Exponential Functions

Example:

In 2000 there were 100 cases of COVID-19 in NY. The population of this disease has grown 50% each year since then. How many people are now infected (in 2020)?

a=100, r=0.5, t=20 (years). A=100(1+0.5)^20. Growth factor(1+/-r)=1.5. There are now 332,525.673 cases in NY.

This is an example of an exponential function. There are two equations that can be used to represent exponential growth or decay.   This equation is primarily used in the banking industry but it is A=a(1+(OR)-r)^t, where A=the final amount, a=initial amount, r=rate, t=time.

The domain of exponential functions is all real numbers. The range is all real numbers greater than zero. The line y = 0 is a horizontal asymptote for all exponential functions. When a > 1: as x increases, the exponential function increases, and as x decreases, the function decreases.

Interesting Facts About Exponential Functions:

There are several real-world problems that you may encounter on a daily basis involving exponential functions. For example, Archaeologists use radio carbon dating to determine how long ago something died. It is also used in ponzi (pyramid) schemes with the person at the top of the pyramid getting the most money from the people on the bottom. Exponential functions, also can be used to determine how much a population has grown or declined over a certain number of years. Most commonly, exponential functions are used in bank accounts to determine how much money one will have after x amount of years (simple interest NOT compounded).

(FORWARD TO 4:45) In the video above, the equation Y=a^n is being used. Where a=10 and is the y-intercept or intial amount and n is the growth/decay factor, which represents the number of weeks in this situation.