Question

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

Accepted Answer

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

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

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Key Concepts

Period of the Tangent Function

Phase Shift

Reflection and Amplitude in Tangent Functions