The monotonicity of the weighted quasi-arithmetic means is discussed from the viewpoint of utility functions and weighting functions. Observing the indices given by utility functions and weighting functions, we find that these indices give similarmonotonicity for the weighted quasi-arithmetic means, and this paper investigates the reason using the direct domain translation adapted to the weighted quasi-arithmetic means. Further, general domain translations are introduced and they are discussed in relation to these indices. Some examples are given for the domain translations and as a special case the translation invariance is presented, and a lot of examples with various utility functions and weighting functions are shown, and their indices and the translated forms are derived.