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The tangent function is one of the primary trigonometric functions, often denoted as \(y = \tan x\). Unlike sine and cosine, which have a wave-like appearance, the graph of the tangent function exhibits a series of repeating vertical asymptotes and sections that increase or decrease sharply. This happens because the tangent function is undefined whenever the cosine component in the denominator equals zero, leading to vertical asymptotes at these points.
To better understand the behavior of the tangent function, it's essential to note the following characteristics:

• The tangent function has vertical asymptotes where it is undefined, occurring at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is any integer.
• It intercepts the x-axis at \(x = n\pi\), where the value of the function is zero.
• Within a period of the tangent function, it transitions from negative infinity to positive infinity or vice versa, which gives it a distinct shape compared to sine and cosine.

This function is significant in trigonometry, calculus, and various applications because it relates angles to the ratio of opposite to adjacent sides in right triangles.

The period of a trigonometric function refers to the length of the interval over which the function completes one full cycle before repeating itself. Each trigonometric function has a distinct period, affecting how its graph appears over the x-axis.
For the basic function \(y = \tan x\), the period is \(\pi\). This means that after every \(\pi\) interval, the tangent function's graph looks the same.
When transformations occur, such as scaling the x-variable, the period of the function is altered. The formula to calculate the new period for a tangent function \(y = \tan(bx)\) is:

• New period = \(\frac{\pi}{b}\)

In the exercise's function \(y = 2 \tan \frac{\pi}{2} x\), the multiplier is \(\frac{\pi}{2}\), resulting in a period of \(2\) calculated as \( \frac{\pi}{\frac{\pi}{2}} = 2\).
This change in period affects how frequently the graph repeats its cycle over the interval.

Graphing trigonometric functions involves mapping the unique features of the functions, such as their periodicity and asymptotic behavior. For tangent functions, this includes plotting vertical asymptotes, zeros, and the slope dictated by any amplitude changes.
To graph \(y = 2 \tan \frac{\pi}{2} x\), recognize the following key features of the graph:

• Vertical asymptotes appear where the function is undefined. For this tangent function, asymptotes are at \(x = -1\) and \(x = 1\).
• The zeros of the function occur at \(x = 0\), where the function crosses the x-axis.
• Amplitudes influence the steepness of the curve, but not the period. The coefficient 2 in the function affects the rapidity of the increase or decrease of the tangent graph.

Begin by plotting the zeros and asymptotes, then sketch the increasing or decreasing sections according to the function's behavior. Keep in mind the period and amplitude adjustments when visualizing the final graph.