8.5 Writing and Graphing Exponential Functions Consider the table of value and accompanying graph below: 1) x ­2 ­1 0 1 2 y ­4 ­1 2 5 8 This function is _______________ There is a constant difference between y­values (slope) Equation: 2) x 0 1 2 3 4 y 1 3 9 27 81 This function is _______________ There is a constant ratio between y­values Equation: Exponential Function ­ a function that grows (or decays) very rapidly y = a(b)x initial (starting) amount growth (or decay) (y­intercept) factor Are the following exponential functions? x 1) y = 2 2) y = x2 x x 3) y = 5(10) 4) y = 4(.3) 5) y = 4x + 2 6) x 1 2 3 4 5 y 4 8 12 16 20 Let's learn how to write a linear and exponential function rule from a table of values: Linear Function: x ­3 ­2 ­1 0 1 2 3 4 y ­3 ­1 1 3 5 7 9 11 y = mx + b Exponential Function: x ­3 ­2 ­1 0 1 2 3 4 y 4 8 16 32 64 128 256 512 y = a(b)x growth (or decay) initial (starting) amount factor (y­intercept) ­3 ­2 ­1 0 1 2 3 4 2 8 32 128 512 2048 8,192 32,768 Exponential Functions (continued) x Exponent (always "x") Recall: y = a(b) Initial amount Growth/Decay Factor (y­intercept) Write the rule for the following functions: 1) 2) 3) Graphing Exponential Functions x Graph: y = (2) x y ­2 ­1 0 1 2 3 x Graph: y = (3) x y ­2 ­1 0 1 2 3 x Graph: y = (4) x y ­2 ­1 0 1 2 3 Graph: y = (.5)x x y ­2 ­1 0 1 2 3 x Graph: y = (.25) x y ­2 ­1 0 1 2 3 Graph: y = (.1)x x y ­2 ­1 0 1 2 3 Exponential Growth and Decay Exponential 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? Exponential Decay Example Starting in the year 2000, City ABC's population of 90,000 decreased by 2.5% every year. Use an exponential decay model to find City ABC's current population in 2013. WARMUP Applications of Exponential 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 out­of­pocket), 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 exponentially 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 six 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. Compound Interest ­ Practice Problems COMPOUND INTEREST FORMULA r nt y = a(1 + )n 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 Example 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 bi­monthly. 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? 8.5 ­ 8.6 QUIZ: Exponential Functions x General format for an Exponential Function: y = a(b) starting point Growth (y­intercept) Exponential Growth Factor y = a(1 + r)t Exponential Decay y = a(1 ­ r) t Compound Interest r nt y = a(1 + )n Compound Interest Worksheet ­ Answers 1) $8,973.56 6) $74,826.89 2) $1,114.91 7) $33,393.66 3) $30,525.87 8) $35,854.85 4) $1,837.56 9) $13,842.19 5) $1,446.34 10) $156.55.