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The cosine function, an essential building block in trigonometry, is a periodic wave that oscillates in a consistent pattern. Its graph resembles a smooth and continuous wave that rises and falls with symmetry around the horizontal axis.

Mathematically, the basic cosine function can be represented as \( f(x) = \text{cos} x \). It reaches its highest and lowest points at regular intervals and these values are always 1 and -1 respectively. When you multiply the cosine function by a number, it increases or decreases its vertical reach but does not affect the periodicity. For example, \( f(x) = 3 \text{cos} x \) will oscillate between -3 and 3, which leads to interesting properties when solving for minimum and maximum values.

Periodic functions are the kinds of functions that repeat their values at regular intervals over time. One period is the length of one complete cycle of the wave, after which the pattern starts again. The cosine function is a prime example of a periodic function.

The concept of periodicity is crucial when analyzing functions in various fields such as physics, engineering, and signal processing. Understanding the period of a function helps with predictions and analysis since it tells you how often certain values will recur. This recurrence is also why finding the minimum value of functions like \( f(x) = 3 \text{cos} 2x \) is more straightforward – it repeats at every integer multiple of \( \text{π} \).

The amplitude of a trigonometric function like sine or cosine refers to the height of its peaks, whether positive or negative, from its central axis (typically the horizontal axis in a graph). In simpler terms, it's how 'tall' or 'short' the waves of the function are.

For the function \( f(x) = a \text{cos} x \), 'a' represents the amplitude. An amplitude increase signifies a stretch in the graph, making the peaks and troughs move further away from the axis. In the context of the given function \( f(x) = 3 \text{cos} 2x \), the amplitude is 3, meaning the function's maximum and minimum values are stretched out to 3 and -3, respectively. The number in front of the \( x \), here a 2, represents frequency and affects the period of the function but not its amplitude.