2. 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
View differences
using spreadsheet
View differences
using spreadsheet
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?
• 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
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
5. 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
3 110.25 * 0.05 = $5.51 $115.76
4
5
View completed table
6. Compounded Interest
• Completed table
At end of
year
Amount of
interest earned
New balance in
account
0 0 $100.00
1 $5.00 $105.00
2 $5.25 $110.25
3 $5.51 $115.76
4 $5.79 $121.55
5 $6.08 $127.63
6 $6.38 $134.01
7 $6.70 $140.71
8 $7.04 $147.75
9 $7.39 $155.13
10 $7.76 $162.89
7. Compounded Interest
• Table of results from
calculator
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 + 0.05 * old balance
• Another way of writing the formula
new balance = 1.05 * old balance
• Why equivalent?
• Growth factor: 1 + 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
Fill in the
rest of the
table
What is the
growth factor?
What is the
growth factor?
11. Decreasing Exponentials
• Completed chart
• Graph
At end of hour Amount Remaining
1 85.00
2 72.25
3 61.41
4 52.20
5 44.37
6 37.71
7 32.06
Amount Remaining
0.00
20.00
40.00
60.00
80.00
100.00
0 1 2 3 4 5 6 7 8
At End of Hour
Mgremaining
Growth Factor = 0.85
Note: when growth factor < 1,
exponential is a decreasing
function
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
( ) 100 0.85 5.0t
M t = × <
13. General Formula
• All exponential functions have the general
format:
• Where
A = initial value
B = growth factor
t = number of time periods
( ) t
f t A B= ×
14. Typical Exponential Graphs
• When B > 1
• When B < 1
( ) t
f t A B= ×
View results of
B>1, B<1 with
spreadsheet
View results of
B>1, B<1 with
spreadsheet
