# How do you graph  y=-1+cos(x-pi)?

May 28, 2018

See below

#### Explanation:

First of all, use the fact that the cosine is an even function, i.e. $\cos \left(x\right) = \cos \left(- x\right)$. So, $\cos \left(x - \pi\right) = \cos \left(\pi - x\right)$

We also know that $\cos \left(\pi - x\right) = - \cos \left(x\right)$

The expression becomes $- \cos \left(x\right) - 1$

This means that, starting from $\cos \left(x\right)$, you apply two transformations:

$\cos \left(x\right) \setminus \to - \cos \left(x\right) \setminus \to - \cos \left(x\right) - 1$

The first transformation is a reflection with respect to the $x$ axis, the second is a vertical translation, one unit down.

So, start from the standard cosine function, reflect it and shift it down to get the desired function:

graph{-cos(x)-1}