PreCalculus Unit 3 Day 6
Exponential Growth and Decay Modeling
Big Ideas: Observed data can be analyzed for patterns that serve as the basis for equations from which
predictions are made.
A system of equations can be used to solve for multiple variables.
Functions can be used to model real life phenomena.
I. Writing Exponential Functions Given Two Data Points
Ex: The number of Apps available at an online store is growing exponentially with time. In 2010, there
were 100,000 apps available, and in 2012, there were 1,250,000.
a. Write the exponential function that gives the number of apps (in 100’s of thousands) as a function
of the year.
b. What is the annual growth rate, based on the data above?
* The problem above is made easier if one of the points is considered a “starting point”, with the
input set to 0. (Though, the question in part a specifically forbids this.
c. Write the exponential function for which f(0) = 1 and f(2) = 12.5
II. Exponential Growth and Decay Models Using the Natural Base. (Using 2 data points and calculations)
A. The form of an exponential growth model is ___________________ with k ________.
B. The form of an exponential decal model is the same equation with k ____________.
How can we convince ourselves that the difference from A to B changes growth to decay?
Ex: The table below shows the U.S. population, in millions for 5 selected years from 1970 to 2010.
year
U.S. Population (mm)
1970
203.3
1980
226.5
1990
248.7
2000
281.4
2010
308.7
a. Use the exponential growth function and the data from the years
1970 and 2010 to write the particular function representing the data.
b. By what year will the U.S. population reach 335 million?
III. Exponential Growth and Decay Models Using the Natural Base. (Using more data points)
A. The model above might have been more precise if we used all data available to us.
B. Take notes on how to enter the full data set into your graphing calculator and how to find the
function that best models the data.
Note: y  ab x is equivalent to y  ae ln b  x
C. Correlation Coefficient - r : A number between -1 and 1 that tells us:
1. Whether y increases or decreases as x increases.
a. If _____________________, y increases as x does.
b. If _____________________, y decreases as x does.
2. How well the model fits the data.
a. _____________________________________________________________
______________________________________________________________
IV: Other Decay or Growth Models
A. Logistic Growth Model –
1. Useful when rate of growth eventually changes, and when there is a limit to
growth.
2. Uses the equation:
f (t ) 
c
1  ae bt
where:
a, b, and c are constants.
c is the _____________________.
y = c is the horizontal asymptote.
Consider why. (p.470)
Ex: The function
f (t ) 
30,000
describes the number of people infected by the flu after its initial
1  20e 1.5t
outbreak in a town of 30,000 people after t weeks.
a. How many people became ill when the outbreak began?
b. How many people were ill after 4 weeks?
c. What is the limiting size of f(t)?
B. Newton’s Law of Cooling
1. Gives the temperature T of something that has been cooling for a time of t from a starting
temperature of T0 in a surrounding temperature of C. k is a negative constant associated with
the object that is cooling.
T  C  T0  C e kt
Classwork: p. 480 #47 (Do this problem first), 1, 5, 9, 13, 19, 32, 37, 43, 45
Homework: Test Review (evens)