6.1 Exponential 6.1 Exponential Growth and DecayGrowth and Decay

6.1 Exponential 6.1 Exponential Growth and DecayGrowth and Decay

With ApplicationsWith Applications

Exponential Expression• An expression

where the exponent is the variable and the base is a fixed number

nb

Multiplier• The base of an

exponential expression

2n

Growth vs. Decay• When b>1, f(x) = bx represents

GROWTH

• When 0<b<1, f(x) = bx represents DECAY

Applications• Exponential Growth and Decay

can be found in many applications• Ex: population growth, stocks,

science studies, compound interest, and effective yield

Basic Growth/Decay Applications:

• When dealing with most growth and decay apps, you have an equation such as:

( )( )ny initial base

• Base is your multiplier

• Growth: multiplier = 100% + rate

• Decay: multiplier = 100% - rate

( )( )ny initial base

Growth/Decay App. WS Problem 1:

• The population of the United States was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

Growth/Decay App. WS

Problem 1:• Growth Application• Initial Population = 248,718,301• Multiplier = 100% + 8% = 108%

= 1.08• Expression to model the problem:248,718,301(1.08)n

Growth/Decay App. WS

Problem 1:• 2010: 2 decades after 1990• n = 2

2

248,718,301(1.08)

248,718,301(1.08)

290,105,026.3

nPop

Pop

Pop

Growth/Decay App. WS

Problem 1:• Round to the nearest hundred

thousand:• 290,100,000 = Population in 2010

Growth/Decay App. WS

Problem 1:• 2025: 3.5 decades after 1990• n = 3.5

3.5

248,718,301(1.08)

248,718,301(1.08)

325,604,866

nPop

Pop

Pop

Growth/Decay App. WS

Problem 1:• Round to the nearest hundred

thousand:• 325,600,000 = Population in 2025

Population Formula WS• This is for homework, but it is for

you to practice writing population formulas.

• YOU DO NOT HAVE TO SOLVE ANYTHING….JUST WRITE THE FORMULAS

Compound Interest Formula

• Another application of exponential growth

• The total amount of an investment A, earning compound interest is:

( ) (1 )ntr

A t Pn

P = Principal, r = annual interest rate, n = # of times interest is compounded per year, t = time in years

Example• Find the final amount of a $500

investment after 8 years, at 7% interest compounded annually, quarterly, monthly, daily.

( ) (1 )ntr

A t Pn

Example (cont)• P = $500• r = 7% = .07• t = 8 years

• Annually, n = 1

Example (cont):Annually

1(8)

( ) (1 )

.07(8) 500(1 )

1(8) $859.09

ntrA t P

n

A

A

Example (cont):Quarterly

• n = 4

4(8)

( ) (1 )

.07(8) 500(1 )

4(8) $871.11

ntrA t P

n

A

A

Example (cont):Monthly

• n = 12

12(8)

( ) (1 )

.07(8) 500(1 )

12(8) $873.91

ntrA t P

n

A

A

Example (cont):Daily

• n = 365

365(8)

( ) (1 )

.07(8) 500(1 )

365(8) $875.29

ntrA t P

n

A

A

Effective Yield• The annually compounded interest

rate that yields the final amount of an investment.

• Determine the effective yield by fitting an exponential regression equation to 2 points.

• Effective Yield = b - 1

Example• A collector buys a painting for

$100,000 at the beginning of 1995 and sells it for $150,000 at the beginning of 2000. Write an equation to model this situation and then find the effective yield.

Example (Cont)• When modeling the situation, you

use the compounded interest formula, and you let n = 1 for compounded annually.

• A(t) = ending value = $150,000• P = initial = $100,000• n = 1 and t = 5 years

Example (Cont)

1(5)

5

150,000 100,000(1 )1

150,000 100,000(1 )

r

r

Example (Cont)• Now we need to find the effective

yield

• First we need 2 points that would model the data:

(0, 100000) and (5, 150000)

Example (Cont)• Plug these points into your calc• STAT, EDIT

Example (Cont)• Then generate the Exponential

Regression: STAT, CALC, 0:ExpReg• y = abx

• a = 100000• b = 1.084• Effective yield = 1.084 – 1 = .084

= 8.4% annual interest rate

Homework:• Finish BOTH WS

• Pg 358 #15, 18, 21, 37, 42, 47, 48

• Pg. 367 #17-23odd, 29-33odd, 47, 49