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

Oct 7, 2015

This function has the same graph of $\cos \left(x\right)$, 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 $\cos \left(x\right)$ and apply one modification at the time:

1. $\cos \left(x\right) \to \cos \left(x - \pi\right)$. A change of this kind, $f \left(x\right) \to f \left(x + k\right)$ 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 $\cos \left(x - \pi\right) = - \cos \left(x\right)$, and observe that $f \left(x\right) \to - f \left(x\right)$ 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 $\cos \left(x - \pi\right) = - \cos \left(x\right)$, then $- \cos \left(x - \pi\right) = - \left(- \cos \left(x\right)\right) = \cos \left(x\right)$. So, you can rewrite your function as $\cos \left(x\right) - 2$, making it much easier.

3. So, the last step to consider is $\cos \left(x\right) \to \cos \left(x\right) - 2$. A change of this kind, $f \left(x\right) \to f \left(x\right) + k$ 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.