Method 1

If you have basic knowledge about what each component of the exponential parent function represents, then you will better be able to formulate a function to fit a given situation. Recall that when a quantity grows by a fixed percent at regular intervals, the pattern can be represented by the growth and decay functions:

Let's quickly check to see if you understand each component represented by the exponential function. Read the situation below; on your own paper put the information given in the boxes in the correct location within the function rule provided.

In the small town of Tinyville, the population in 2000 was 8,075. A big corporation is planning to gradually move its headquarters to Tinyville, and the city council is expecting the town to grow by 4% per year.

Check Your Answer

Now try your hand at applying this first method to find a model to fit an exponential growth situation.

One of the most common examples of exponential growth deals with bacteria, which can grow at alarming rates. Laboratory technicians often prepare cultures to help identify various bacteria present in blood samples taken from patients. A culture permits the bacteria to reproduce and create large, more easily identified populations.

Suppose we start with a single bacteria cell from a patient and suppose the cells double every hour. To begin with there is one cell. After one hour, there would be two cells. After 2 hours, there would be 4 cells; after 3 hours, there would be 8 cells, etc. Since you know the initial amount and the growth rate, you can formulate an exponential function that models the situation above. Just plug the given information into the exponential growth function y = a(1 + r)^x. Then, just for fun, see if you can determine how many cells there will be after one 24-hour period! You will probably be astonished at the number of bacteria cells!

Check Your Answer

If an amount doubles, the rate of increase is 100%. So any of the following equivalent exponential functions would correctly model this situation: y = 1(1 + 1.00)^x or y = 1(2)^x or just y = 2^x

To find the amount of bacteria after 24 hours, substitute "24' for "x' in the equation y = 2^xand you get 16,777,216 cells. Isn't exponential growth fascinating?