How do you graph y = -tan(x-pi) -3?

Oct 21, 2015

Shift the graph of $\tan \left(x\right)$ to the right of $\setminus \pi$ units, then reflect it with respect to the $x$-axis, then shift it downwards of $3$ units.

Explanation:

Assuming you know the graph of $\tan \left(x\right)$, let's analise step by step what changes you make, and their consequences.

First step: from $\tan \left(x\right)$ to $\tan \left(x - \setminus \pi\right)$. This changes is one of the form $f \left(x\right) \to f \left(x + k\right)$. This kind of changes means a horizontal translation of $k$ units, to the left if $k$ is positive, to the right if $k$ is negative.

So, in you case, we start from the graph of $\tan \left(x\right)$, and shift it to the right of $\setminus \pi$ units.

Second step: from $\tan \left(x - \setminus \pi\right)$ to $- \tan \left(x - \pi\right)$. This is one of the most simple and intuitive changes: if you go from $f \left(x\right)$ to $- f \left(x\right)$, you simply change the sign of every single value of the function, resulting in a reflection with respect to the $x$-axis.

Third step: from $- \tan \left(x - \pi\right)$ to $- \tan \left(x - \pi\right) - 3$: this is a change of the form $f \left(x\right) \to f \left(x\right) + k$. This means that you need to add a certain number to the values of the function, resulting in a vertical shift, which will be upwards if $k$ is positive, and downwards if $k$ is negative. So, in your case, you need to shift the graph of $- \tan \left(x - \pi\right)$ down of three units.