Frequency Translation by Band
What you are describing refers to Pitch Shifting.

A naive approach to pitch shifting is based on the shifting property of the Fourier Transform (and more formally here). According to the shifting property, multiplying a signal with a sinusoid in the time domain results in shifting in the frequency domain.

But, convincing pitch shifting takes a little bit more effort than that because it does not only matter to shift the frequencies of part of the spectrum to a new range, it is also necessary to preserve their relative phase for the sound to be recognisable. If you are trying to pitch shift musical instruments (or vocals), then it gets a little bit more challenging, because of the way pitch is working. In that case, you not only have to shift the components in the frequency domain but also preserve their intervals. In other words, it is not only enough to shift those formants that you show in your illustration, you also need to make sure that the new frequencies they end up into, have the same "distances" between them as the ones they are being pitch shifted from. This is not as straightforward as it...sounds.

Pitch shifting in this way implies a linear pitch scale. But the pitch scale is non-linear and going up in pitch (higher octaves) does not result in equally proportional frequencies. 

Because of these reasons, more complex pitch shifting algorithms contain a "processing stage" between analysing and resynthesizing the pitch shifted version of the signal in their input. This processing stage attempts to take into account these relationships between the components of the spectrum according to some approach. For example, directly working out these relationships to the new range as the phase vocoder does, or attempting to fit a model to the spectrum and use it to reproduce it at a different range as spectral modelling does.

Hope this helps.