Population growth is the increase in the number of people in a population. The population growth rate is the rate at which the number of individuals in a population increases in a given time period.

For example, suppose the population of a certain city was 2.5 million in 2000. Then, every year after that, the population has grown by 2%. This is an example of exponential growth. Notice that the rate of growth is 2% or 0.02 and it is constant. This is important since the rate of growth cannot change.

Following this pattern, suppose that x is the number of years since 2000, 2,500,000 is the starting amount and 1.02 is the rate or growth factor. Then, y = 2,500,000(1.02)^x.

Comparing this exponential function with y = ab^x, we see that a = 2,500,000, b = 1.02.

Compound Interest

When interest is added to the principal at the end of each period, it counted into principal. The total amount due at the end of the period is called the compound amount. For example, David lends L$5,000 to his friend at annual interest of 6% compounded annually. Notice that the rate of growth is 6% or 0.06 and it is constant. Let us find the exponential function.

1 year after:
5,000 + 5,000 × 0.06 = 5,000(1+ 0.06) = 5,000(1.06) = 5,000(1.06)1.

2 years after:
5,000(1.06) + 5,000(1.06) × 0.06 = 5,000(1.06) [1+ 0.06] = 5,000(1.06) (1.06)
                                                                                  = 5,000(1.06)2.

Following this pattern, suppose that x is the number of years, 5,000 is the starting amount and 1.06 is the rate or growth factor. Then, the exponential function will be y = 5,000(1.06)^x. Comparing this exponential function with y = ab^x, we can see that a = 5,000, b = 1.06.