Precalculus Notes
Section 3.2
Exponential and Logistic Models
Video Notes http://tinyurl.com/3-2-Video-Notes
Exponential Change Model
If a population (or any other quantity) is changing at a constant percentage rate r each year (or other
time period), then P(t )  P0 1  r  where P0 is the initial population (or quantity), r is expressed as a
t
decimal and t is time in years.
Important to Note:
If r  0 , P(t ) is an exponential growth function and r is the percentage growth rate.
1 + r is the growth factor.
If r  0 , 1 + r < 1, P(t ) is an exponential decay function and r is the percentage decay rate.
1 + r is the decay factor.
Example 1, page 290
a) Tell whether P(t )  782, 248 1.0136t is an exponential growth or decay function and give its growth or
decay factor and its growth or decay rate.
Exponential __________ function
growth/decay factor: ___________________
Growth/decay rate: ____________________
b) Tell whether P(t )  1, 203,368  0.9858t is an exponential growth or decay function and give its growth
or decay factor and its growth or decay rate.
Exponential __________ function
Growth/decay rate: ____________________
growth/decay factor: ___________________
Example 2, p.291
What is the exponential function with an initial value of 12 and is increasing at 8% per year?
P(t )  ________________________
or
f ( x)  ______________________________
Example 3, p.291
Suppose a culture of 100 bacteria is put into a Petri dish and the culture doubles every hour. Predict
when the number of bacteria will be 350,000.
The initial value is _______________________.
Tells us we have exponential ___________________
The growth factor is _________________.
 The exponential function is P(t )  ___________ and we need to solve (graphically)
P(t )  ____________  ___________
The two graphs intersect at__________________, so there will

y

be 350,000 bacteria after ___________ hours.







 


x







Another exponential growth model example (not in the book)
Suppose you have a bacterial infection, but you only have 50,000 bacteria in your body when you come
to school at 8 a.m., so you feel fine. The bacteria have a growth factor of 1.84 (they grow by a rate of
84% each hour). When there are 4,000,000 bacteria in your body, you’ll feel awful. What time of day
can we expect this to happen?
The exponential function for this situation is B(t )  ______________________ and the function we
need to solve (again, graphically) is ____________________  _____________________ .
Dividing both sides by the initial value gives:_____________________________
Graph this function (at right) and find the intersection point.

y

The graphs intersect at ____________________ , so you’re going to


feel bad _____________ hours after 8 a.m.. That is, just after

___________ .


x











Example 4, p.292
A substance has a half-life of 20 days. How long will it take for 5 grams to decay to 1 gram?
t
1 k
Half-life problems can be modeled by the function f ( x)  a    ,
2
where a is the initial value, k is the substance’s half-life and t is the time the substance decays.
________________________ is the equation representing the situation described.
______________________can be solved graphically.
The graphs intersect at _________________ days,
y


so after ____________ days, there should only be 1 gram
x
         
remaining.

Example 8, p.294
A rumor spreads at a 1200 student school and the speed is expressed as the following equation:
S (t ) 
1200
1  39  e0.9t
How many students heard the rumor on day one, and when did student 1,000 hear the rumor.
At the end of day 0, S (0) 
1000 
1200
 _____________________ students have heard the rumor.
1  39  e0.90
1200
can be solved graphically.
1  39  e0.9t
The graphs intersect at ______________ , so 1000 students will
have heard the rumor during the ___________ day.