# How do you graph y=5+tan(x+pi)?

Feb 13, 2017

Start from a "basic cycle" for the $\tan$ function to obtain the results below.

#### Explanation:

Starting with the "basic cycle" for $\tan \left(\theta\right)$ i.e. for $\theta \in \left[- \frac{\pi}{2} , + \frac{\pi}{2}\right]$

Then consider what values of $x$ would place $\left(x + \pi\right)$ in this same range, i.e. $\left(x + \pi\right) \in \left[- \frac{\pi}{2} , + \frac{\pi}{2}\right]$
(Yes; I know: because $\tan$ has a cycle length of $\pi$ we could recognize that the cycle will repeat at exactly the same place, but let's do this for the more general case.)
$\left(x + \pi\right) \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right] \textcolor{w h i t e}{\text{X"rarrcolor(white)("X}} x \in \left[- \frac{3 \pi}{2} , - \frac{\pi}{2}\right]$
Giving us the "basic cycle" for $\tan \left(x + \pi\right)$:

Adding $5$ to this to get $y = 5 + \tan \left(x + \pi\right)$ simply shifts the points upward $5$ units:

In the image below, I have added the "non-basic cycles" as well as the "basic cycle" used for analysis: