Problem 11 - Precalculus Essentials

Question

Graph two periods of the given tangent function. $$y=\tan (x-\pi)$$

Step-by-Step Solution

Answer

The graph of the function \(y=\tan (x-\pi)\) features asymptotes at \( x = \pi \pm \frac{\pi}{2} + k\pi \), crosses the x-axis at \( x = (k+1)\pi \) and includes a curve between the asymptotes for two full periods.

Step 1: Identify the Period and Asymptotes

The period of the function \(y = \tan (x-\pi)\) is \( \pi \). Asymptotes will be at \( x - \pi = \pm \frac{\pi}{2} + k\pi \), in other words it is at \( x = \pi \pm \frac{\pi}{2} + k\pi \) for all integer \( k \)

Step 2: Identify the Zeros

The zeros of the function are at \( x - \pi = k\pi \), or \( x = k\pi + \pi = (k+1)\pi \). This means the function crosses the x-axis at \( x = (k+1)\pi \).

Step 3: Plot the function

Plot the function \(y=\tan (x-\pi)\) by drawing the appropriate curves between and approaching the asymptotes. Create markings at the zeros of the function. Remember to plot two full periods of the function.