# How do you identify the period and asympotes for y=-2tan(pitheta)?

Apr 11, 2018

See below.

#### Explanation:

If we express the tangent function in the following way:

$y = a \tan \left(b x + c\right) + d$

Then:

$\setminus \setminus \setminus \boldsymbol{a} \setminus \setminus \setminus = \text{the amplitude}$

$\boldsymbol{\frac{\pi}{b}} \setminus \setminus = \text{the period}$

$\boldsymbol{\frac{- c}{b}} = \text{the phase shift}$

$\setminus \setminus \setminus \setminus \boldsymbol{d} \setminus \setminus \setminus = \text{the vertical shift}$

For given function we have:

$b = \pi$

So period is:

$\frac{\pi}{\pi} = \textcolor{b l u e}{1}$

The function will have vertical asymptotes everywhere it is undefined.

We know:

$\tan \left(\theta\right)$ is undefined at $\frac{\pi}{2}$ , $\frac{\pi}{2} + \pi$ and so on. We can write this in a general way as follows:

$\frac{\pi}{2} + n \pi$

Where $n$ in an integer.

We now solve:

$\pi \theta = \frac{\pi}{2} + n \pi$

$\theta = \frac{1}{2} + n$

So vertical asymptotes occur everywhere $\theta = \frac{1}{2} + n$

For:

$n \in \mathbb{Z}$

The graph confirms these findings: