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Understanding the concept of periodicity is central to trigonometry. For trigonometric functions like the cosine function, periodicity refers to the characteristic that the function repeats its values at regular intervals. For instance, the standard cosine function, represented as \(y = \cos x\), has a period of 360 degrees. This means that every 360 degrees, the function values repeat. When we modify this function, such as \(y = \cos (kx)\), the period changes. The formula to determine the new period is \( \text{Period} = \frac{360}{|k|}\), where \(k\) is the coefficient of \(x\). Applying this formula helps in understanding how the graph of the function is altered.

Horizontal compression is a key transformation for trigonometric functions. It occurs when we multiply the variable \(x\) by a constant greater than 1. For example, in the function \(y = \cos (2x)\), the 2 in the argument of the cosine function causes the graph to compress horizontally by a factor of 2. To see the effect on the period, use the formula \( \text{Period} = \frac{360}{|k|} \). For \(y = \cos (2x)\), we get \( \text{Period} = \frac{360}{2} = 180 \) degrees. Hence, the graph repeats every 180 degrees, instead of the original 360 degrees. This means the graph will look 'squeezed' together when compared to the standard cosine function.

Horizontal stretch is another transformative concept for trigonometric functions. This occurs when we multiply the variable \(x\) by a constant between 0 and 1. Consider the function \(y = \cos (\frac{1}{4} x)\). The \(\frac{1}{4}\) causes the graph to stretch horizontally by a factor of 4. Using the period formula, \( \text{Period} = \frac{360}{|k|}\), we find that \(y = \cos (\frac{1}{4} x)\) has a period of \( \text{Period} = \frac{360}{\frac{1}{4}} = 1440 \) degrees. Hence, the graph stretches out, repeating every 1440 degrees. This transformation makes the graph appear 'stretched' out compared to the original cosine function.

Trigonometric transformations involve various modifications to trigonometric functions, affecting their amplitude, period, phase shift, and vertical shift. Two important transformations are horizontal compression and stretch:
- Horizontal Compression: When \(k\) is greater than 1, as seen in \(y = \cos (2x)\), the function compresses horizontally.
- Horizontal Stretch: When \(k\) is between 0 and 1, as in \(y = \cos (\frac{1}{4} x)\), the function stretches horizontally.
These transformations can be analyzed by the period formula, \( \text{Period} = \frac{360}{| k |}\), making it straightforward to understand and predict the behavior of the transformed function. For example, \(y = \cos (-3 x)\) involves a horizontal compression by 3 and has a period of 120 degrees, while the negative sign indicates a reflection, which doesn't change the graph for cosine functions.