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Understanding the amplitude of a trigonometric function, such as the cosine function in our example, is essential to grasp their behavior. Amplitude is the height of the wave from the center line to a peak or from the center line to a trough. It determines how 'tall' or 'short' the waves of the function are.

In mathematical terms, if a trigonometric function is given in the form of
\( y = A \cdot \text{trig}(bx+c) \)
where \( \text{trig} \) stands for either sine or cosine, the absolute value of \( A \) is the amplitude. For our function \( y = 3 \cos 2x \), the amplitude is 3. This equates to the maximum vertical distance from the graph's center line, which lies at \( y = 0 \) for a standard trigonometric function without a vertical shift.

The concept of period in trigonometric functions is another fundamental aspect to understand. The period is the length it takes for the function to complete one full cycle before repeating itself.

Using \( y = A \cdot \text{trig}(bx+c) \), where \( \text{trig} \) is again sine or cosine, the period \( T \) is calculated using the formula \( T = \frac{2\pi}{|b|} \). If we apply this to our example function \( y = 3 \cos 2x \), the period would be \( \frac{2\pi}{2} \), which simplifies to \( \pi \). So, the graph of our function completes its wave pattern every \( \pi \) units along the x-axis.

Phase shift represents the horizontal translation of a trigonometric graph. When a trigonometric function includes a horizontal translation, the graph shifts left or right from the standard position.

The formula for a function with phase shift is \( y = A \cdot \text{cos}(b(x - c)) + D \), where \( c/b \) is the phase shift. If \( c \) is positive, the graph shifts to the right, and if \( c \) is negative, it shifts to the left. In our example \( y = 3 \cos 2x \), since there is no \( c \) value, this means there is no phase shift, and the graph starts at the standard initial point of a cosine wave, which is a peak at \( (0, A) \).

Transformations apply to all kinds of functions, including trigonometric ones. They can stretch, compress, reflect, or translate the graph. If we review our function, each transformation can be deciphered:

The factor '3' in \( y = 3 \cos 2x \) vertically stretches the graph by a factor of 3, producing a higher amplitude. The factor '2' associated with 'x' results in a frequency increase, halving the period of the function. There are no factors outside the trigonometric function adding or subtracting, meaning there's no vertical shift or phase shift in this instance.

When graphing these functions, it's vital to take into consideration each of these transformations as they significantly alter the appearance and characteristics of the wave. For the cosine function specifically, reflecting across the x-axis or a vertical shift will change the starting and ending points within one period, making a clear understanding of these transformations crucial.