1. Intro to Exponential Functions

2. Contrast Linear Functions Change at a constant rate
View differences using spreadsheet
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. Contrast 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

4. 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

5. 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

6. Compounded Interest
Completed table

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

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

9. 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

10. 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?

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

12. 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

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

14. Typical Exponential Graphs
When B > 1
When B < 1
View results of B>1, B<1 with spreadsheet