Intro to Exponential Functions

Lesson 4.1

Contrast

LinearFunctions

• Change at a constant rate• Rate of change (slope) is a constant

ExponentialFunctions

• Change at a changing rate• Change at a constant percent rate

General Formula

• All exponential functions have the general format:

• Where A = initial value B = growth factor t = number of time periods

( ) tf t A B

Contrast

• Suppose you have a choice of two different jobs at graduation Start at $30,000 with a 10% per year increase Start at $40,000 with $1000 per year raise

• Which should you choose? One is linear growth One is exponential growth

1 30,000 40,000

2 33,000 41,000

3 36,300 42,000

4 39,930 43,000

5 43,923 44,000

6 48,315 45,000

7 53,147 46,000

8 58,462 47,000

9 64,308 48,000

10 70,738 49,000

11 77,812 50,000

12 85,594 51,000

13 94,153 52,000

14 103,568 53,000

Which Job?

• How do we get each nextvalue for Option A?

• When is Option A better?• When is Option B better?

• Rate of increase a constant $1000

• Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.10

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

Compounded Interest

• Completed table

At end of year

Amount of interest earned

New balance in account

0 0 $100.001 $5.00 $105.002 $5.25 $110.253 $5.51 $115.764 $5.79 $121.555 $6.08 $127.636 $6.38 $134.017 $6.70 $140.718 $7.04 $147.759 $7.39 $155.1310 $7.76 $162.89

( ) tf t A B

Compounded Interest

• Table of results from calculator Set y= screen

y1(x)=100*1.05^x Choose Table (Diamond Y/F5)

• Graph of results

Exponential Modeling

• Population growth often modeled by exponential function

• Half life of radioactive materials modeled by exponential function

Growth Factor

• Recall formulanew balance = old balance + 0.05 * old balance

• Another way of writing the formulanew balance = 1.05 * old balance

• Why equivalent?

• Growth factor: 1 + interest rate as a fraction

Assignment

• Lesson 3.1A• Page 112• Exercises

1 – 23 odd

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 hourAt 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

tableWhat is the

growth factor?

What is the growth factor?

Decreasing Exponentials

• Completed chart

• Graph

At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06

At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 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

Mg

rem

ain

ing

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

Mg

rem

ain

ing

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

Solving Exponential Equations Graphically

• For our medication example when does the amount of medication amount to less than 5 mg

• Graph the functionfor 0 < t < 25

• Use the graph todetermine when

( ) 100 0.85 5.0tM t

Typical Exponential Graphs

• When B > 1

• When B < 1

( ) tf t A B

View results of B>1, B<1 with

Excel

View results of B>1, B<1 with

Excel

Assignment

• Lesson 4.1• Pg 136• Exercises 1 – 53 EOO