Graph each function over a one-period interval.
y = 2 tan (¼ x)

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Identify the basic function and its transformation: The given function is \(y = 2 \tan\left(\frac{1}{4}x\right)\). Here, the coefficient 2 is a vertical stretch, and the argument \(\frac{1}{4}x\) affects the period of the tangent function.

Recall the period of the basic tangent function \(\tan(x)\) is \(\pi\). For \(y = \tan(bx)\), the period is given by \(\frac{\pi}{|b|}\). In this case, \(b = \frac{1}{4}\), so the period is \(\frac{\pi}{\frac{1}{4}} = 4\pi\).

Determine the one-period interval for the function: Since the period is \(4\pi\), one period can be taken as \(x\) in \([0, 4\pi]\) or any interval of length \(4\pi\). For example, \([-2\pi, 2\pi]\) is also valid.

Identify the vertical asymptotes of the tangent function within one period. The tangent function has vertical asymptotes where its argument equals \(\frac{\pi}{2} + k\pi\), for integers \(k\). Solve \(\frac{1}{4}x = \frac{\pi}{2} + k\pi\) to find the asymptotes in terms of \(x\).

Plot key points: At \(x=0\), \(y=2 \tan(0) = 0\). At \(x\) values where \(\frac{1}{4}x = \frac{\pi}{4}\) or \(\frac{3\pi}{4}\), calculate \(y\) values to understand the shape. Use the vertical asymptotes to sketch the graph of \(y=2 \tan\left(\frac{1}{4}x\right)\) over one period.

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Period of the Tangent Function

The standard period of the tangent function, tan(x), is π. When the function is transformed as tan(bx), the period changes to π divided by the absolute value of b. Understanding this helps determine the interval over which to graph one full cycle of y = 2 tan(¼ x).

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Introduction to Tangent Graph

Amplitude and Vertical Stretch

Although tangent functions do not have a maximum or minimum amplitude, the coefficient outside the function, here 2, vertically stretches the graph. This affects the steepness of the curve but does not change the period or asymptotes.

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Stretches and Shrinks of Functions

Vertical Asymptotes of Tangent

Tangent functions have vertical asymptotes where the cosine function equals zero, causing the function to approach infinity. For y = 2 tan(¼ x), these asymptotes occur at values of x where ¼ x equals (2k+1)π/2, which is crucial for accurately sketching the graph.

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