SUGGESTED LEARNING STRATEGIES: Shared Reading, Summarize/Paraphrase/Retell, Create Representations, Look for a Pattern, Quickwrite, Note Taking Suppose your neighbor, Margaret Anderson, has just won the state lottery and her first payment will be $50,000. Margaret is interested in options that involve spending part of her winnings and saving the balance so that she can accumulate a nest-egg at the end of the 20-year period. The tasks that follow will help you analyze Margaret s situation. 1. If Margaret saves only her first lottery payment of $50,000 and deposits it in a savings account paying 5% interest compounded annually, determine how much money she will have in her account at the end of the each year given in the table below. Year Years Since 2004 Account Balance 2004 0 $50,000 2005 1 $52,500 2006 2 $55,125 2007 2008 2009 2014 3 4 5 10 $57,881.25 $60,775.31 $63,814.07 $81,444.73 2. What patterns do you notice in the table? Answers may vary. Each year, the amount is 1.05 times the previous year. The amount of increase is not constant, but increases each year. CONNECT TO HISTORY ACTIVITY 2.6 Harry Casey was the first winner of the Pennsylvania lottery in 1972. He won $1 million, which was paid in 20 annual installments of $50,000. Harry retired immediately, spent $50,000 each year, received his last check in 1991, and was broke by the spring of 1992. Investigative Activity Focus Compound interest Exponential functions Formulas and the effects of parameters e Materials Graphing calculator Chunking the Activity #1 2 #13 15 #25 26 #3 #16 #27 #4 6 #17 #28 #7 #18 #29 31 #8 #19 22 #9 12 #23 24 First Paragraph Shared Reading, Summarize/Paraphrase/Retell Exponential functions are multiplicative. That is, when a change in the input is constant, there is a constant multiplicative change in the output. The general form of an exponential function can be expressed as f (x) = a (b) x, where a is a non-zero constant, and b is a positive constant, b 1. 3. What is the constant multiplier for the exponential function representing the data in the table? 1.05 2 Look for a Pattern, Quickwrite Note Taking 3 Look for a Pattern Unit 2 Functions and Their Graphs 117 1 Create Representations Students must first recognize that the 5% increase is compound interest, not simple interest. Some students may increase the total amount by $2500 each year; if so, a further explanation of compound interest will be necessary. Methods used to complete the table will vary and a class discussion of these methods will be helpful. Students will encounter a situation with extraneous decimals for the amount of money for the years 2007 and beyond as well as for many other questions in this activity. Banks do not round amounts to the nearest cent instead, they truncate. Discrepancies due to rounding will be minor, and these differences can lead to a meaningful discussion about how these discrepancies occur. Unit 2 Functions and Their Graphs 117
Continued 4 Quickwrite 5 Create Representations 6 Create Representations, Discussion Group If students have difficulty writing the function, lead a class discussion in formulating the amount of money after t years, 50,000(1.05 ) t. The purpose of this question is to involve students in the process of developing a generalization. Students should recognize that the amount of money for a particular year is the previous year's amount plus 5% of the previous year's amount. Symbolically, A(2) = A(1) + A(1) (0.05). The argument below may assist with your explanation. 7 Think/Pair/Share, Quickwrite Students graph A(t) = 50,000(1.05 ) t on their calculators; check that they use an appropriate viewing window. To answer this question, some may use the TRACE feature, others may use the VALUE command. Other students may evaluate the function using the TABLE feature or an equivalent process. The time to double the initial amount is 15 years. It is important for students to realize that, because interest is compounded annually, the number of years must be a whole number. 7a. $81,444.73 Amount Balance ($) 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of Years (t) MATH TERMS In an exponential function, the constant multiplier or scale factor is known as an exponential growth factor when the constant is greater than 1. The constant multiplier is known as an exponential decay factor when the constant is between 0 and 1. 118 SpringBoard Mathematics with Meaning TM Precalculus SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Discussion Group, Think/Pair/Share, Quickwrite, Mark Text, Summarize/Paraphrase/Retell, Debrief 4. Explain how to find the constant multiplier for a set of data. Explanations may vary. Sample answer: Divide the value of the dependent variable by the previous value of the dependent variable. 5. Complete the table below. Years Since 2004 Change in Account Value from Previous Year Annual Growth Factor 1 $2500.00 1.05 2 3 4 5 $2625.00 $2756.25 $2894.06 $3038.76 1.05 1.05 1.05 1.05 6. As described in Question 1, the amount of money in Margaret s account, her account balance A(t), is a function of the number of years t that have elapsed since 2004. Write an expression that defines A. A(t ) = 50,000 (1.05) t 7. Using a graphing calculator, graph A(t) in an appropriate viewing window and answer the following. a. How much money will be in Margaret s account after 10 years? $81,444.73 b. How long will it take Margaret to double her initial investment (that is, to have at least $100,000 in her account)? t = 14.207, so it would take 15 years. The parameters of the function A represent particular features of the situation. The $50,000 value represents the amount of money that was deposited to open Margaret s savings account. This value is known as the initial amount, or the principal P. For a 5% interest rate, the value 1 + 0.05 or 1.05 represents the amount by which the current balance is multiplied to get the following year s balance. For any interest rate r, 1 + r is the annual growth factor. 8. Using parameters P and r, define a general function A(t) where t is the number of years since the principal was deposited in Margaret s savings account. A(t ) = P (1 + r ) t 6. (Continued) Year 1: 50,000 + 50,000(0.05) = 50,000(1 + 0.05) = 50,000(1.05) Year 2: 50,000(1.05) + 50,000(1.05)(0.05) = 50,000(1.05)(1 + 0.05) = 50,000(1.05 ) 2 Year 3: 50,000(1.05 ) 2 + 50,000(1.05 ) 2 (0.05) = 50,000(1.05 ) 2 (1 + 0.05) = 50,000(1.05 ) 3. Therefore, for year t, A(t)=50,000(1.05 ) t. 118 SpringBoard Mathematics with Meaning Precalculus
SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Quickwrite, Group Presentation 9. Write a function for Margaret s account balance at the same annual interest rate of 5%, but with a principal of $30,000. A(t ) = 30,000(1.05) t 10. How long would it take for Margaret to double her investment if she deposits $30,000 instead of $50,000? t = 14.207, so it would take 15 years. Continued Principal, Interest, Rate, Growth Factor Mark Text, Summarize/ Paraphrase/Retell 8 Create Representations, Debriefing Suggested Assignment p. 124, #1 11. From the results of Questions 7b and 10, and any other principal amounts you choose to investigate, what conclusion can you make regarding the doubling time for any principal amount P at an annual interest rate of 5%? It would take 15 years for any principal amount to double at 5% annual interest. t 12. Use the function A(t) = P(1.05) to explain why your conclusion in Question 11 must be true. For the principal to double, A(t ) must equal 2P. Substituting, 2P = P (1.05) t Dividing, 2 = 1.05 t, so the value of P does not affect the doubling time. 1.05 14 = 1.980 and 1.05 15 = 2.079, so the answer is reasonable. 13. Write a function for Margaret s account balance at the annual interest rate of 4% with a principal of $50,000. A(t ) = 50,000 (1.04) t 14. How long would it take to double Margaret s initial investment of $50,000 if the annual interest rate is 4%? t = 17.672, or about 18 years 15. Suppose Margaret invests her money in an account that offers a 5% annual interest rate compounded annually. Find the amount of money Margaret would have in her account after 20 years if she makes the following initial investments. a. $10,000 b. $25,000 c. $50,000 A(t ) = 10,000 (1.05) 20 A(t ) = 25,000 (1.05) 20 A(t ) = 50,000 (1.05) 20 A(t ) = $26,532.98 A(t ) = $66,332.44 A(t) = $132,664.89 MINI-LESSON: Rule of 70 Unit 2 Functions and Their Graphs 119 If an amount of money is growing at r % per year (or month), then the doubling time is approximately 70 r years (or months). For example, if the interest rate is 7% per year, the doubling time is 70 7 = 10 years. While the Rule of 70 is used for interest rates of less than 10%, the Rule of 72 is used for interest rates higher than 10%. The rules are derived by solving the following equation for t, as shown below. 2P = P(1 + r ) t, t = ln 2 Since ln 2 is approximately 0.693, and ln ln (1 + r) (1 + r) is approximately r for small values of r, the quotient approximates 70 r. UNIT 2 PRACTICE p. 153, #40 9 Create Representations a Look For a Pattern, Quickwrite Individually, then through small groups and a whole class discussion, students should realize that the doubling time is 15 years, regardless of the initial principal. Students may be surprised at this fact and several examples will alleviate any skepticism they may express. b Quickwrite, Group Presentations When the initial principal P is doubled, its value is 2P. Therefore, the equation 2P = P(1.05 ) t can be solved to determine the doubling time. Dividing both sides of the equation by P yields the equation 2 = (1.05) t. This equation is independent of P and can be solved for t. At this point, it would be valuable to have students explore the expression (1.05) t on their calculators. They will discover that (1.05) t = 2 when t 14.207, but since the determination is based upon whole-number years, the doubling time is 15 years. You may choose to discuss the Rule of 70 and the Rule of 72. These rules are used by financial analysts to approximate the amount of time it will take to double an amount of money for a given interest rate. Unit 2 Functions and Their Graphs 119
Continued d Students should see that by lowering the interest rate the doubling time increases. f Think/Pair/Share Students will look at different investments and analyze the situation to determine which parameter interest or initial investment has a greater impact on earnings g Create Representations, Quickwrite h Group Presentation, Debriefing Based on their answers to the previous questions, students should recognize that the interest rate, not the initial deposit, determines the time needed for the investment to double. Suggested Assignment p. 124, #2 5 UNIT 2 PRACTICE p. 153, #41 44 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Quickwrite, Group Presentation, Debrief 16. Suppose Margaret invests her money in an account that offers a 4.5% annual interest rate. Find the amount of money Margaret would have in her account after 20 years if she makes the following initial investments. a. $10,000 b. $25,000 c. $50,000 $24,117.14 $60,292.85 $120,585.70 17. Consider two investments made at the same time. In the first investment, $50,000 is deposited in an account that offers an annual interest rate of 5% compounded annually. In the second investment, $30,000 is deposited in an account that offers an annual interest rate of 8.5% compounded annually. a. Use a graphing calculator. Graph the balance in each account for the first 20 years of the investments. Write a function for each investment and state an appropriate viewing window on which to graph the functions. A 1 (t ) = 50,000 (1.05) t A 2 (t) = 30,000( 1.085) t Possible window [0, 20] by [30,000, 150,000] b. Over the first 20 years, for which years is the amount of money greater in the account that began with an investment of $50,000? For which years is the amount of money greater in the account that began with an investment of $30,000? A 1 (t) is greater for the first 15 years. A 2 (t ) is greater for years 16 20. 18. Over a long period of time, which parameter, principal or interest, has a greater effect on the amount of money in an account that has interest compounded yearly? Explain your reasoning. Explanations may vary. When money is left to accumulate for a long period, the interest rate has a greater effect. A smaller principal at a higher interest rate will exceed a larger principal at a lesser interest rate due to the compounding of interest. 120 SpringBoard Mathematics with Meaning TM Precalculus 120 SpringBoard Mathematics with Meaning Precalculus
Continued First Paragraph Shared Reading, Summarize/Paraphrase/Retell j Create Representations Students determine the amounts of money in an account that has interest compounded quarterly. Some students may need additional assistance in understanding that three months represents one compounding period, six months represents two compounding periods, and so on. In order to complete the table for t years, students will need to recognize that not only has the base of the exponential expression changed, but also that the exponent has changed. Watch for students who use t as the exponent in the last row of the table. Remind them that t years is 4 times the number of compounding periods. l Debriefing SUGGESTED LEARNING STRATEGIES: Shared Reading, Summarize/ Paraphrase/Retell, Create Representations, Debrief Most savings institutions offer compounding intervals other than annual compounding. For example, a bank that offers quarterly compounding computes interest on an account every quarter; that is, every three months. Instead of computing the interest once each year, interest is computed four times each year. If a bank advertises that it is offering 8% annual interest compounded quarterly, 8% is not the growth factor. Instead, the bank will use 8% = 2% to determine the quarterly growth 4 factor. In this example, 8% is the nominal interest rate, and 2% is the quarterly interest rate. 19. What is the quarterly interest rate for an account with a nominal rate of 5%, compounded quarterly? 5% 4 = 1.25% 20. Suppose that Margaret invested $50,000 in the account described in Question 19. a. In the table below, determine Margaret s account balance after the specified times since her initial investment. Time Since Initial Investment 3 months 6 months 9 months 1 year 4 years t years Account Balance $50,625.00 $51,257.81 $51,898.53 $52,547.26 $60,994.47 $50,000 (1.0125) 4t b. Write the amount A in the account as a function of t, the number of years since the investment was made. A(t ) = 50,000 ( 1 + 0.05 4 ) 4t 21. How long will it take for Margaret s investment, as described in Question 20, to double? t = 13.95, which is about 14 years 22. How much less time is required for Margaret s initial amount of money to double when the rate is 5% compounded quarterly versus a rate of 5% compounded annually? It takes 1 fewer year for the amount of money to double. Unit 2 Functions and Their Graphs 121 Unit 2 Functions and Their Graphs 121
Continued m This is the first question in which students are asked to put the compound interest formula into a graphing calculator. Students must be careful to enter the function properly, particularly when writing the exponent. Students may incorrectly key in 50,000(1+0.05/4)^4 X instead of 50,000(1+0.05/4)^(4 X). This will cause the calculator to raise the expression to the fourth power and then multiply by x. When students input the functions properly, from either graphs or a table, they will see that there will be $1000 more in A 2 than in A 1, in about 12.2 years. Since the interest is compounded quarterly, 12.2 years should be rounded to 12.25 years. Students can use TRACE or TABLE to help answer this question. n Quickwrite, Look for a Pattern Students should notice that as the number of compounding periods increases, the amount of money in the account increases. However, the rate at which the amount of money increases, decreases as the compounding periods increase. It may be necessary to explore additional compounding periods for students to recognize this fact. If a student initiates a discussion, this would be an appropriate point at which to discuss limits. SUGGESTED LEARNING STRATEGIES: Quickwrite, Look for a Pattern, Create Representations 23. For the compounding periods given below, determine the amount of money that Margaret would have in an account after 20 years if the principal is $50,000 and the nominal rate is 5%. a. Yearly b. Quarterly c. Monthly $132,664.88 $135,074.24 $135,632.01 d. Daily (assume that there are 365 days in a year) 24. What is the effect of the compounding period on the amount of money after 20 years as the number of times the interest is computed each year increases? Observations may vary. As the compounding period shortens and the number of times interest is compounded each year increases, the amount of money increases. But shortening the compounding period greatly results in less of an increase in the balance. 25. Consider an initial investment of $1 and an interest rate of 100%. Find the amount of money in this account after one year with the following number of compounding periods per year. Record your answers to four decimal places in the table. Compounding Periods Per Year 1 10 100 1,000 10,000 100,000 1,000,000 Acount Balance $2.0000 $2.5937 $2.7048 $2.7169 $2.7181 $2.7183 $2.7183 $135,904.78 26. As the number of times the account in Question 25 is compounded per year increases, what appears to be happening to the amount of money in the account after one year? Observations may vary. The account balance after 1 year appears to be reaching a limit of about $2.7183 as the number of compounding periods gets very large. o Create Representations For this hypothetical situation, be certain that students understand that although money carries only two decimal places, four decimal places are now being considered. 122 SpringBoard Mathematics with Meaning TM Precalculus p Quickwrite, Look for a Pattern The numbers in the table appear to be approaching a limit, which is the approximate number 2.7183. This expression does have a limit, and the limit of this expression, as n approaches infinity, is the number e. This is an optimal opportunity to introduce the number e and discuss its history and uses. Suggested Assignment p. 124, #6 7 UNIT 2 PRACTICE p. 153, #45 46 122 SpringBoard Mathematics with Meaning Precalculus
SUGGESTED LEARNING STRATEGIES: Note Taking, Interactive Word Wall, Create Representations, RAFT The exponential function A(t) = Pe rt, where P is the principal, r is the interest rate, t is time, and e is a constant with a value of 2.718281828459, is used to calculate a quantity (most frequently money) that is compounded continuously (that is, the number of compounding periods approaches infinity). 27. Find the amount of money in an account after 20 years if the principal is $50,000 and the nominal rate is 5% compounded continuously. Compare this answer to your answers in Question 23. A = Pe rt = $135,914.09. The amount is greater than any found in Question 23, but not much greater than the amount found by compounding daily. 28. Margaret would like information on a few different investment options. She wants to invest either all or half the amount of her first $50,000 lottery check. Write a proposal to Margaret giving her advice on where to invest her money. Include an explanation of why you are making these recommendations. Include options for both a $50,000 and a $25,000 initial investment. Use the following account information to help make your recommendations. Big Bucks Bank: Serious Savings: Infinite Investments: Annual rate of 4% on amounts greater than or equal to $30,000 Annual rate of 3.7% on amounts less than $30,000 Nominal rate of 3.67% compounded weekly Nominal rate of 3.5% compounded continuously Answers may vary. Example: Margaret, if you invest $50,000 for, say, 10 years, Big Bucks is the best investment: Big Bucks Serious Savings Infinite investments 50,000 (1.04) 10 50,000 ( 1 + 0.0367 52 ) 52 10 50,000 0.035 10 $74,012.21 $72,160.55 $70,953.37 But if you decide to invest $25,000, Serious Savings is the best investment. Big Bucks Serious Savings Infinite Investments 25,000 (1.037) 10 25,000 ( 1 + 0.0367 52 ) 52 10 25,000(e) 0.035 10 $35,952.37 $36,080.28 $35,476.69 CONNECT TO STATISTICS In 1683, Jacob Bernoulli looked at the problem of continuously compounded interest and tried to find the limit of ( 1 + 1 n ) n as n. Bernoulli used the Binomial Theorem to show that this limit had to lie between 2 and 3. In 1731, Leonhard Euler first used the notation e to represent this limit; he gave an approximation of the irrational number e to 18 decimal places. The number e is believed to be the first number to be defined using a limit and has since been calculated to thousands of decimal places. This number is very important in advanced mathematics and frequently appears in statistics, science, and business formulas. Unit 2 Functions and Their Graphs 123 Continued First Paragraph Note Taking, Interactive Word Wall q Create Representations Students are asked to find the amount of money in an account after 20 years, using the formula for continuous compounding, A = P e rt. Students may need assistance finding the constant e on their calculators. Some students may recognize that this is only about $10 more than the amount of money accrued from daily compounding, leading to interesting discussion possibilities in the classroom. r RAFT In their letter to Margaret, students should include two separate proposals: one for $50,000 and one for $25,000. For both proposals, students may choose to evaluate each investment possibility for a fixed time period, or they may choose to use effective rates to make the comparisons. The effective rates for Big Bucks Bank are the same as the annual rates; 4% on amounts greater than or equal to $30,000, and 3.7% on amounts less than $30,000. The formula E = (1+ r n) n -1 can be used to find that the effective rate for Serious Savings is 3.7368%. For Infinite Investments, the effective rate of 3.5620% can be found by solving the equation P e 0.035(1) = P(1 + r ) 1 for r. Whether students evaluate each investment possibility for a fixed time period or compare effective rates, they should conclude that for the $50,000 proposal, Big Bucks Bank is the best option. For the $25,000 proposal, Serious Savings is the best option. s This item asks students to look at exponential decay. Unit 2 Functions and Their Graphs 123
Continued t Activating Prior Knowledge This question is designed to have students work through a half-life problem. Some students may need to see examples in other contexts to remind them of their work in Algebra 2. Suggested Assignment p. 124, #8 10 UNIT 2 PRACTICE p. 153, #47 48 CONNECT TO FINANCE Depreciation is the reduction in the value of an asset due to usage, passage of time, wear and tear, technological outdating or obsolescence, depletion or other such factors. MATH TERMS The half-life of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half. SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge 29. Margaret plans to purchase a boat that will cost her $10,000. The boat continuously depreciates at an annual rate of 17%. Write an exponential function for the value. How much will the boat be worth in 15 years? y = 10000 e -0.17x ; $780.81 30. How long will it take for the boat to be worth half of its original value? 10,000 e ( 0.17x) < 5000; 4.077 years 31. Use a graphing calculator to graph a function with 17% growth and a function with 17% depreciation. Compare and contrast the graph of the exponential growth function with that of the exponential decay function. Answers may vary. The decay function is the inverse of the growth function, so it is the growth function flipped across the y -axis. 1. $40,722.36 2. $45,152.78 3. $35,422.72 4. $67,957.05 5. $42,261.47 6. y = 500(1.02 ) t 7. About 35 years 8. y = 25,000(0.88 ) t 9. Never. Explanations may vary. The function value is always greater than 0, even for very large values of t. 10. Exponential functions are multiplications by a constant factor, geometric sequences have a common ratio multiplied by each number to obtain the next number. The graph of a geometric sequence will look like that of an exponential function. Use notebook paper to write your answers. 1. Determine the balance in an account after 10 years that had an initial investment of $25,000 at 5% interest compounded annually. 2. Determine the balance in an account after 20 years that had an initial investment of $25,000 at 3% interest compounded annually. 3. Determine the balance in an account after 10 years that had an initial investment of $25,000 at 3.5% interest compounded quarterly. 4. Determine the balance in an account after 20 years that had an initial investment of $25,000 at 5% interest compounded continuously. 5. Determine the balance in an account after 15 years that had an initial investment of $25,000 at 3.5% interest compounded continuously. 124 SpringBoard Mathematics with Meaning TM Precalculus 6. The population of deer on an island is growing exponentially. The first year the population was measured there were 500 deer. Five years later there were 552. Create an exponential function that represents the number of deer on the island given the years since the initial population count. 7. How long will it take for the number of deer to double? 8. A new car was purchased for $25,000. It depreciates continuously at a rate of 12%. Create an exponential function that represents the value of the car after t years of ownership. 9. When will the car have a value of $0? Explain. 10. MATHEMATICAL How do exponential REFLECTION functions relate to geometric sequences? 124 SpringBoard Mathematics with Meaning Precalculus