91Ó°ÊÓ

The tangent function, known as \( y = \tan(x) \), is unique in its periodic nature compared to sine and cosine functions. Its periodicity is \( \pi \) radians, meaning the function repeats its values every \( \pi \) units along the x-axis. This is fundamentally different from the sine and cosine functions, which have a period of \( 2\pi \).

The repetition implies that every segment of the graph from any point \( x \) to \( x + \pi \) will look identical to any other segment of the same length anywhere on the graph. For example, if you graph \( y = \tan(x) \) from \( x = 0 \) to \( x = \pi \), and then graph from \( x = \pi \) to \( x = 2\pi \), you will notice these segments look the same.

When a tangent function undergoes a transformation, for example, by shifting or stretching, its periodicity remains \( \pi \), but those changes will affect where each period starts and ends.

A horizontal shift in a trigonometric function results in moving the entire graph left or right on the coordinate plane. For the function \( y = \tan(x - \pi) \) given in the exercise, the horizontal shift moves the standard tangent graph to the right by \( \pi \) units.

This means that every key point of the function, like zeros and asymptotes, will also shift right by \( \pi \). Therefore, if the unshifted tangent graph has zeros at \( x = 0, \pi, 2\pi, \ldots \), after applying the shift, these will move to \( x = \pi, 2\pi, 3\pi, \ldots \).

This shift is crucial when sketching the function because it determines the new starting point of the period on the graph. It's often helpful to first identify the new positions of both the zeros and vertical asymptotes after the shift is applied, making it simpler to sketch each period accurately.

Vertical asymptotes are a key feature in graphing the tangent function. Asymptotes are invisible lines that indicate where a function's value becomes undefined or tends towards infinity. For the tangent function \( y = \tan(x) \), these occur at odd multiples of \( \frac{\pi}{2} \) (like \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \)).

When we introduce a horizontal shift, such as in \( y = \tan(x - \pi) \), these asymptotes also move in accordance with the shift. In this case, they will shift to positions like \( x = \frac{3\pi}{2} \) and \( x = \frac{5\pi}{2} \) within the given range for two periods.

Visualizing these asymptotes is critical as they divide the periods of the tangent function. Each segment between two asymptotes represents one period. When sketching, it's beneficial to draw the asymptotes lightly as guides before plotting the actual graph to ensure accuracy and clarity.