1. Exponential Functions and Models
Lesson 5.3

2. Exponential Functions
Contrast
Linear Functions
Change at a constant rate
Rate of change (slope) is a constant
Exponential Functions
Change at a changing rate
Change at a constant percent rate

3. Definition An exponential function
Note the variable is in the exponent
The base is a
C is the coefficient, also considered the initial value (when x = 0)

4. Explore Exponentials
Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5
Determine the exponential function x
f(x) 1
0.75 2 3 4

5. Explore Exponentials Graph these exponentials
What do you think the coefficient C and the base a do to the appearance of the graphs?

6. Contrast Linear vs. Exponential
Suppose you have a choice of two different jobs at graduation
Start at $30,000 with a 6% per year increase
Start at $40,000 with $1200 per year raise
Which should you choose?
One is linear growth
One is exponential growth

7. Which Job? How do we get each next value for Option A?
Year
Option A
Option B 1
$30,000
$40,000 2
$31,800
$41,200 3
$33,708
$42,400 4
$35,730
$43,600 5
$37,874
$44,800 6
$40,147
$46,000 7
$42,556
$47,200 8
$45,109
$48,400 9
$47,815
$49,600 10
$50,684
$50,800 11
$53,725
$52,000 12
$56,949
$53,200 13
$60,366
$54,400 14
$63,988
$55,600
How do we get each next value for Option A?
When is Option A better?
When is Option B better?
Rate of increase a constant $1200
Rate of increase changing
Percent of increase is a constant
Ratio of successive years is 1.06

8. Amount of interest earned
Example
Consider a savings account with compounded yearly income
You have $100 in the account
You receive 5% annual interest
At end of year
Amount of interest earned
New balance in account 1
100 * 0.05 = $5.00
$105.00 2
105 * 0.05 = $5.25
$110.25 3
* 0.05 = $5.51
$115.76 4 5
View completed table

9. Compounded Interest
Completed table

10. Table of results from calculator
Compounded Interest
Table of results from calculator
Set Y= screen y1(x)=100*1.05^x
Choose Table (♦ Y)
Graph of results

11. Assignment A
Lesson 5.3A
Page 415
Exercises 1 – 57 EOO

12. Compound Interest
Consider an amount A0 of money deposited in an account
Pays annual rate of interest r percent
Compounded m times per year
Stays in the account n years
Then the resulting balance An

13. Exponential Modeling
Population growth often modeled by exponential function
Half life of radioactive materials modeled by exponential function

14. Growth Factor
Recall formula new balance = old balance * old balance
Another way of writing the formula new balance = 1.05 * old balance
Why equivalent?
Growth factor: interest rate as a fraction

15. Decreasing Exponentials
Consider a medication
Patient takes 100 mg
Once it is taken, body filters medication out over period of time
Suppose it removes 15% of what is present in the blood stream every hour
At end of hour
Amount remaining 1
100 – 0.15 * 100 = 85 2
85 – 0.15 * 85 = 72.25 3 4 5
Fill in the rest of the table
What is the growth factor?

16. Decreasing Exponentials
Completed chart
Graph
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing function

17. Solving Exponential Equations Graphically
For our medication example when does the amount of medication amount to less than 5 mg
Graph the function for 0 < t < 25
Use the graph to determine when

18. General Formula All exponential functions have the general format:
Where
A = initial value
B = growth rate
t = number of time periods

19. Typical Exponential Graphs
When B > 1
When B < 1

20. Using e As the Base We have used y = A * Bt Consider letting B = ek
Then by substitution y = A * (ek)t
Recall B = (1 + r) (the growth factor)
It turns out that

21. Continuous Growth
The constant k is called the continuous percent growth rate
For Q = a bt
k can be found by solving ek = b
Then Q = a ek*t
For positive a
if k > 0 then Q is an increasing function
if k < 0 then Q is a decreasing function

22. Continuous Growth
For Q = a ek*t Assume a > 0
k > 0
k < 0

23. Continuous Growth For the function what is the continuous growth rate?
The growth rate is the coefficient of t
Growth rate = 0.4 or 40%
Graph the function (predict what it looks like)

24. Converting Between Forms
Change to the form Q = A*Bt
We know B = ek
Change to the form Q = A*ek*t
We know k = ln B (Why?)

25. Continuous Growth Rates
May be a better mathematical model for some situations
Bacteria growth
Decrease of medicine in the bloodstream
Population growth of a large group

26. Example
A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year.
What is the formula P(t), the population in year t?
P(t) = 22000*e.071t
By what percent does the population increase each year (What is the yearly growth rate)?
Use b = ek

27. Assignment B
Lesson 5.3B
Page 417
Exercises 65 – 85 odd