Our general exponential growth and decay functions:

Growth:  y = a(1 + r)^t

Decay:  y = a(1 - r)^t

Remember, a is our initial Amount, r is our Rate of change, and t is for Time, Toss, or whatever parameter is changing.

Example:

nA microbiologist discovered a new strain of bacteria that can double its population every half hour.  There are 100 bacteria in the colony to begin with.


•Write a formula to model this situation
     This is exponential growth.  Our initial amount is 100 bacteria, and we know that every half an hour our population is doubling.  This means it is increasing by 100%.
               y = 100(1 + 1.0)^t

•Find the number of bacterial cells after 0.5 hour, 1 hour, and 1.5 hours.
   Our time 0 (t = 0) is before the first set of doubling occurs.  Since we're working with half hours, our t = 1 value will mean that 30 minutes have gone by.  t = 2 indicates an hour.
               y = 100(1 + 1.0)^1 = 200 bacteria
               y = 100(1 + 1.0)^2 = 400 bacteria
               y = 100(1 + 1.0)^3 = 800 bacteria

Double check - do these numbers for bacterial count make sense?

•How many bacterial cells were there after 45 minutes?
    For each t value, half an hour passes (30 minutes).  In order to find the population of bacteria at 45 minutes, we have to decide what our t value will be.  45 minutes is exactly half way between 30 and 60 minutes, or t = 1 and t = 2, so we need to evaluate the expression when t = 1.5
                   y = 100(1 + 1.0)^1.5 = 282.84 bacteria 
Since we're not counting partial bacteria cells, we would round to 282 bacteria cells.  


Don't forget the word problem page and the exponential functions page.  We'll be going over them on Tuesday.
Have an awesome 4 day weekend!

Links:
Video Example
Extra Problems