Question

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

Answer 1

See below

Explanation:

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

We also know that `cos(pi-x) = -cos(x)`

The expression becomes `-cos(x)-1`

This means that, starting from `cos(x)`, you apply two transformations:

`cos(x) \to -cos(x) \to -cos(x)-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}