Shifted functions in transient analysis

Hi, I'm studying transient analysis on my own and have a question about the expression or equation used when describing a shifted function. Let's say we have a simple sine wave that starts at t = 0 as shown in the first graph. We describe this as V(t) = sin(t), for instance. Now if we shift the waveform to the right so that it starts at t = 2 the expression for the function changes to V(t) = sin(t - 2) as in the second graph, and if we shift the waveform to the left so that it starts at t = -2 then the function is described as V(t) = sin(t + 2).

Why must we replace "t" with "t - 2", that is, t minus 2, if the function is to start at t equal to positive 2 and vice versa, why is the expression "t + 2", or t plus 2 used to denote that the function starts at t minus 2 seconds from the origin?

In other words the notation in the equations is counter-intuitive, yet it is correct. My question is: why is it correct, why must it be written that way?

Cifrocco,

Because, although the t (time) goes in the positive direction, it means a later time. So we have to subtract the time to make the graph look like it did back at at "time zero", if the graph is delayed. The same reasoning can be made for starting the graph earlier than time zero. Why don't you can also ask why a mirror image appears backward to the viewer?

Ratch

Thanks for your answer Ratchit, I appreciate you taking the time.