Exponential functions are good for modeling population growth under ideal conditions, because then the growth is proportional to the population size. For a simple example we can consider a population of bacteria that doubles every hour. The table below shows the population size at the specified time.
As we can see, after every 1 hour that passes the population is multiplied by 2. This makes an exponential function the perfect model. Now, we know that our function is of the form shown below.
y=ab^x
We can evaluate it at two different points to find the values for a and b. For example, we can try with (0,50).
Now that we know that a=50 and b=2, we can write our function.
y = 50 (2)^x
Furthermore, we can identify that the value a=50 represents the initial population and the value b gives us an idea of how fast is the population growing, in this case b=2. It indicates that every hour the population is doubled.
