Given the periodic function $x(t) = x(t+T)$, I would like to ask the following:
The textbook says that if the above function is valid then it is also valid for any $t$, that is $t+(k-1)T$. If I could use an animation of a Cartesian axis to get a 'feeling' of it, what does it mean visually the $(k - 1)$? What is the difference if it is $k + 1$ or if it is just $k$ alone? Maybe a shift ("delay/advance") in $t$ axis? which direction? What is the meaning of $k-1$? The $k-1$ is confusing me. (Edited)
Let $T$ be the period of a function $x$· In general, the graph of $x(t+T)$ is that of $x(t)$ shifted $T$ units to the left, but since $x$ is periodic with period $T$, translation of $T$ unit to the left and to the right are the same. Since translation of $T$ unit to the left does not change the graph, such action performed $k-1$ times (gives the graph of $x(t+(k-1)T)$ and) leaves the graph unchanged.