function’s period by a scalar value. For example, if the parent cosine function’s period is 4, multiplying it by 2 will make the new cosine graph’s period 8.

## Midline

The midline of a cosine graph is the horizontal line that the graph crosses at the origin. This value can be modified by changing the parent cosine function’s vertical shift. For example, if the parent cosine function is shifted upwards by 10 units, the new cosine graph will be shifted upwards by 20 units.

These transformations can be applied to the parent cosine function to create new cosine functions. As such, they are useful when analyzing the relationship between two cosine graphs, or when attempting to determine the properties of a new cosine graph.