Exponential Growth and Decay

Exponential GrowthWhen you have exponential growth, the numbers are getting large very quickly. The “b” in your exponential function is always greater than 1.The “b” is called the growth factor.

Exponential DecayWhen you have exponential decay, the y values are getting small very quickly.The “b” in your exponential function is always less than 1.The “b” is called the growth factor.

Figuring The Growth Factor With Interest

When the interest is compounded annually, that means once every year. Therefore, the “x” here will be the number of years.

Example 1A certain town has a population of 13,000 people. The town grows 10% annually. Write an equation to model the

increases in population. Use your equation to find how many

years it will take for the town’s population to double.

Example 1Remember your “a” or initial amount is 13,000.The growth factor “b” 1 + 0.10 = 1.10.“x” is the number of years because this growth happens annually.

Example 1y = 13000 • 1.1x

This is the exponential function asked for.Now, enter that function into y=.Scroll down the Y1 column until you see 26,000 or the closest you can get to that (because that is when the 13000 has doubled).Go to the X column. 8 represents 8 years.It will take 8 years for the population to double.

Example 2Suppose your grandmother deposits $1000 in an account for you the day you were born. The money is left alone. It earns 5% annually until you turn 25 years old. Write an exponential function to show the growth

in the account. y = 1000 • 1.05x

How much money in the account after 25 years? $3386.35 How long does it take for the money to double? About 15 years

Compound InterestWhen a bank pays interest on both the principal (the amount you put in the bank) and the interest (the amount the bank has already paid you for having your money in the bank), the bank is paying compound interest. An interest period is the length of time over which interest is calculated.

Compound Interest$1000 is invested at a rate of 3% compounded quarterly for 5 years. Write the function

Take the 3% and divide it by 4 (Quarterly means 4 times a year) = 0.75%

Make it a decimal and add it to one 1 + 0.0075 = 1.0075

y = 1000 • 1.0075x

Replace x with 20 (5 years, 4 times a year) $1161.18

$18,000 invested at a rate of 4.5% compounded semi annually for 6 years

Find the growth factor 4.5 divided by 2 (Semi annually means twice a

year) 4.5/2 = 2.25% Change it to a decimal and add it to one 1 + 0.0225 = 1.0225

Write the function y = 18000 • 1.0225x

Replace x with 12 (twice a year for 6 years) $23,508.90

To Find The Decay Factor With Percents

Take the percent that it is decaying.Make it a decimal.Subtract it from 1.If it is decaying 7%: Change 7 % to 0.07 Subtract it from 1 1 – 0.07 = 0.93 0.93 is the decay factor

Example 3A town with the population of 270,000 loses 2.3% of its residents every year. What will the population be after 20 years? Find the decay factor:

Change 2.3% to a decimal (0.023) Subtract it from 1 (1 – 0.023 = 0.977)

Write the function: 270,000 • 0.977x

Replace x with 20 and round to the nearest whole person. 169,533 people

ExamplesGrowth or Decay? Give the rate of growth or decay as a percent.y = 15 • 1.25x

y = 248 • 0.9x

y = 150 • 1.5x

y = 2 • 0.87x

25% Growth10% Decay50% Growth13% Decay

Examples (2)Write the growth or decay factor used to model each percent of increase or decrease in an exponential function.5% growth8% decay6.3% growth3.25% decay0.8% growth

1.050.921.0630.96751.008