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

1 Answer
Oct 7, 2015

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.