The amplitude of a sine or cosine function is often defined to be the maximum distance from the $x\text{-}$axis to the graph.

This definition works for $f(x)=\sin(x)$ and $f(x)=\cos(x)$ just because the distance from the $x\text{-}$axis to either the maximum or minimum value of the function is the same. However, if the function is translated up or down the amplitude doesn't change but this definition can be misleading in that case.

For this reason, and to avoid confusion, it is better to define the amplitude as half the distance from the maximum value to the minimum value of the sine or cosine function.

This way, the amplitude doesn't depend on the function being translated or not.

If the function is not sine or cosine, there are alternative defintions to talk about the amplitude. One of them, is the peak-to-peak amplitude, which is the distance from the highest value to the lowest value.