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

Verified step by step guidance

1. 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.
2. Recall that the tangent function has a period of \(\pi\). Since the problem asks for two periods, determine the interval length to graph: \(2 \times \pi = 2\pi\).
3. 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 \(x = \frac{\pi}{4} + 2\pi\).
4. 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.
5. 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.

Key Concepts

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.