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Understanding tangent function transformations is vital when graphing equations like \(y=3 \tan \frac{x}{2}\). Similar to other trigonometric functions, the tangent function can be manipulated with scaling, shifting, and reflecting.

In our exercise, the coefficient '3' in front of the tangent function indicates a vertical scaling. This means the height of its peaks and the depths of its troughs are multiplied by three, making them more pronounced. In contrast, a horizontal transformation is observed by the \frac{x}{2} term inside the tangent function. This modification actually stretches the function horizontally, meaning the period of the function—the distance between repeating patterns—is increased.

Visualizing these changes is key:

• The vertical stretch makes the graph's movement from peak to valley more drastic.
• The horizontal stretch makes each wave of the tangent longer before it repeats.

It's essential to carefully analyze both kinds of transformations to accurately plot the graph of a tangent function.

A vertical asymptote is a line the function approaches but never touches or crosses. The basic \(y=\tan(x)\) function has vertical asymptotes at \(x=n\pi + \frac{\pi}{2}\), for any integer 'n'.

What happens during tangent function transformations? The locations of these asymptotes can also shift. In our exercise, since we have \(y=3 \tan \frac{x}{2}\), the horizontal stretch affects the position of the asymptotes. Now, they occur every \(2\pi\) rather than \(\pi\), resulting in asymptotes at \(x = -\pi, \pi, 3\pi\), within the given range. These are points where the graph shoots to infinity, making them critical to note when sketching the curve of a transformed tangent function.

Remember, the existence of vertical asymptotes and their correct placement are crucial for the graph’s accuracy of any tangent function.

The period of a function is the length of the smallest interval over which the function's values repeat. For the basic \(y=\tan(x)\) function, this interval is \(\pi\). However, a transformation inside the function, like the \(\frac{x}{2}\) in our exercise, modifies this period.

To understand the effect, we use the formula for the period of a transformed tangent function: \[\text{Period} = \frac{\pi}{|b|}\] where 'b' is the coefficient of 'x' in the tangent function. In our case, the period becomes \(\frac{\pi}{1/2} = 2\pi\), doubling the length of the basic tangent's cycle.

This change in period helps us determine the correct spacing between peaks, valleys, and asymptotes. The expanded period especially means fewer cycles over a given x-interval compared to the untransformed tangent, which students must correctly illustrate to ensure their graph's veracity.