Exponentials

1. Example of an Exponential Function

   A biologist grows bacteria in a culture.  If initially there were three grams of bacteria, after one day there are six grams of bacteria, and after two days, there are twelve grams, how many grams will there be at the end of the week?
   

   Solution:

   We draw a t chart
   
   

   t	P(t)
   0	3 = 3(20)
   1	6 = 3(21)
   2	12 = 3(22)

   
   We see that the general formula is

   P(t) = 3(2^t)

   Hence after one week, we calculate

   P(7) = 3(2⁷) = 384 grams of bacteria.

   We call P(t) and exponential function with base 2.
   
   
   

2. Graphing Exponentials 

   Below is the graph of y = 2^x.    It turns out that for any b > 1 the graph of y = b^t looks similar.

   

    
   Notice that 
   

   1.  the left horizontal asymptote at 0
      

   2. The y-intercept is 1 
      

   3. The graph is always increasing.  
      

   Shifting techniques can also be used to graph variations of this curve.
   

   Example 

   Graph y = 2^-x 

   Solution:  

   We see that the graph is reflected about the y-axis:

   
   

    

3. Three Properties of Exponents  
   
   

   1. b^x b^y = b^x+y
      

   2. b^x / b^y = b^x-y
      

   3. (b^x)^y = b^xy
      

   
   

   Definition         b-x  = 1 / bx

   
   Example

   Simplify  

           3⁴(-3)^-1/[(3²)³]

   
   Solution

         3⁴(-3)^-1/[(3²)³] = 3⁴(-3)^-1/ 3⁶ 
   
            = -3⁴ /(3¹3⁶)  =  -3⁴ / 3⁷
   
           =  -1/3³ = -1/27

   
   
   

4. Applications

   Money and Compound Interest

   We have the formula for compound interest

    

   A = P(1 + r/n)nt

   where A corresponds to the amount in the account after t years in a bank that gives an annual interest rate r compounded n times per year.

    

   Example
   
   Suppose we have $2,000 to put into a savings account at a 4% interest rate compounded monthly.  How much will be in the account after 2 years?

   We have 
   
           P = 2,000, r = .04, n = 12 and t = 2
   
   We want A.

   A = 2000(1 + .04/12)¹²⁽²⁾  = $2,166.29.

    

   Continuous Interest

   For continuously compounded interest, we have the formula:  
   
   

   A = Pert

   
   Inflation Example
   
   With an 8% rate of inflation in the health industry, how much will health insurance cost in 45 years if currently I pay $200 per month?
   

   Solution
   
   We have 
   
           r = .08, P = 200, and t = 45 
   
   So that

   A = 200e^(.08)(45) = $7319 per month! 

Back to the College Algebra Part II (Math 103B) Site

Back to the LTCC Math Department Page

e-mail Questions and Suggestions