exponential_functions_for kids tointro.ppt

11

1. Intro to Exponential Functions

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

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
What is the
growth factor?

11. Decreasing Exponentials
•Completed chart
•Graph
At end of hourAmount 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
M
g

r
e
m
a
i
n
i
n
g
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 5.0
t
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