Why is it the case that adding a constant to a function shifts its graph vertically, but adding a constant to a probability distribution shifts its graph horizontally?
Similarly, why is it the case that multiplying a function by a constant stretches its graph vertically, but multiplying a probability distribution by a constant stretches its graph horizontally?
I understand that shifting or stretching a probability distribution vertically would violate the condition that the area under its graph must be $1$ (PMF/PDF) or that its graph must approach $1$ (CDF), but I'm curious as to what difference in the geometric setup accounts for this differing behavior.