In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = −tan(x − π/4)

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Identify the basic function and its transformations. The given function is \(y = -\tan\left(x - \frac{\pi}{4}\right)\), which is a tangent function shifted horizontally and reflected vertically.

Recall the period of the basic tangent function \(\tan x\) is \(\pi\). Since there is no coefficient multiplying \(x\) inside the function (other than 1), the period remains \(\pi\).

Determine the horizontal shift (phase shift). The function is shifted to the right by \(\frac{\pi}{4}\) because of the term \((x - \frac{\pi}{4})\) inside the tangent.

Account for the vertical reflection. The negative sign in front of the tangent means the graph is reflected over the x-axis, so all \(y\) values of \(\tan\left(x - \frac{\pi}{4}\right)\) are multiplied by \(-1\).

To graph two full periods, plot the function from \(x = \frac{\pi}{4}\) to \(x = \frac{\pi}{4} + 2\pi\). Mark the vertical asymptotes where the tangent function is undefined, which occur at \(x = \frac{\pi}{4} + \frac{\pi}{2} + k\pi\) for integers \(k\), and sketch the curve accordingly with the reflection and shift.

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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. For y = tan(x), one full period spans an interval of length π. When graphing two full periods, you need to cover an interval of length 2π along the x-axis.

Phase Shift

A phase shift occurs when the input variable x is replaced by (x − c), shifting the graph horizontally by c units. In y = −tan(x − π/4), the graph is shifted π/4 units to the right. This affects the location of key features like asymptotes and zeros.

Reflection and Amplitude in Tangent Functions

The negative sign in front of the tangent function, as in y = −tan(x − π/4), reflects the graph across the x-axis, reversing its increasing and decreasing behavior. Unlike sine and cosine, tangent has no amplitude since it is unbounded, but reflections affect the direction of the curve.