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The period of a trigonometric function is the length of the smallest interval over which the function repeats. For the tangent function, the standard period is \(\pi\), which means that the function will have the same values at \(x\) and at \(x + \pi\). The function given in the exercise, \(f(x)=3 \tan \left(\frac{x}{2}\right)\), exhibits a transformation that affects its period. Specifically, the horizontal stretch factor of \(\frac{1}{2}\) causes the period to double from \(\pi\) to \(2\pi\). This concept is crucial in sketching the function, as it dictates the 'wavelength' of the tangent curve.

Understanding the period helps you predict where the function will start repeating itself, which is particularly important when you're asked to sketch it over a certain interval. Since the period in this case is \(2\pi\), students should note that the pattern of the tangent curve within any \(2\pi\) interval will be identical to the pattern in any other \(2\pi\) interval.

Vertical asymptotes are lines to which the graph of a function approaches but never touches or crosses. They represent values for which the function is undefined. For the tangent function, vertical asymptotes occur at \(x = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer. The given function, \(f(x)=3 \tan \left(\frac{x}{2}\right)\), has its vertical asymptotes at \(x = 2n\pi\) due to the horizontal stretch that alters the function's behavior.

When sketching the tangent function, it's essential to accurately plot these asymptotes because they delineate the boundaries between which the function exists. Any errors in placing the vertical asymptotes will significantly distort the shape of the graph. Remember, the function's curve approaches these lines without limit, but it will never intersect them.

Graphing transformations involve changing the position, shape, or orientation of a graph. These can include horizontal stretches or compressions, vertical stretches or shrinks, translations, and reflections. For the exercise \(f(x)=3 \tan \left(\frac{x}{2}\right)\), transformations include a horizontal stretch by a factor of \(2\), due to the \(\frac{x}{2}\) term, and a vertical stretch by a factor of \(3\), given by the coefficient before the tangent function.

When sketching, one should first consider the transformations that affect the period and then address vertical stretches, which will change the distance that the curve peaks and troughs from the x-axis. The horizontal stretch will be reflected in the new position of the asymptotes and the period, while the vertical stretch will alter how ‘tall’ or ‘steep’ the peaks and troughs are.

The tangent function has several key characteristics that are important for sketching its graph. It's periodic, with a standard period of \(\pi\), and exhibits symmetry about the origin, meaning that it is odd and has 180-degree rotational symmetry. The function increases without bound as it approaches its vertical asymptotes from the left and decreases without bound as it approaches from the right. It crosses the origin, and between its asymptotes, it looks like a rising or falling line depending on the direction the curve is headed.

For the function \(f(x)=3 \tan \left(\frac{x}{2}\right)\), these characteristics still apply but with the modified period and vertical stretch. It's important to correctly apply these properties when plotting the graph to ensure an accurate representation of the function. Consider how the curve should pass symmetrically through the origin and remember that its value increases indefinitely as it approaches the vertical asymptote from one side, and decreases indefinitely as it approaches from the other side.