# How to properly scale, time-reverse and shift a signal?

I'm not sure if this one is on-topic here, but I'll give it a shot anyway.

I know that if I have some sort of signal, like for example $y(t)=x(t)$, and I want a signal with twice the frequency, I can write that as $z(t)=x(2t)$.

I also know that if I have signal $y(t)=x(t)$ and I want to shift it to right, I can write the shifted signal as $z(t)=x(t-1)$.

I also know that if I have signal $y(t)=x(t)$ and I want to time-reverse it, I can write the time-reversed signal as $z(t)=x(-t)$.

What confuses me is if I have a signal which has already been shifted, time-reversed or scaled and I need to shift it or scale it again.

For example I have a signal which is $y(t)=x(2t-1)$ and I need a signal which has been time-reversed, should that be $z(t)=x(-2t-1)$ or $z(t)=x(-2t+1)$? Same thing for scaling: If I have $y(t)=x(2t-1)$ and want to scale it by a factor of 2, should the result be $z(t)=x(4t-1)$ or $z(t)=x(4t-2)$?

Note that this is a homework question, so I'm looking for an answer which will explain the principle behind the transformations.

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It usually helps to look at what is happening graphically. I don't have the time right now to fully go into it, maybe someone else does, but if I were you I would start by drawing an arbitrary function and performing the operations you mentioned. – Kellenjb Apr 14 '11 at 20:32
@vKellenjb Good idea, I'll check out Wolfram Alpha. – AndrejaKo Apr 14 '11 at 20:37
x(−t) is a reversal of time, not an inversion of the signal. Inversion would be -x(t). – endolith Apr 14 '11 at 20:58
@endolith You're right. We use term "time inversion" for "reversal of time" and in my case, I dropped the "time" part where I mustn't have. – AndrejaKo Apr 14 '11 at 21:01
It would be cool if an inverting amplifier could predict the future though. – endolith Apr 14 '11 at 21:02