Chapter 4, Problem 105CR

To determine

To draw: The graph of the function.

Explanation of Solution

Given information: The given equation is $f(x)=-\tan \frac{\pi x}{4}$ .

Calculation: The given equation is $f(x)=-\tan \frac{\pi x}{4}$ .

Period for $\tan \theta$ is depicted by $\frac{\pi }{k}$ ,where $k$ is the absolute value of $\theta$ .

Period $\frac{\pi }{\frac{\pi }{4}}=\pi \times \frac{4}{\pi }=4$ .

An asymptote occurs when the angle $=-\frac{\pi }{2}$ .

Then $\begin{array}{l}\frac{\pi x}{4}=-\frac{\pi }{2}\\ x=-\frac{\pi }{2}\times \frac{4}{\pi }=-2\end{array}$

Another asymptote occurs when the angle $=\frac{\pi }{2}$ .

Then $\begin{array}{l}\frac{\pi x}{4}=\frac{\pi }{2}\\ x=\frac{\pi }{2}\times \frac{4}{\pi }=2\end{array}$

Calculate some points between a pair of asymptote,then use them to find other points:

  $\begin{array}{l}\mathrm{put}x=-2\\ \mathrm{then}y=-\tan \frac{-2\pi }{4}=-tan(-\frac{\pi }{2})=tan(\frac{\pi }{2})=\mathrm{not}\mathrm{defined}\\ \mathrm{put}x=-1\\ \mathrm{then}y=-tan(\frac{-1\pi }{4})=tan(\frac{\pi }{4})=1\\ \mathrm{put}x=0\\ \mathrm{then}y=-tan\frac{0\pi }{4}=-tan0=-1\times 0=0\\ \mathrm{put}x=1\\ \mathrm{then}y=-tan(\frac{1\pi }{4})=-tan(\frac{\pi }{4})=-1\\ \mathrm{put}x=2\\ \mathrm{then}y=-\tan \frac{2\pi }{4}=-tan(\frac{\pi }{2})=-tan(\frac{\pi }{2})=\mathrm{not}\mathrm{defined}\end{array}$

Now pair the respective $x\mathrm{and}y$ values to form coordinates points.

Plot the coordinates points and connect the points to draw the graph function: