Intro to Exponential Functions
Lesson 3.1
Contrast
View differences
using spreadsheet
• Change at a constant
rate
Linear
Functions • Rate of change (slope) is
a constant
• Change at a changing
Exponential rate
Functions • Change at a constant
percent rate
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
Which Job?
Year
• 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
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
Example
• Consider a savings account with
compounded yearly income
 You have $100 in the account
 You receive 5% annual interest
At end of
year
View
completed
table
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
Compounded Interest
New balance in
Amount of
At end of
account
interest earned
year
• Completed table
0
1
2
3
4
5
6
7
8
9
10
0
$5.00
$5.25
$5.51
$5.79
$6.08
$6.38
$6.70
$7.04
$7.39
$7.76
$100.00
$105.00
$110.25
$115.76
$121.55
$127.63
$134.01
$140.71
$147.75
$155.13
$162.89
Compounded Interest
• Table of results from
calculator
 Set y= screen
y1(x)=100*1.05^x
 Choose Table (Diamond Y)
• Graph of results
Exponential Modeling
• Population growth often modeled by exponential
function
• Half life of radioactive materials modeled by
exponential function
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
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
Fill in the
rest of the
table
At end of hour
Amount remaining
1
100 – 0.15 * 100 = 85
2
85 – 0.15 * 85 = 72.25
3
4
5
What is the
growth factor?
Decreasing Exponentials
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
• Completed chart
Growth Factor = 0.85
Note: when growth factor < 1,
exponential is a decreasing
function
Amount Remaining
Mg remaining
• Graph
100.00
80.00
60.00
40.00
20.00
0.00
0
1
2
3
4
5
At End of Hour
6
7
8
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
M (t )  100  0.85  5.0
t
General Formula
• All exponential functions have the general
format:
f (t )  A  B
• Where
 A = initial value
 B = growth factor
 t = number of time periods
t
Typical Exponential Graphs
• When B > 1
f (t )  A  Bt
• When B < 1
View results of
B>1, B<1 with
spreadsheet
Assignment
• Lesson 3.1A
• Page 112
• Exercises
1 – 23 odd
• Lesson 3.1B
• Pg 113
• Exercises
25 – 37 odd