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Understanding the concept of periodicity in trigonometric functions, especially the tangent function, is crucial for graphing it accurately. Periodicity refers to the characteristic of a function where its values repeat after a fixed interval, known as the period. For the basic tangent function, represented by the equation \(y = \tan(x)\), this period is \(\pi\), which means the function's values repeat every \(\pi\) radians.

In the given exercise, the period of the tangent function is modified due to the presence of a coefficient within the function's argument. The coefficient \(\frac{1}{2}\) affects the period by stretching it, so the equation \(y = -3 \tan(\frac{1}{2}x)\) has a period of \(2\pi\) instead of the basic \(\pi\). These subtle changes in the function's period are often the source of mistakes when sketching the graph, so it is very important to always calculate the new period when given a modified tangent function.

When graphing trigonometric functions, the coefficients can tell you a lot about the transformation the basic function undergoes. For instance, in the exercise, the coefficient -3 in front of the tangent function indicates two things: the graph will be flipped vertically and it will be stretched.

The negative sign tells us that the graph of \(y = \tan(x)\) will be flipped over the x-axis, which in mathematical terms, is known as a reflection. It means that where the graph of the basic tangent function goes up, the modified graph will go down, and vice versa.

The absolute value of the coefficient, 3, indicates a vertical stretch. Vertical stretching occurs when each y-value on the original graph is multiplied by a factor greater than 1, which in this case is 3. Consequently, this makes the highs higher and the lows lower within the same period interval, effectively 'stretching' the graph out vertically. When sketching, it’s essential to factor in these changes to ensure your graph accurately reflects both the stretching and the flipping.

Sketching trigonometric functions like the tangent function can seem challenging, but it becomes manageable when approached systematically. Graphing involves identifying the key characteristics of the function such as its period, amplitude (which does not apply to tangent functions as they have no maximum or minimum values), reflections, and vertical shifts.

In the step-by-step solution of our exercise, these modifications—such as the doubling of the period and the vertical stretching by a factor of 3 combined with the flip—were considered. Asymptotes are particularly important when sketching the tangent function. These are lines that the graph approaches but never touches, occurring at the points of discontinuity on the x-axis, which for the basic tangent are at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). In our modified function, their location will be adjusted according to the new period and the function’s horizontal scaling.

By following these steps and understanding how each transformation affects the graph, you can sketch the tangent function, or any trigonometric function, with confidence and precision. Always remember to clearly show these characteristics on your graph: asymptotes, period intervals, and the effects of vertical stretching and flipping.