Warm-Up A population of mice quadruples every 6 months. If a mouse nest started out with 2 mice, how many mice would there be after 2 years? Write an equation and then use it to solve the problem.

Introduction to Exponential Functions

Exponential Functions An exponential function is a function in which the independent variable appears in an exponent. An exponential function has the form 𝑓 𝑥 =𝑎 𝑏 𝑥 , where 𝑎≠0, 𝑏≠1, and 𝑏>0. a represents the initial/original/principal amount b represents the rate of increase or decrease x represents the time period

Exponential Growth and Decay Exponential growth occurs when a quantity increases by the same rate in each time period. Exponential decay occurs when a quantity decreases by the same amount in each time period.

Example A population of insects doubles every month. This particular population started out with 20 insects. Find the population after 6 months. 𝑦=20 ∙2 𝑥 𝑦=20∙ 2 6 𝑦=20∙64 𝑦=1280

Example Flourine-20 has a half-life of 11 seconds. Find the amount of Flourine-20 left from a 40-gram sample after 44 seconds. 𝐴=40 (0.5) 4 𝐴=2.5 grams

Growth/Decay Rates Given as Percentages Note: When the rate is given as a percentage, you must convert the percent to a decimal. Add or subtract the rate from 1, depending on if it is a growth or decay. For exponential growth, use the formula 𝑦= 𝑎(1+𝑟) 𝑡 For exponential decay, use the formula 𝑦= 𝑎(1−𝑟) 𝑡

What changes about the formula when the rate is given as a percentage? Example: The original value of a painting is $1400, and the value increases by 9% each year. Write an exponential growth function to model this situation. Then find the value of the painting after 25 years. 𝑦=1400 (1+0.09) 𝑥 𝑦=1400 (1.09) 25 𝑦=1400 8.62 𝑦=$12,072.31

Exponential Growth/Decay and Money Money is not free to borrow  Interest is how much is paid for the use of money.

So how much does it cost to borrow money?? Different places charge different amounts at different times. In general, interest is charged as a percent of the amount borrowed.

Example Let’s say that Alex wants to borrow $1000 and the local bank offers the loan with 10% interest. To borrow the $1000 for 1 year will cost 1000+1000 0.10 So Alex borrows $1000, but must pay back $1100.

There are generally special words used when borrowing money, as shown below… Alex is the Borrower. The bank is the Lender. The Principal of the loan is $1000. The Interest is $100.

More Than One Year… What if Alex wanted to borrow the money for 2 years? If the bank charges “Simple Interest” then Alex just pays another 10% for the extra year. So after 2 years, Alex will pay $1200

But what if the bank says “If you paid me everything back after one year, and then I loaned it to you again… I would be loaning your $1100 for the second year!” Then Alex would pay $110 interest in the second year, not jus $100. Such a way of calculating interest is called compounding.

Compound Interest Compound interest is the interest earned or paid on both the principal and previously earned interest.

How much would Alex owe on a 5 year loan with 10% interest compounded annually?? Loan at Start Interest Loan at End 0 (Birth of the loan) $1000 $100 $110 1 $1100

Formula for Compound Interest The formula for compound interest is as follows: Growth: A= 𝑃(1+ 𝑟 𝑛 ) 𝑛𝑡 A is the balance after t years P is the principal/original amount r is the rate (given as a decimal) n is the number of times interest is compounded per year t is the time in years

Periodic Compounding It is also possible to have yearly interest with several compounding’s within the year. For example, 6% interest with monthly compounding does not mean 6% per month, it means 0.5% per month (6% divided by 12 months) For a $1000 loan, the final amount owed could be worked out as follows: 1000 (1+ 0.06 12 ) 12 =$1,061.68

Example $1000 is invested at a rate of 3% compounded quarterly for 5 years 𝐴=1000 (1+ .03 4 ) 4(5) 𝐴=1000 (1.0075) 20 𝐴≈1161.18

Example $18,000 is invested at a rate of 4.5% compounded annually for 6 years. 𝐴=18,000( 1+ 0.045 1 ) 1∙6 𝐴=18,000( 1.045) 6 𝐴≈$23,440.68