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Trigonometric functions are the foundation of trigonometry, a branch of mathematics that deals with the relationships between angles and sides of triangles. This category includes functions like sine, cosine, and tangent, which are used to relate the angles of a triangle to the lengths of its sides. These functions are periodic, meaning they repeat their values in regular intervals.

When we describe the behavior of these functions, we're observing how their values change as the angle, typically measured in radians or degrees, increases. Understanding trigonometric functions is crucial because they have applications in various fields such as physics, engineering, and navigation.

Diving into the characteristics of the tangent function, we find that it exhibits some intriguing traits distinguishing it from its trigonometric counterparts. Unlike the sine and cosine functions, the tangent function does not oscillate between fixed maximum and minimum values. Instead, it sweeps through all real numbers, extending from negative infinity to positive infinity and passing through zero.

The lack of a fixed amplitude means the tangent graph looks like a sequence of increasing curves without any horizontal bounds. As a result, it is critical to note that the tangent function will produce very large or very small values for angles near its vertical asymptotes, which occur at odd multiples of \(\frac{\pi}{2}\).

Asymptotes are lines that a curve approaches infinitely close but never touches or crosses. For the tangent function, its asymptotic behavior is one of its defining features. The vertical asymptotes occur at points where the cosine, the denominator in the tangent ratio \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), is zero. These points are located at \(\frac{\pi}{2}\) and every odd multiple thereof.

On a graph, these asymptotes are key to sketching the accurate shape of the tangent curve. They represent the boundaries between which the function rises or falls sharply, and they are also the points at which the function is undefined because you cannot divide by zero.

The period of a trigonometric function is the smallest interval over which the function's values repeat. For the tangent function, this interval is \(\pi\), which is notably just half of the period of the sine and cosine functions. What this indicates is that after every \(\pi\) radians, the tangent function will reset and start its pattern anew. The function's periodic nature is essential when solving trigonometric equations or analyzing waveforms in physics, since it enables us to predict the function's behavior beyond the basic \(\pi\) interval.

This completion of the tangent's cycle every \(\pi\) radians is why it's critical to understand period when interpreting the function's behavior in applied mathematics or when graphing the function over a span of multiple periods.