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When discussing the amplitude of the function \(y = 3 \cot 2x\), it's important to highlight that the amplitude refers to the "height" of the wave. In trigonometric functions like sine and cosine, amplitude is commonly defined as the maximum displacement from the horizontal axis. However, for the cotangent function, amplitude isn't traditionally defined in the same way.
Instead, when we mention amplitude in the context of \(y = 3 \cot 2x\), we refer to the coefficient in front of the cotangent function, which is 3 in this case. This factor signifies how the cotangent function is stretched or compressed vertically. Specifically, it indicates that the wave approaches or "hits" the extremum values of \(y = 3\) and \(y = -3\).
This helps visualize the vertical scaling of the function and gives an idea of how "tall" the peaks and "deep" the troughs are, even though cotangent inherently does not have a peak like a sine wave.

In the function \(y = 3 \cot 2x\), frequency involves understanding how many times the wave pattern repeats over a specific interval. Generally, with periodic functions, frequency is derived from the term inside the trigonometric function. Here, the expression \(2x\), implies a frequency modification.
While frequency in the context of sound might tell us about how often a sound wave cycles per second (measured in Hertz), for trigonometric functions like cotangent, it tells us how often the wave completes a full cycle within a certain interval on the x-axis. For \(\cot x\), a full period usually encapsulates intervals of \(\pi\) radians. By adjusting this through \(2x\), the entire function completes twice its cycle within its usual full period, affecting how fast or slow the function repeatedly oscillates up and down.
This modification makes the wave span more compactly along the x-axis, intensifying the number of wave cycles within the usual span.

The concept of a period in a trigonometric function describes the length required for the function to complete one full cycle before it starts repeating itself. For the equation \(y = 3 \cot 2x\), the period is influenced by the coefficient of \(x\) within the function.
The period of a standard cotangent function, \(\cot(x)\), is \(\pi\). However, when looking at \(\cot(2x)\), the formula for finding the period changes slightly. You calculate the period by dividing the standard period (\(\pi\)) by the number with \(x\), which is 2 in this equation. Therefore, \(\text{Period} = \pi / 2\).
In practical terms, this means that it only takes \(\pi/2\) units for the wave to complete a full cycle and begin anew. Therefore, when sketching the graph, every segment \(\pi/2\) along the x-axis marks where a full waveform is completed, repeated consistently along the graph. This reduced period means the graph is denser or more compressed horizontally.