Which function should be used to model data that grows quickly at first, slows down, and approaches a limit?

The logistic function is the appropriate choice for modeling data that displays rapid growth initially, followed by a slowing growth rate as it approaches a limit. This type of growth is often seen in populations or systems where there are resource constraints, causing the growth to eventually stabilize as it nears a carrying capacity.

The logistic function has a characteristic "S"-shaped curve, where the rate of growth is high when the population or quantity is small, then slows down as it gets closer to the maximum limit or carrying capacity, which reflects the environment's ability to sustain that growth. This behavior contrasts with linear functions, which grow at a constant rate, and polynomial or exponential functions, which either grow indefinitely or at an accelerating rate without leveling off. Therefore, using a logistic function effectively captures the behavior of systems where growth is initially rapid but must eventually stabilize.