 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

A.CED.A.2^

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ^Equations using all available types of expressions, including simple root functions

F.IF.B.4'''

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. '''Include rational, square root and cube root; emphasize selection of appropriate models.

F.IF.C.8'''

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. '''Include rational and radical; Focus on using key features to guide selection of appropriate type of model function