Suppose you have a function Vertical translations look like : they correspond to a shifting of the range of a function, or its output. Horizontal translations look like : they correspond to a shifting of the domain of a function, or its input.
There are a few functions where you can't tell the difference between the two translations: straight lines, for example: could be viewed either way. Or, if you had a slope of then the line could have a positive vertical translation of from the line , or it could be viewed as in which case it's a horizontal translation of to the left.
I'll give you one application of translations: modeling switches in electrical circuits. The Heaviside step function is defined as
I can think of a switch turning on when I start a clock timer at as the function . But now, suppose I want to flip a switch for one second, and then switch it back off? I could model that as a horizontally translated difference of step functions: We might want to model that if we're trying to solve the differential equation governing the behavior of the circuit.