Define LTI system from a single input output pair
To find the response of a system, you need to know the input, the output 

The equation is this (where y(s) is the output):

 

So if you know the input and output functions, you can take the laplace tranform and find the transfer function.

Shifting simply adds a delay, and won't change the transfer function, as the delay will happen in the output and will be canceled out. 

\$\mathcal{L}U(t) f(t-a)=e^{-as}F(s) \$

Scaling will scale, and will also be canceled out, as multiplying the input by c will also multiply the output. 

To get the response of the system, you need something that operates on all frequencies, like a dirac delta function, a sine sweep or a step input function. 

The method for estimating the transfer function with data is called system identification