6.02 Applications of exponential functions

Lesson

Exponential growth and decay

An exponential function is the appropriate model to use when a quantity is increasing or decreasing at a rate that depends on the quantity present.

For example, in the final rounds of a sports competition, the number of competing teams is halved at every stage. Thus, if $16$16 teams reached the first semifinal, there would be $8$8 in the second semifinal, and so on. The number of teams playing drops from $16$16 to $8$8 and then to $4$4 and finally, $2$2 and we see that the reduction in the number of teams playing at each stage depends on the number in the previous round.

This is an example of exponential decay. The rate of decrease gets progressively smaller. Many processes show the opposite pattern and exhibit exponential growth. In this, the rate of increase increases progressively.

Remember!

If we are looking at an exponential function of the form $y=ab^x$y=abx, then

• $b$b is tells us whether the function is growing (increasing) or decaying (decreasing)
• If $b>1$b>1, it is growth

Outcomes

A2.IF.A.1

Identify and interpret key characteristics of functions represented graphically, with tables and with algebraic symbolism to solve problems.

A2.IF.A.2

Translate between equivalent forms of functions.