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

Verified step by step guidance

1. 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.
2. 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 \(\frac{\pi}{\frac{1}{4}} = 4\pi\).
3. 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.
4. 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 \(x\).
5. 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.

Key Concepts

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).

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.

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.