Period, Cycle, and Frequency

A function is periodic if there is some fixed number p so that f(x + p) = f(x) for every x in the domain. That p is the period: the length of one complete cycle, after which the graph's shape copies itself exactly, forever in both directions. This is a strong promise. If you know what happens on one interval of length p, you know what happens everywhere, because the function is just that same piece pasted over and over. That is why one period carries all the information the function will ever show you.

To find a period from a graph, hunt for the shortest stretch after which the curve starts tracing an identical copy of what it just did: same heights in the same order, same shape, not just a similar wiggle. Mark a clear feature, like a peak or a place where the graph crosses zero going upward, then find the next x-value where that exact feature happens again. The gap between them is the period. Frequency flips this: it is the number of cycles per unit of time, so frequency = 1/period. A short period means fast repetition and high frequency; a long period means slow repetition and low frequency.

This idea sets up the rest of the unit. Sine and cosine are the standard examples of periodic functions, with period 2 π in their basic form, and every transformation you apply next, stretching, shrinking, shifting, reflecting, changes that period in a predictable way. Before touching any formula for a transformed sine or cosine, you need this plain definition of period and frequency so you can check your transformed graph actually repeats the way the formula claims.