Graph two periods of the given tangent function. y = tan(x − π/4)

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Identify the basic form of the tangent function given: \(y = \tan(x - \frac{\pi}{4})\). This is a horizontal shift of the standard tangent function \(y = \tan x\) by \(\frac{\pi}{4}\) units to the right.

Recall that the tangent function has a period of \(\pi\). Since the problem asks for two periods, determine the interval length to graph: \(2 \times \pi = 2\pi\).

Find the new domain interval for two periods considering the phase shift. Since the function is shifted right by \(\frac{\pi}{4}\), the interval for two periods can be from \(x = \frac{\pi}{4}\) to \(x = \frac{\pi}{4} + 2\pi\).

Identify the vertical asymptotes of the function. For \(y = \tan(x - \frac{\pi}{4})\), asymptotes occur where the argument of tangent is \(\frac{\pi}{2} + k\pi\), so solve \(x - \frac{\pi}{4} = \frac{\pi}{2} + k\pi\) for \(x\) to find asymptotes within the interval.

Plot key points between asymptotes by evaluating \(y = \tan(x - \frac{\pi}{4})\) at values such as the midpoints between asymptotes, then sketch the curve showing the characteristic shape of the tangent function over two periods.

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주요 개념

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Period of the Tangent Function

The tangent function has a fundamental period of π, meaning its values repeat every π units. When graphing, identifying one full period helps in plotting the function accurately. For y = tan(x − π/4), the period remains π, but the graph is shifted horizontally.

Horizontal Phase Shift

A phase shift occurs when the input variable x is adjusted inside the function, such as x − π/4. This shifts the graph horizontally by π/4 units to the right. Understanding this shift is essential to correctly position the tangent curve on the x-axis.

Asymptotes of the Tangent Function

Tangent functions have vertical asymptotes where the function is undefined, occurring at x = π/2 + kπ for integers k. For y = tan(x − π/4), these asymptotes shift accordingly. Recognizing asymptotes helps in sketching the graph and understanding its behavior near undefined points.