What is the minimum y value on the graph of y=cosx+5

We can look at this graph y=cosx+5 as a translated graph. This is the same as the standard graph y=cosx being shifted 5 units upward.

The function y=cosx has the minimum y value of - 1, meaning the smallest number the graph can have on the y axis is - 1. Adding 5 to - 1, we have 4.

Therefore, the minimum y value of y=cosx+5 is y=4.