8-8: E XPONENTIAL G ROWTH AND D ECAY Essential Question: Explain the difference between exponential growth and decay

8-8: EXPONENTIAL GROWTH AND DECAYEssential Question: Explain the difference between exponential growth and decay

8-8: EXPONENTIAL GROWTH/DECAY

Exponential growth y = a ● bx

Note that this is the same as any old exponential function, except that the base in exponential growth is always greater than 1.

Why?

Starting amount(when x = 0)

Base (greater than 1)

Exponent


Example 1: Modeling Exponential Growth Since 1995, the daily cost of patient care in

hospitals in the United States has increased about 4% per year. In 1995, such hospital costs were an average of $968 per day.

Part A: Write an equation to model the cost of hospital care since 1995. y = a ● bx

Let x = the number of years since 1995 Let y = the cost of hospital care Let a = the initial cost in 1995, $968 Let b = the growth factor

100% + 4% = 104% = 1.04 y = 968 ● 1.04x


Part B: Use your equation to estimate the approximate cost per day in 2015 y = 968 ● 1.04x

Since our function starts in 1995, 2015 is 20 years later

y = 968 ● 1.0420

y = $2121.01 The cost per day in 2015 will be about $2121.


Your Turn Suppose your school district has 4512 students

this year. The student population is growing 2.5% each year.

A) Write an equation to model the student population y = 4512 ● 1.025x

B) What will the student population be in 3 years? About 4859 students


When a bank pays interest on both the principal and the interest an account has already earned, the bank is paying compound interest. An interest period is the length of time over which interest is calculated.

Example 2: Compound Interest Suppose your parents deposited $1500 in an

account paying 3.5% interest compounded annually (once a year) when you were born. Find the account balance after 18 years.

y = a ● bx

y = 1500 ● 1.03518

y = $2786.23


Your Turn Suppose your parents invested $1500 at 4%

interest compounded annually instead. How much would the account be after 18 years? $3038.72


Exponential decay y = a ● bx

Note that this is the same as exponential growth, except the base is between 0 and 1.

Why?

Starting amount(when x = 0)

Base (between 0 and 1)

Exponent


Example 3: Modeling Exponential Decay Since 1980, the number of gallons of whole milk

each person in the US drinks each year has decreased 4.1%. In 1980, each person drank an average of 16.5 gallons of whole milk per year.

Part A: Write an equation to model the amount of milk consumed since 1980. y = a ● bx

Let x = the number of years since 1980 Let y = the number of gallons drunk Let a = the initial amount in 1980, 16.5 Let b = the decay factor

100% - 4.1% = 95.9% = 0.959 y = 16.5 ● 0.959x


Part B: Use your equation to estimate the approximate consumption of milk in 2020 y = 16.5 ● 0.959x

Since our function starts in 1980, 2020 is 40 years later

y = 16.5 ● 0.95940

y = 3.1 The average consumption in 2020 will be about

3.1 gal/person


Your Turn In 1990, the population of Washington, DC was

about 604,000 people. Since then, the population has decreased about 1.8% per year.

A) Write an equation to model the student population y = 640,000 ● 0.982x

B) What was the population in 2010? About 420,017 people


Assignment Worksheet #8-8 Problems 1 – 9, odds

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8-8: E XPONENTIAL G ROWTH AND D ECAY Essential Question: Explain the difference between exponential growth and decay