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Warm-Up οA 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 ο An exponential function is a function in which the independent variable appears in an exponent. ο An exponential function has the form π π₯ = ππ π₯ , where π β 0, π β 1, and π > 0. οa represents the initial/original/principal amount ο b represents the rate of increase or decrease ο x represents the time period Exponential Growth and Decay ο Exponential growth occurs when a quantity increases by the same rate in each time period. ο Exponential decay occurs when a quantity decreases by the same amount in each time period. Example οA population of insects doubles every month. This particular population started out with 20 insects. Find the population after 6 months. π¦ = 20 β 2π₯ π¦ = 20 β 26 π¦ = 20 β 64 π¦ = 1280 Example ο Flourine-20 has a half-life of 11 seconds. Find the amount of Flourine-20 left from a 40-gram sample after 44 seconds. π΄ = 40(0.5)4 π΄ = 2.5 grams Growth/Decay Rates Given as Percentages ο 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. ο For exponential growth, use the formula π¦ = π(1 + π)π‘ ο For exponential decay, use the formula π¦ = π(1 β π)π‘ 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. π¦ = 1400(1 + 0.09)π₯ π¦ = 1400(1.09)25 π¦ = 1400 8.62 π¦ =$12,072.31 Exponential Growth/Decay and Money ο Money is not free to borrow ο ο Interest is how much is paid for the use of money. So how much does it cost to borrow money?? ο Different places charge different amounts at different times. ο In general, interest is charged as a percent of the amount borrowed. Example ο ο ο Letβs 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. ο There are generally special words used when borrowing money, as shown belowβ¦ ο Alex is the Borrower. The bank is the Lender. The Principal of the loan is $1000. The Interest is $100. More Than One Yearβ¦ ο What if Alex wanted to borrow the money for 2 years? ο If the bank charges βSimple Interestβ then Alex just pays another 10% for the extra year. So after 2 years, Alex will pay $1200 ο But what if the bank says βIf you paid me everything back after one year, and then I loaned it to you againβ¦ I would be loaning your $1100 for the second year!β ο Then Alex would pay $110 interest in the second year, not jus $100. ο Such a way of calculating interest is called compounding. Compound Interest ο 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?? Year Loan at Start Interest Loan at End 0 (Birth of the loan) $1000 $100 $110 1 $1100 Formula for Compound Interest ο The formula for compound interest is as follows: ο Growth: ο 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 ο ο ο ο π π A = π(1 + )ππ‘ Periodic Compounding ο ο ο It is also possible to have yearly interest with several compoundingβs 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: 0.06 12 1000(1 + ) = $1,061.68 12 Example ο $1000 is invested at a rate of 3% compounded quarterly for 5 years .03 4(5) π΄ = 1000(1 + ) 4 π΄ = 1000(1.0075)20 π΄ β 1161.18 Example ο $18,000 is invested at a rate of 4.5% compounded annually for 6 years. 0.045 1β6 π΄ = 18,000(1 + ) 1 π΄ = 18,000(1.045)6 π΄ β $23,440.68