I have gathered my understanding of Laplace transforms from my Diff EQ course and the following YouTube video: https://www.youtube.com/watch?v=n2y7n6jw5d0
From what I understand, the "peaks" in the Laplace transform represent the way the original function can be formed by combining exponential and sinusoidal functions.
Then it makes perfect sense to me why functions of the form $x^n$ have a single peak at $0$ (the video also mentions functions with a peak at $0$ have infinite integrals, which makes sense.)
My confusion is to how the Laplace transforms of $x^0$, $x^1$, $x^2$... differ. I ran this in MatLab but it was hard to tell what the difference between these functions were.
I understand what the Laplace transforms themselves are ($n!/s^n$) and why they are those values, but I do not understand how the Laplace transform represents these functions when they seem to be very similar functions. That is my main point of confusion.