Question

How do you graph `y = -2 - cos(x-pi)`?

Answer 1

This function has the same graph of `cos(x)`, but translated down by two units.

Explanation:

When you must graph a composed function, the idea is to recognize every step, and understand the way it affect the graph of a function. So, let's start from the fundamental function `cos(x)` and apply one modification at the time:

1. `cos(x) -> cos(x-pi)`. A change of this kind, `f(x)->f(x+k)` means to translate the graph of the function horizontally. If `k` is positive, we shift to the left, otherwise we shift to the right. Since in your case `k=-pi`, we shift the graph to the left. Note: for this change, you could also have used the identity `cos(x-pi)=-cos(x)`, and observe that `f(x)-> -f(x)` consists in a horizontal flip (symmetry with respect to the `x`-axis.

2. Now we have to change sign again. Since we just noted that `cos(x-pi)=-cos(x)`, then `-cos(x-pi)=-(-cos(x))=cos(x)`. So, you can rewrite your function as `cos(x)-2`, making it much easier.

3. So, the last step to consider is `cos(x)->cos(x)-2`. A change of this kind, `f(x)->f(x)+k` means to translate the graph of the function vertically. If `k` is positive, we shift upwards, otherwise we shift downwards. Since in your case `k=-2`, we shift the graph downwards.