gives the following outputthe preceding graph shows the difference between counting six operations or seven operations we can see how the two cases divergeand this is important when we are talking about the specifics of an application howeverwhat we are more interested in here is way to characterize growth rates we are not so much concerned with the absolute valuesbut how these values change as we increase in this way we can see that the two lower curves have similar growth rateswhen compared to the top ( curve we say that these two lower curves have the same complexity class this is way to understand and describe different runtime behaviors we will formalize this performance metric in the next section asymptotic analysis there are essentially three things that characterize an algorithm' runtime performance they areworst case use an input that gives the slowest performance best case use an input that givethe best results average case assumes the input is random to calculate each of thesewe need to know the upper and lower bounds we have seen way to represent an algorithm' runtime using mathematical expressionsessentially adding and multiplying operations to use asymptotic analyseswe simply create two expressionsone each for the best and worst cases