8.5 Writing and Graphing Exponential Functions

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1 8.5 Writing and Graphing Eponential Functions Consider the table of value and accompanying graph below: 1) y This function is There is a constant difference between y values (slope) Equation: 2) y This function is There is a constant ratio between y values Equation:

2 Eponential Function a function that grows (or decays) very rapidly y = a(b) initial (starting) amount (y intercept) growth (or decay) factor Are the following eponential functions? 1) y = 2 2) y = 2 3) y = 5(10) 4) y = 4(.3) ) y = ) y

3 Let's learn how to write a linear and eponential function rule from a table of values: y Linear Function: y = m + b y Eponential Function: y = a(b) initial (starting) amount (y intercept) growth (or decay) factor ,192 32,768

4 Eponential Functions (continued) Recall: y = a(b) Eponent (always "") Initial amount (y intercept) Growth/Decay Factor Write the rule for the following functions: 1) 2) 3)

5 Graphing Eponential Functions Graph: y = (2) y Graph: Graph: y = (3) y y = (4) y Graph: y = (.5) y Graph: y = (.25) y Graph: y = (.1) y

6 Eponential Growth and Decay Eponential Growth You have $100 to deposit in the bank. The bank will pay you 5% interest at the end of the year. How much money will you have?

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8 Eponential Decay Eample Starting in the year 2000, City ABC's population of 90,000 decreased by 2.5% every year. Use an eponential decay model to find City ABC's current population in 2013.

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14 Applications of Eponential Functions: Compound Interest It is said that the two best friends of any investor are compound interest and time. In fact, Albert Einstein called compound interest the greatest mathematical discovery of all time. Compound interest simply means that you are earning interest (payment from a bank) not only on your principal (how much you invest out of pocket), but also on previously accrued interest. EXAMPLE: You invest $100 and earn $6 interest. You now have $106. Any new interest you earn will now be based on your full $106 instead of just the original $100. Obviously, the more money you have to invest and the longer you have to invest it, the more eponentially powerful the effect of compound interest. Let's investigate compound interest: Scenario 1: You deposit $100 in Bank A and they offer to pay you 12% interest at the end of the year. How much money will you have at year's end? Scenario 2: You deposit $100 at Bank B and they offer to pay you 6% interest every si months (for a total of 12% for the entire year). How much money will you have at year's end? Scenario 3: You deposit $100 at Bank C and they offer to pay you 3% interest every three months (for a total of 12% for the entire year). How much money will you have at year's end? Which option is best? Conclusion: The more times interest is paid (or "compounded"), the more money you will earn. That's the magic of compound interest.

15 Compound Interest Practice Problems COMPOUND INTEREST FORMULA r n y = a(1 + ) nt y = ending amount (also called the "balance") a = initial amount (also called the "principal") r = interest rate (written as a decimal) n = number of times you are compounding per year t = number of years invested Eample Problems 1) You deposit $1,000 in an account that pays 6% annual interest. Find the balance in the account after 8 years if the interest is compounded monthly. 2) You invest $1,245 with a bank that pays 2.25% annual interest, compounded quarterly. Find the balance in the account after 20 years. 3) You invest $500 in an account that pays 3.5% annual interest, compounded bi monthly. Find the balance in the account after 12 years. 4) Nine years ago, you invested some money with a bank that offered 7.5% annual interest, compounded daily. If your ending balance today is $6,459, how much money did you initially invest? 5) You want to make an investment with a bank that pays 11% annual interest, compounded monthly. If your goal is to have $18,540 at the end of five years, how much money should you invest now? *6) You invested $1,950 in an account that paid a great interest rate, compounded quarterly. If your ending balance after 3 years was $3,540, what was the interest rate on your investment?

16 QUIZ: Eponential Functions General format for an Eponential Function: y = a(b) Eponential Growth y = a(1 + r) t starting point (y intercept) Growth Factor Eponential Decay y = a(1 r) t Compound Interest y = a(1 + ) nt n r

17 Compound Interest Worksheet Answers 1) $8, ) $74, ) $1, ) $33, ) $30, ) $35, ) $1, ) $13, ) $1, ) $156.55