Foundations of Mathematics Simple Interest

Transcription

1 1.1 Simple Interest Principal, P, is the amount of money invested or loaned. Interest, I, is the money earned on an investment or paid on a loan. Maturity is the contracted end date of an investment or a loan. Simple interest is determined only on the principal of an investment and can be calculated using the formula, where I is the interest, P is the principal, r is the annual interest rate expressed as a decimal, and t is the time in years. Solve a Simple Interest Problem Example 1. Aaron is 18 years old and needs money to pay for college. When he was born, his grandparents bought him a $500 Canada Savings Bond (CSB) with a term of 10 years. They chose a CSB as an investment because they like the security of loaning money to the government. The interest earned was determined using a fixed interest rate of 6% per year on the original investment and was paid at the end of each year until Aaron s 10 th birthday. a) Determine the simple interest earned each year. Simple Interest - interest earned/paid on original investment/loan b) Determine the future value of the investment at maturity. Future Value - The amount, A, that an investment or a loan will be worth and can be determined using the formula or Interest rates are communicated as a percent for a time period. Since most often the time period is per year or per annum (abbreviated as /a), a given percent is assumed to be annual unless otherwise stated. For example, an interest rate of 4% means 4%/a or 4% interest per year. 3

2 Try. Marty invested in a $2,500 guaranteed investment certificate (GIC) at 2.5% simple interest, per annum, with a term of 10 years. a) How much interest will accumulate over the term of Marty s investment? b) What is the future value of his investment at maturity? Unless otherwise stated, an interest rate is assumed to be annual, or per annum. Even though interest rates are usually annual, interest can be paid out at different intervals, such as annually, semi-annually, monthly, weekly, and daily. Represent the Growth of a Simple Interest Investment Example 2. Sunny invests $15,000 in a savings account. She earns a simple interest rate of 8%, paid semi-annually on her investment. She intends to hold the investment for 4.5 years. Determine the future value of the investment at maturity. Try. Determine the future value of each investment if it earns simple interest. a) 1.25% interest paid quarterly for 4 years on $10,000 b) 0.5% interest paid weekly for 2 years on $25,000 4

3 Rate of Return is the ratio of money earned (or lost) on an investment relative to the original amount of money invested or loaned. Determine the Duration of a Simple Interest Investment and the Rate of Return Example 3. Pauline invested her summer earnings of $5,000 at 8% simple interest, paid annually. a) How long will it take for the future value of the investment to grow to $7800? b) What is Pauline s rate of return? Try. Principal of $1 000 is invested at 2.5% simple interest, paid annually, for 7 years. What is the rate of return? Determine the Interest Rate on a Simple Interest Investment Example 4. Danielle invested $ in a simple interest CSB that paid interest annually. If the future value of the CSB is $ at the end of the term, what interest rate does the CSB earn? Given that rate, how much would the investment be worth if she withdrew after four years instead? 5

4 1.2 Exploring Compound Interest Simple Interest is determined only on the principal of an investment. Compound Interest is earned or paid on both the principal and the accumulated interest. Compare Simple Interest with Compound Interest Example 1. Compare the future values of the two investments over 3 years, in which one pays compound interest while the other pays simple interest, assuming that the principal, interest rate, and term are the same. Sample Simple Interest Investment Sample Annual Compound Interest Investment p = $2000 r = 8% t = 3 years p = $2000 r = 8% t = 3 years Term (Year) Value at Start of Interest Rate Interest Earned Value at End of Term (Year) Value at Start of Interest Rate Interest Earned Value at End of Year ($) Year ($) ($) Year ($) Year ($) ($) Example 2. Both Edwin and Lena received a $1000 prize in a story-writing contest. Edwin bought a $1000 simple interest GIC with his prize money. It has a 5-year term and earns 3.6% paid annually. Lena bought a $1000 compound interest GIC with her prize money. It also has a 5-year term and earns 3.6% paid annually. How do the future values of Edwin s and Lena s investments compare at maturity? 6

5 1.3 Compound Interest and Future Value (Part 1) Comparison Between Simple Interest and Compound Interest Example 1. Both Eugene and Francine received a $2000 prize in a math contest. Eugene bought a $2000 simple interest GIC (guaranteed investment certificate) with his prize money. It has a 5-year term and earns 2.6% paid annually. Francine bought a $2000 compound interest GIC (guaranteed investment certificate) with his prize money. It has a 5-year term and earns 2.6% paid annually. Compare the future values of Eugene s and Francine s investments at maturity. Simple Interest Compound Interest Year Value The future value of an investment that earns compound interest can be determined using the compound interest formula A = P (1 + i ) n, where A is the future value, P is the principal, i is the interest rate per compounding period (expressed as a decimal), and n is the number of compounding periods. Try. Ron purchased a 10-year GIC for $3,000. The GIC earns 5.6% interest, compounded annually. What will be the future value of the GIC at maturity? 7

6 Four common compounding frequencies are given in the table on the right. The table shows how the interest rate per compounding period (i ) and the number of compounding periods (n) are determined. Compounding Frequency Times per Year Interest Rate per Compounding Period (i ) Number of Compounding Periods (n) annually 1 annual interest rate number of years semi-annually 2 quarterly 4 monthly 12 annual interest rate 2 annual interest rate 4 annual interest rate 12 number of years 2 number of years 4 number of years 12 Determine the Future Value of an Investment with Semi-annual Compounding Example 2. Max has invested a $23,000 in an account that earns 13.6%, compounded semi-annually. The interest rate is fixed for 10 years. What is the future value of the investment after 10 years? Try. Determine the future value and the total interest earned for $1,400 invested for 15 years at 8.6% compounded semi-annually. Determine the Future Value of an Investment with Quarter-annual Compounding Example 3. Determine the future value and the total interest earned for $2,300 invested for 6 years at 7.5% compounded quarter-annually. Try. Determine the future value and the total interest earned for $520 invested for 8 years at 4.5% compounded quarter-annually. 8

7 The compound interest earned (I ) on an investment at the end of any compounding period is the difference between the value of the investment at that time (A) and the original principal (P): Calculate Interest Earned Example 4. Determine the total interest earned if $6,500 is invested in an account paying 6% compounded monthly for five years. Try. Determine the total interest earned if $520 is invested in an account paying 4.5% compounded monthly for 8 years. The more frequent the compounding and the longer the term, the greater the impact of the compounding on the principal and the greater the future value will be. Compare Interest on Investments with Different Compounding Periods Example 5. Hanna wants to invest $3000 so that she can renovate her living room in about 3 years; she has the following investment options (semi-annual/ monthly/ weekly/ daily) at 4.8%: Principal (P ) Interest Rate (r ) Period/Year $3000 semi-annual $3000 monthly $3000 weekly $3000 daily Calculation Accumulated (A) 9

8 1.3 Compound Interest and Future Value (Part 2) Use TVM Solver on TI-83 Graphing Calculator Step 1. Use TVM Solver Press APPS Select #1 Finance Select #1 TVM Solver Step 2. Input all givens N = I% = PV = PMT = FV = P/Y = C/Y= PMT: END BEGIN Step 3. Determine the unknown Go to the unknown Press ALPHA Press Enter For Single-Payment Investment N = # of Years I% = Interest Rate PV = Present Value (Use negative value) PMT = Monthly Payment (Use zero) FV = Future Value P/Y = # of Payments Per Year (Use 1) C/Y = # of Compounding Periods Per Year PMT: END BEGIN = Payment at the end/beginning Use TI-83 to Determine Future Value Example 1: Jennifer invested $4,300 in a 10-year Canada Savings Bond (CSB) that will earn 3.8% compounded annually. What is the future value of Jennifer s investment after 10 years? 10

9 Example 2. Max has invested a $23,000 in an account that earns 13.6%, compounded semi-annually. The interest rate is fixed for 10 years. What is the future value of the investment after 10 years? Example 3. Determine the future value and the total interest earned for $2,300 invested for 6 years at 7.5% compounded quarter-annually. Try. Determine the future value and the total interest earned if $520 is invested in an account paying 4.5% compounded monthly for 8 years. 11

10 A simple formula for estimating the doubling time of an investment; 72 is divided by the annual interest rate as a percent to estimate the doubling time of an investment in years. The Rule of 72 is most accurate when the interest is compounded annually. Estimate Doubling Time for Investments Example 4: Chris invests $5000 by purchasing Canada Savings Bonds, which earns 9%, compounded semi-annually. Estimate and determine the doubling time. Try. Use the Rule of 72 to estimate the doubling time and then determine the doubling time. 12

11 1.4 Compound Interest Present Value The present value of an investment that earns compound interest can be determined using TI-83 graphing calculator or the formula 1 Determine the Present Value of Investments Earning Compound Interest Example 1: Ginny is 18 years old. She has inherited some money from her parents. Ginny wants to invest some of the money so that she can buy a home, when she turns 30. She estimates that she will need about $170,000 to buy a home. a) How much does she have to invest now, at 6.5% compounded annually? b) What is the ratio of future value to present value? (NOTE: NOT the rate of return) c) How would the ratio change if the interest rate decreased to 6% but was compounded semi-annually? 13

12 To compare investments, usually with the same term or principal, the ratio of the future value to the present value can also be determined using the form: 1 Determine the Present Value of Investments that is Compounded Quarterly Example 2: Lana and Matt are computer scientists. They researched the costs to set up a software company. They estimate that $ will be enough. They plan to set up the company in 3 years and have invested money at 9.6%, compounded quarterly, to save for it. a) How much money should they have invested? b) How much interest will they earn over the term of their investment? Try. Mike invested in a 10-year GIC that matures this year. Mike s investment is currently worth $13,009 and has been earning 9.6%, compounded monthly. How much money should he have invested? 14

13 Example 3: Determine an Unknown Interest Rate and Unknown Term Laura has invested $15,500 in a Registered Education Savings Plan (RESP). She wants her investment to grow to at least $50,000 by the time her newborn enters university, in 18 years. a) What interest rate, compounded annually, will result in a future value of $50,000? Round your answer to two decimal places. b) Suppose that Laura wants her $15,500 to grow to at least $ at the interest rate from part a). How long will this take? Try. An investment of $400 grew to $625 in 10 years. What was the annual interest rate if the interest was compounded monthly? Try. An investment of $250 grew to $1000 at 6% interest, compounded semi-annually. Estimate how long it took for the investment to grow, and then verify your estimate. 15

14 1.5 Investments Involving Regular Payments Regular-Payment VS Single Payment Some investments, such as CSBs, lock in money for specified periods of time, thus limiting access to the money, but offer higher interest rates. Other investments, such as savings accounts, are accessible at any time but offer lower interest rates. For Regular-Payment Investment N = Number of Payments I% = Interest Rate PV = Present Value (Use zero) PMT = Monthly Payment (Use negative value) FV = Future Value P/Y = Number of Payments Per Year C/Y = Number of Compounding Periods Per Year PMT: END BEGIN = Payment at the end / beginning Determine the Future Value of an Investment Involving Regular Deposits Example 1: Dora deposits $500 into her savings account at the end of every 6 months. The account earns 3.8%, compounded semi-annually. a) How much money will be in the account at the end of 5 years? b) How much of this money will be earned interest? Try. If Dora s deposits were only $400 every 6 months instead of $500 every 6 months, and that the interest rate on her account remains 3.8%, compounded semi-annually. a) At the end of 5 years, how much less would the future value of the account be? b) How much interest would Dora earn? 16

15 Determine the Interest Rate of a Regular Payment Investment Example 2: Jeremiah deposits $750 into an investment account at the end of every 3 months (regular deposits). Interest is compounded quarterly, the term is 3 years, and the future value is $ What annual rate of interest does Jeremiah s investment earn? Try. Determine the interest rate each situation below, assuming the term remains 3 years, and the future value remains $ a) Jeremiah made payments of $800 every 3 months. b) Jeremiah made payments of $750 every 6 months, and interest was compounded semi-annually. 17

16 Determine the Regular Payment Amount of an Investment Example 3: Celia wants to have $300,000 in 20 years so that she can retire. Celia has found a trust account that earns a fixed rate of 10.8%, compounded annually. a) What regular payments must Celia make at the end of each year to meet her goal of $300,000? b) How much interest will she earn over the 20 years? Try. Determine Celia s payment amount in each situation. a) The payment frequency is every 6 months for 20 years (assume compounding is also every 6 months. b) The payment frequency is every month for 20 years (assume compounding is also every month). Regular-Payment VS Single Payment Investment The future value of a single deposit has a greater future value than a series of regular payments of the same total amount. Small deposits over a long term can have a greater future value than large deposits over a short term because there is more time for compound interest to be earned. 18

17 Determine the Term of a Regular Payment Investment Example 4: Louis makes regular $1,000 payments into an investment account at the end of every 6 months. His investment earns 3.5%, compounded semiannually. How many years will it take for his investment to grow more than $18,000? Example 5. How long will it take for $20,000 payments every 3 months to be $1,080, if the interest rate is 4.75%, compounded quarterly? Try. How long will it take for $1000 payments every 6 months to grow to more than $ if the interest rate is 7.5%, compounded semi-annually? 19

18 1.6 Solving Investment Portfolio Problems An investment portfolio can be built from different types of investments, such as single payment investments (for example, CSBs and GICs) and investments involving regular payments. Some of these investments, such as CSBs, lock in money for specified periods of time, thus limiting access to the money, but offer higher interest rates. Other investments, such as savings accounts, are accessible at any time but offer lower interest rates. Investments that involve greater principal amounts invested or greater regular payment amounts when contracted tend to offer higher interest rates. Determine the Future Value and Doubling Time of an Investment Portfolio Example 1: Phyllis started to build an investment portfolio for her retirement. She purchased a $500 Canada Savings Bond (CSB) at the end of each year for 10 years. The first five CSBs earned a fixed rate of 4.2%, compounded annually. The next five CSBs earned a fixed rate of 4.6% compounded annually. Three years ago, she also purchased a $4000 GIC that earned 6%, compounded monthly. a) What was the value of Phyllis s portfolio 10 years after she started to invest? b) Phyllis found a savings account that earned 4.9%, compounded semiannually. She redeemed her portfolio and invested all the money in the savings account. Estimate and determine how long will it take her to double her money? Rule of 72 20

19 Try. Andy is 17 years old and in Grade 12. When he was born, his parents deposited $100 at the end of each month into a savings account, earning an average annual interest rate of 3%, compounded monthly. On his 7th birthday, his grandparents bought him a 10-year $5000 GIC that earned 4%, compounded annually. He plans to start a 4-year history degree next year. He plans to redeem both investments now and combine them into one investment account that earns 4.2%, compounded quarterly, for one year until he starts school. a) How much will Andy s parents investment be worth when he redeems it? b) How much will his grandparents investment be worth when he redeems it? c) How much will Andy s new investment account be worth when he starts school? 21

20 Compare the Rates of Return of Two Investment Portfolios Example 2: Jason and Malique are each hoping to buy a house in 10 years. They want their money to grow so they can make a substantial down payment. Jason s Portfolio: Malique s Portfolio: A 10-year $2000 GIC that earns 4.2%, compounded semi-annually A savings account that earns 1.8% compounded weekly, where he saves $55 every week. A 5-year $4000 bond that earns 3.9%, compounded quarterly, which he will reinvest in another bond at an interest rate of 4.1%. A tax-free savings account (TFSA) that earns 2.2%, compounded monthly, and has a current balance of $5600 The purchase, at the end of each year, of a 10-year $500 CSB that earns 3.6%, compounded annually A savings account that earns 1.6%, compounded monthly, where she saves $200 every month. a) Determine the rate of return for Jason s investment portfolio. b) Determine the rate of return for Malique s investment portfolio. Rate of return is a useful measure for comparing investment portfolios. 22

21 2.1 Analyzing Loans (Part 1) The large majority of commercial loans are compound interest loans, although simple interest loans are also available. The cost of a loan is the interest charged over the term of the loan. It can be determined by Cost = Total Payment - Principal. A loan can involve regular payments (monthly or biweekly) over the term of the loan or a single payment (lump-sum) at the end of the term. NOTE: Borrow only what you can afford. Solve for the Future Value of a Loan with a Single Loan Payment Example 1: Tina s employer lent her $10,000 at a fixed interest rate of 6%, compounded annually. The loan is to be repaid in a lump-sum payment at the end of 5 years. a) How much will Tina need to pay her employer on the maturity date? What is the cost of her loan? b) Suppose the interest was compounded monthly instead. How much will Tina need to pay her employer on the maturity date? What is the cost of her loan? 23

22 For the lump-sum payment option, N = Number of Years, PMT = 0, P/Y = 1. Loans can be paid off at anytime. Try. A school ordered $1020 in books. Suppose that the bookstore offered the school a loan at 4%, compounded monthly, for 1.5 years. How much would the school need to repay the loan in a lump-sum payment on the maturity date? What is the cost of the loan? Example 2: Stan borrowed $1500 at 7.2%, compounded monthly, to buy a riding lawn mower for his summer business. He arranged to pay off the loan in 4 months, with a single payment. What amount did Stan need to pay on the maturity date? What is the amount of total interest Stan paid? Try. If Stan chooses to pay off the loan in 8 months with a lump-sum payment, what amount did Stand need to pay on the maturity date? What is the cost of the loan? 24

23 Solve for the Present Value and Interest of a Loan with a Single Payment Example 3: Anna wants to get a home improvement loan for renovation. Her bank charges 3.6%, compounded quarterly. She has a GIC that will mature in 5 years. When her GIC reaches maturity, Anna wants to use all the money to repay the loan with one lump-sum payment of no more than $20,000. a) How much can she borrow now? b) How much interest will she pay? Try. Matt needs a loan that he will not have to pay back for 18 months. The interest rate for the loan is 4.9%, compounded quarterly. On the maturity date, Matt wants to make a lump-sum payment of no more than $12,000. What is the most that Matt can borrow? How much interest will Matt pay on his loan? 25

24 2.1 Analyzing Loans (Part 2) For the regular payment (monthly, be-weekly, etc.) option, N = Total Number of Regular Payments, FV = Remaining Amount to Pay Off, P/Y = Number of Regular Payments per Year Solve for the Term and Total Interest of a Loan with Regular Payments Example 1: Lex borrowed $ at 5% interest, compounded monthly, to fund his research into a viable kryptonite weapon. Lex negotiated regular loan payments of $350 at the end of each month until the loan is paid off. a) In which month will Lex have at least half of the loan paid off? b) How long will it take Lex to pay off the loan? c) How much interest will Lex have paid by the time he has paid off the loan? 26

25 Total Payment = (Number of Payments) x (Amount of Each Payment) Total Interest Paid = Total Payment - Principal Try. Amber paid $1025 for her prom dress. She used her mother's credit card, which charges 18.9% compounded daily. Amber plans to make $50 payments each month. a) When will Amber have paid half the cost of her dress? b) How long will it take Amber to repay the total amount? c) How much interest will Amber pay? 27

26 To own this To pay off, you need a down payment and the help of mortgage. your mortgage, it takes (Number of Payments) x (Amount of Each Payment). So by the time when you finished your mortgage, you have paid (down payment) + (mortgage) + (cost). Solve for the Payment and Interest of a Loan with Regular Payments Example 2: John is negotiating with his bank for a mortgage on a house. He needs to make a 10% down payment on the purchase price of $225,000. Then the bank will offer a mortgage of the remaining, at 3.75%, compounded semiannually, with a term of 20 years and with monthly mortgage payments. a) What is the amount of the down payment? b) What is the amount John needs for his mortgage? c) What will the monthly mortgage payment be? What will the cost (interest paid) of the mortgage be? d) How much will he pay altogether? 28

27 Unlike loans, mortgage cannot be paid off at anytime unless you pay the penalty. There will be a chosen duration that the mortgage will be paid back in, which is called the amortization. It is usually 25, 30 or 35 years, depending on your regular payment amount. The interest rate is less likely to stay the same throughout the entire amortization. You will need to re-negotiate the rate with the bank once every few years and that s called the term. It is usually 3, 4, 5 or 6 years, depending on the economy and rate fluctuation. Try. Justin and Jen bought a house for $ They made a 20% down payment and negotiated a mortgage at 3.82% per annum compounded semi-annually. The mortgage is amortized over 25 years. a) Determine the amount of the down payment. b) Determine the amount to be financed. c) Determine the monthly mortgage payment. d) What would their house really cost if the term was continuously renegotiated to be the same throughout the amortization period? 29

28 Example 3. Ryan borrowed $5 000 at 7.25%, compounded quarterly, for 2 years. He decided to make regular monthly payments over the 2 years. What is the cost of the loan? How much did he pay altogether? Relate Payment and Compounding Frequency to Interest Period Example 4: Bill has been offered the following two loan options for borrowing $8000. Which option is better? a) Option A: He can borrow at 4.06% interest, compounded annually, and pay off the loan in annual payments of $ at the end of each year. b) Option B: He can borrow at 4.06% interest, compounded weekly, and pay off the loan in weekly payments of $34.62 at the end of each week. 30

29 2.2 Exploring Credit Card Use Credit cards have a credit limit, which is the maximum amount you can borrow. The credit limit varies from person to person, based on credit history. Incentives or promotions are sometimes offered to entice people to use credit cards. For example, an immediate cash rebate may be offered on the first purchase using a credit card. Low interest rates, rewards, or no annual fees may also be offered. But there's always a catch Credit that is offered in conjunction with a special offer or promotion must be considered very carefully. There may be conditions for how the loan is paid back, which may result in unexpected costs or penalties. NOTE: Credit cards usually have a minimum amount that must be paid each month, based on a percent of the outstanding balance. If there is no outstanding balance from the previous month and the new balance is paid off in full by the payment due date, no interest is charged. 31

30 Example 1: Explore Credit Card Use Jayden saw the new sound system he wanted on sale for $ He had to buy it on credit and had two options: Use his new bank credit card, which has an interest rate of 14.5%, compounded daily. (Because this credit card is new, he has no outstanding balance from the previous month.) Apply for the store credit card, which offers an immediate rebate of $100 on the price but has an interest rate of 19.3%, compounded daily. As with most credit cards, Jayden would not pay any interest if he paid off the balance before the due date on his first statement. However, Jayden cannot afford to do this. Both cards require a minimum monthly payment of 2.1% on the outstanding balance, but Jayden is confident that he can make regular monthly payments of $110. Which credit card is the better option for Jayden, and why? 32

31 Example 2. Lauren and Morgan are buying new equipment for their home office for $ They can afford payments of $1250 each month. Whose credit card should they use, if neither card has an outstanding balance? Explain. Lauren s credit card has an interest rate of 17.2%, compounded daily, but she will receive a 2% cash back on purchases at the end of a year. Morgan s credit card has an interest rate of 16.5%, compounded daily. 33

32 Try. The new drum set Jayson wants is on sale for $ , plus taxes of 13%. He can afford monthly payments of $250. He has two credit options: Use the store credit card, which charges 18.2% interest, compounded daily. As an incentive, the store will pay the taxes. Use his bank credit card, which charges 12.9% interest, compounded daily, and has no outstanding balance. Which credit card should he use? Why? 34

33 2.3 Solving Problems Involving Credit (Part 1) Solve a Credit Problem that Involves Overall Cost and Number of Payments Example 1: Meryl and Kyle are buying furniture worth $1075 on credit. They can make monthly payments of $75 and have two credit options. a) Option A: The furniture store credit card, which is offering a $100 rebate off the purchase price and an interest rate of 18.7%, compounded daily. b) Option B: A new bank credit card, which has an interest rate of 15.4%, compounded daily, but no interest for the first year. c) Which option should they choose? Explain. 35

34 Example 2. Madison wants to visit her parents in Regina at Easter. The return airplane ticket costs $1736. Madison has two options for payment: A bank loan with an interest rate of 5.6%, compounded monthly A credit card that offers 0% interest for 3 months and then 16.2%, compounded daily She plans to make monthly payments of $250. Which option should she choose? Explain. 36

35 Line of credit is similar to a loan. You are pre-approved and can borrow only what you need up to the credit limit and pay interest only on the amount borrowed. Solve a Credit Problem that Involves Payment Amount and Overall Cost Example 3: Ed wants to buy a car and needs to use credit to finance it. The cost, with taxes and shipping, is $24,738. Ed wants to repay his loan in 4 years using monthly payments and has two credit options: a) His secured line of credit at 1.7%, compounded monthly, above the Bank of Canada rate, which is currently 0.5%. b) The dealership s financing plan at 2.5%, compounded daily. c) Which option should he choose? Why? 37

36 Example 4: Chris wants to upgrade his backpacking equipment to prepare for a summer of trekking. The cost of the new equipment is $1265, and he has two options to pay for it: A new credit card with an introductory offer of 3% off the first purchase and an interest rate of 14.5%, compounded daily A line of credit with an interest rate of 6.8%, compounded daily a) Chris intends to pay off what he borrows in 1 year, with monthly payments. How much would his payments be for each option? b) Should he use the new credit card or his line of credit? Explain. 38

37 2.3 Solving Problems Involving Credit (Part 2) There are many factors that determine the best credit option, such as the interest charged, the total payment, the amount of each payment, and the length of time it takes to pay off the loan. All of these factors must be considered carefully before making a decision. Solve a Credit Problem that Involves Interest Amount and Rate Example 1: Jon s $475 car insurance payment is due. He does not have enough cash now but he will by the end of next month. So he is considering these two credit options: a) Borrow the money from a payday loan company for a $100 fee if it is paid back in full within 2 months. b) Use his credit card. The interest charged for cash advances is 19.99%, compounded daily. He can afford to pay the required $5 minimum payment after the first month and then plans to pay off the balance in full at the end of the second month. c) Which is the better option for Jon? Explain. d) What annual interest rate would equate to the fee charged by the payday loan company? Payday loans must also be considered carefully, since the fee for borrowing is often high. 39

38 Solve a Debt Consolidation Problem that Involves an Interest Amount Example 2: Nicki wants to be debt-free in 5 years. She has two credit cards on which she makes monthly payments: Card A has a balance of $ and an interest rate of 18.5%, compounded daily. Card B has a balance of $ and an interest rate of 19%, compounded daily. a) Nicki has qualified for a line of credit at her bank with an interest rate of 9.6%, compounded monthly, and a credit limit of $6000. She plans to pay off both credit card balances by borrowing the money from her line of credit. b) Card A has a balance of $ and an interest rate of 18.5%, compounded daily. Card B has a balance of $ and an interest rate of 19%, compounded daily. c) How much interest will she save? 40

39 2.4 Buy, Lease, or Rent? Lease is a contract for purchasing the use of property, such as a building or vehicle, from the leaser for a specified period. Equity is the difference between the value of an item and the amount still owing on it; it can be thought of as the portion owned. Asset is an item or a portion of an item owned; also known as property. Solve a Problem that Involves Leasing, Buying or Renting a Vehicle Example 1: Mandy needs a vehicle for work, on average, 12 days each month. She has three options: a) She could lease a vehicle, which requires a down payment of $4000 and lease payments of $380 per month plus tax. She would need insurance at $1220 each year (which could be paid monthly) and would have to pay for repairs and some maintenance, which would average $50 each month. For the 4-year lease she is looking at, she would have no equity in the vehicle at the end of the term. b) She could buy a vehicle for $ and finance it for 4-year term at 4.5% interest, compounded monthly. She would have the same insurance, repairs, and maintenance costs that she would have with leasing. However, the equity of the vehicle would be considered an asset. c) She could rent at $49.99 per day, plus tax, with unlimited kilometres. d) Which option is the most economical choice? 41

40 Solve a Problem that Involves Leasing or Buying a Water Heater Example 2: The water heater in Tom s home stopped working, so he needs a new one. Tom works for minimum wage. After paying his monthly expenses, he has $35 saving left every month. He has an unused credit card that charges 18.7%, compounded daily. He has two options: a) He could buy a water heater for $712.99, plus an installation fee of $250, using his credit card. He could afford to pay no more than $35 each month. What are costs of buying? b) Tom could lease from his utility company for $17.25 per month. This would include parts and service. What are the costs with leasing? c) To lease or to buy, which option is better? When deciding whether to rent, buy (with or without financing), or lease, each situation is unique. Since each situation is unique, it is impossible to generalize about whether renting, leasing, or buying is best. A cost and benefit analysis should take everything into account. Costs include initial costs and fees, shortterm costs, long-term costs, disposable income, the cost of financing, depreciation and appreciation, penalties for breaking contracts, and equity. Benefits include convenience, commitments, flexibility, and personal needs or wants, such as how often you want to buy a new car. 42

41 Appreciation is an increase in the value of an asset over time. Depreciation is a decrease in the value of an asset over time. Solve a Problem that Involves Leasing or Buying Office Space Example 3: Lance started his own business 2 years ago. His business has grown quickly, and his office is no longer big enough. He is considering these two options: a) He could sign a 3-year lease on office space, with monthly rental payments of $2,000. What are the costs of leasing over 15 years? b) He could purchase a property for $285,000 and renovate so it could be used as an office. A 5% down payment would be required, and he would take out a 15-year mortgage at 5%, compounded semi-annually, with monthly payments. Assume the property has appreciated by 34.6% over the 15 years. What are the costs of buying over 15 years? c) Which option is better if Lance decides to retire and sell his business in 15 years? 43

42 Solve a Problem that Involves Renting or Buying a House Example 4: Two couples made different decisions about whether to rent or buy a house. After 3 years, both couples decide to move. a) Donald and Patricia are renting a house for $1,600 per month. They plan to renew the lease yearly. Determine their cost over 3 years. b) Helen and Tim bought a house for $249,900. They have negotiated a mortgage of 95% of the purchase price with a 5% down payment. The mortgage is compounded semi-annually at 5.5%, has a 20-year term, and requires monthly payments. The house has depreciated by 10% over the 3 years. Determine their cost over 3 years. c) Compare each couple's situation after 3 years. 44

43 6.1 Exploring the Graphs of Polynomial Functions A polynomial function consist of one or more terms, which are separated by + or signs. The degree of a polynomial function is the value of the highest exponent in the function. If a polynomial function includes a term with no variable, this term is called a constant term. Determine the Degree and the Constant Example 1. Determine the degree and the constant of each polynomial function. a) 4 5 b) 2 4 c) 3 d) A number that multiplies the variable in a polynomial is called a coefficient. The leading coefficient is the number that multiplies the term with the highest power. Determine the Leading Coefficient Example 2. Determine the leading coefficient of each polynomial function. a) 4 5 b) 2 4 The terms in a polynomial function are normally written so that the powers are in descending order. For example, Example 3. Write a polynomial function in descending order that satisfies the following conditions. a) degree 2, leading coefficient 3 b) degree 2, leading coefficient 7, two terms c) degree 1, leading coefficient 1 d) degree 0 e) degree 3, constant term 8 45

44 Domain The domain is the set of all possible x-values which will make the function "work" and will output real y-values. Range The range of a function is the complete set of all possible resulting y-values of the dependent variable. End behaviour The end behaviour of a polynomial is the description of the shape of the graph, from left to right, on the coordinate plane. Turning Point A turning point is any point where the graph of a function changes from increasing to decreasing or from decreasing to increasing. Polynomial functions are named according to their degree. Polynomial functions of degrees 0, 1, 2, and 3 are called constant, linear, quadratic, and cubic functions, respectively. Characteristics of Polynomial Functions Example 4. Determine the type of function, the degree, the x-intercepts, y- intercepts, end behaviour, range and number of turning points for each type of function. red Blue Red Type of Function Degree Number of x-int blue Number of y-int Domain Range End Behaviour Number of Turning Points Blue Red 46

45 blue green Blue Red Green Type of Function Degree Number of x-int Number of y-int red Domain Range End Behaviour Blue Red Green Number of Turning Points green red blue Blue Red Green Type of Function Degree Number of x-int Number of y-int Domain Range End Behaviour Blue Red Green Number of Turning Points 47

46 Blue Red blue Type of Function Degree Number of x-int Number of y-int red Domain Range Blue Red End Behaviour Number of Turning Points Example 5. Which of the following graphs might represent polynomial functions? 48

47 6.2 Characteristics of the Equations of Polynomial Functions Standard Form Linear function is Quadratic function is Cubic function is +d (where a 0) Observe the Characteristics of the Graph of a Given Polynomial Function Example 1. Sketch the graph on the grid using the graphing calculator window x:[ 8, 8, 2] y:[ 20, 20, 5] a) How is the constant term in a polynomial function related to the y- intercept of the graph of the function? b) How does the sign of the leading coefficient affect the end behaviour of the graph of each type of polynomial function? 49

48 The degree of a polynomial function determines the shape of the function. The graphs of polynomial functions of the same degree have common characteristics. Reason about the Characteristics of the Graph of a Given Polynomial Function Using Its Equation Example 2. Predict the number of possible x-intercepts, y-intercept, domain, range, end behaviour, number of possible turning points of each function using its equation. a) 3 5 # of x-intercepts # of y-intercepts domain range end behaviour # of turning pts b) # of x-intercepts # of y-intercepts domain range end behaviour # of turning pts c) # of x-intercepts # of y-intercepts domain range end behaviour # of turning pts 50

49 Connect Polynomial Functions to Their Graphs Example 3. Match each graph with the correct polynomial function. Justify your reasoning i) ii) iii) iv) v) vi) 51

50 Reason about the Characteristics of the Graphs of Polynomial Functions Example 4. Sketch the graph of a possible polynomial function for each set of characteristics below. a) Range: 2, and y-intercept: 4 b) Range: and turning points: one in quadrant III and another in quadrant I Try. Sketch the graph of a possible polynomial function for each set of characteristics below. a) extending from quadrant II to quadrant b) range of y 6, x-intercepts of 2 and 6 IV, degree 1, y-intercept of 3 52

51 6.3 Modelling Data with a Line of Best Fit Interpolation Interpolation is the process used to estimate a value within the domain of a set of data, based on a trend. You can graph the scatter plot and interpolate using Technology (TI 83). Step 1. Enter the data Press STAT key Select EDIT Clear any numbers that are written in L1, L2 Under Column L1, enter the data (x-values) Under Column L2, enter the data (y-values) Step 2. Choose window Press WINDOW and adjust Xmin, Xmax, Ymin, Ymax Graph Step 3. Obtain the function Make sure Plot1 is highlighted to see the scatter plot L1 & L2 deleted: STAT Select 5 SetUpEditor Enter Press STAT key Select CALC Select #4 LinReg (or #5 QuadReg or #6 Cubic Reg) Press Y= key, clear Y 1 = VARS Select #5 Statistics Select EQ Select #1 RegEQ Use Technology to Determine a Linear Model for Continuous Data Example 1. The one-hour record is the farthest distance travelled by bicycle in 1h. The table below shows the world-record distances and the dates they were accomplished. Xmin = Ymin = Xmax = Ymax = a) Use technology to create a scatter plot and to determine the equation of the line of best fit. Round to three decimal places. b) Interpolate a possible world-record distance for the year 2006, to the nearest hundredth of a kilometre. Linear regression results in an equation that balances the points in the scatter plot on both sides of the line. A line of best fit can be used to predict values that are not recorded or plotted. Predictions can be made by reading values from the line of best fit on a scatter plot or by using the equation of the line of best fit. 53

52 Extrapolation Extrapolation is the process used to estimate a value outside the domain of a set of data, based on a trend. Use Linear Regression to Solve a Problem that Involves Discrete Data Example 2. Matt buys T-shirts for a company that prints art on T-shirts and then resells them. When buying the T-shirts, the price Matt must pay is related to the size of the order. Five of Matt s past orders are listed in the table below. Xmin = Ymin = Xmax = Ymax = Matt has misplaced the information from his supplier about price discounts on bulk orders. He would like to get the price per shirt below $1.50 on his next order. a) Use technology to create a scatter plot and determine an equation for the linear regression function that models the data. Round to three decimal places. b) What do the slope and y-intercept of the equation of the linear regression function represent in this context? c) Use the linear regression function to extrapolate the size of order necessary to achieve the price of $1.50 per shirt. Try. Consider the data in the table. Use technology to create a scatter plot and to determine the equation of the line of best fit. x y a) Determine, to the nearest tenth, the value of y when x is b) Determine, to the nearest tenth, the value of x when y is

53 6.4 Modelling Data with a Curve of Best Fit Curve of best fit Curve of best fit is a curve that best approximates the trend on a scatter plot. Use Technology to Solve a Quadratic Problem Example 1. Audrey is interested in how speed plays a role in car accidents. She knows that there is a relationship between the speed of a car and the distance needed to stop. She has found the following experimental data on a reputable website, and she would like to write a summary for the graduation class website. Xmin = Xmax = Ymin = Ymax = a) Plot the data on a scatter plot. Determine the equation of a quadratic regression function that models the data. Round to three decimal places. b) Use your equation to compare the stopping distance at 30 km/h with the stopping distance at 50 km/h, to the nearest tenth of a metre. c) Determine the maximum speed that a car should be travelling in order to stop within 4 m, the average length of a car. 55

54 Example 2. The concentration (in milligrams per litre) of a medication in a patient's blood is measured as time passes. Susan has collected the following data and is attempting to express the concentration as a polynomial function of time. Xmin = Ymin = Xmax = Ymax = Time (T hours) Concentration (C mg/l) a) On a graphing calculator, enter the data in two lists. Time in List 1 and Concentration in List 2. Create a scatter plot of the data and use the quadratic regression feature to determine the polynomial function,, that best fits the data. Round the parametres a, b, and c to 2 decimal places. b) The doctor has decided that the patient needs a second dose of medication when the concentration in the blood is less than 10 mg/l. If the first dose of medication was given at 9:00am, at what time should the second dose be given? Try. Consider the data in the table. Use technology to create a scatter plot and to determine the equation of the line of best fit. x y a) Determine, to the nearest tenth, the value of y when x is 10.6 b) Determine, to the nearest tenth, the value of x when y is 9.8. Technology uses polynomial regression to determine the curve of best fit. Polynomial regression results in an equation of a curve that balances the points on both sides of the curve. A curve of best fit can be used to predict values that are not recorded or plotted. Predictions can be made by reading values from the curve of best fit on a scatter plot or by using the equation of the curve of best fit. 56

55 Use Technology to Solve a Cubic Regression Function Example 3. The following table shows the average retail price of gasoline, per litre, for a selection of years in a 30-year period beginning in Xmin = Ymin = Xmax = Ymax = a) Use technology to graph the data as a scatter plot. What polynomial function could be used to model the data? Explain. b) Determine the cubic regression equation that models the data. Use your equation to estimate the average price of gas in 1984 and c) Estimate the year in which the average price of gas was 56.0 /L. Try. Consider the data in the table. Use technology to create a scatter plot and to determine the equation of the line of best fit. x y a) Determine, to the nearest tenth, the value of y when x is 10.6 b) Determine, to the nearest tenth, the value of x when y is

56 Example 4. Consider the data in the table. Create a scatter plot from the data using a graphing calculator. x y a) Use the cubic regression feature of a calculator to determine a cubic function that models the data. Round to three decimal places. b) Use the cubic regression equation to determine the value of x when y = 90. c) Use the linear regression feature of a calculator to determine a leaner function that models the data. Round to three decimal places. d) Use the linear regression equation to determine the value of x when y = 90. e) Which model appears to be the better for the data? 58

57 7.1 Exploring the Characteristics of Exponential Functions Exponential Function (Increasing) An exponential function is a function of the form, or, where 0 and 1. Investigate the Characteristics of the Graphs of Exponential Functions (Increasing) Example 1. Graph each exponential function. Determine the number of x- intercepts, the y-intercept, the end behaviour, the domain, and the range. a) 10 as 1 10 x f(x) number of x-intercepts: y-intercept: domain: range: end of behaviour: b) 2 5 or 2 5 x f(x) number of x-intercepts: y-intercept: domain: range: end of behaviour: 59

58 Exponential Function (Decreasing) An exponential function is a function of the form, where 0 and 0 1. Investigate the Characteristics of the Graphs of Exponential Decay Functions Example 2. Graph each exponential function. Determine the number of x- intercepts, the y-intercept, the end behaviour, the domain, and the range. a) as 1 x f(x) number of x-intercepts: y-intercept: domain: range: end of behaviour: b) 8 x f(x) number of x-intercepts: y-intercept: domain: range: end of behaviour: 60

59 Try. Graph each function using technology. Determine the number of x- intercepts, the y-intercept, the end behaviour, the domain, and the range. a) 3 4 b) 8 number of x-int: y-int: end haviours: domain: range: number of x-int: y-int: end haviours: domain: range: 61

60 7.2 Relating the Characteristics of an Exponential Function to Its Equation Connect the Characteristics of an Increasing Exponential Function to Its Equation and Graph Example 1. State the number of x-intercepts, the y-intercept, end behaviour, domain, and range for each function, without graphing the function. Predict whether the function is increasing or decreasing. Verify your answers by graphing. a) 2 5 number of x-int: y-int: end behaviour: domain: range: increasing or decreasing b) as number of x-int: y-int: end behaviour: domain: range: increasing or decreasing 62

61 Connect the Characteristics of a Decreasing Exponential Function to Its Equation and Graph Example 2. State the number of x-intercepts, the y-intercept, end behaviour, domain, and range for each function, without graphing the function. Predict whether the function is increasing or decreasing. Verify your answers by graphing. Number of x-ints y-int End behavio ur Domain Range Incr OR Decr Try. State the number of x-intercepts, the y-intercept, end behaviour, domain, and range for each function, without graphing the function. Predict whether the function is increasing or decreasing. Verify your answers by graphing. Number of x-ints y-int End behavio ur Domain Range Incr OR Decr

62 Match an Exponential Equation with Its Corresponding Graph Example 3. Which exponential function matches each graph below? Provide your reasoning. Try. Match each function with the corresponding graph below. 64

63 7.3 Modelling Data Using Exponential Functions You can graph the scatter plot and interpolate using Technology (TI 83). Step 1. Enter the data Press STAT key Select EDIT Clear any numbers that are written in L1, L2 Under Column L1, enter the data (x-values) Under Column L2, enter the data (y-values) Step 2. Choose window Press WINDOW and adjust Xmin, Xmax, Ymin, Ymax Graph Step 3. Obtain the function Press STAT key Select CALC Select #0 ExpReg Make sure Plot1 is highlighted to see the scatter plot L1 & L2 deleted: STAT Select 5 SetUpEditor Enter Press Y= key, clear Y 1 = VARS Select #5 Statistics Select EQ Select #1 RegEQ Create Graphical and Algebraic Models of Given Data Example 1. Simon, a biologist, is investigating a new bacteria culture which could help strengthen a person's immune system. He isolates fifty cells and records the growth in the number of cells over a period of five hours. His results are shown in the table and graph below. Numbers Number of of Bacteria hours (x) (y) a) Determine if the data can be represented by an exponential model. b) Use regression to determine the exponential function that best models the data. Round a and b to three decimal places. c) Determine the numbers of bacteria, to the nearest whole number, when 8. 65

64 Example 2. Angela invests $2000 in GIC that increases in value every 3 months. The table below shows the value of the investment during the first 18 month. Months (x) Value in Dollars (y) $2000 $2012 $ $ $ $ $ a) Use regression to determine the exponential function that best models that data. Give a to the nearest whole number, and b to the nearest thousandth. Xmin = Ymin= Xmax= Ymax = b) Determine the value of the investment after two years. Example 3. The following data represents the winning times, to the nearest minute, for the men's Olympic Marathon in some of the Olympics in the twentieth century. Year (x) Time in Minutes (y) a) Use regression to determine the exponential function that best models that data. Give a to the nearest whole number, and b to the nearest ten thousandth. Xmin = Ymin= Xmax= Ymax = b) Estimate the winning time by the Finnish Athlete in the 1924 Olympics. c) Estimate the winning time by the Czech Athlete in the 1952 Olympics. Try. The following data gives the population in a town over a period of fifty years. Year (x) Population (y) a) Use regression to determine the exponential function that best models that data. Give a to the nearest whole number, and b to the nearest thousandth. Xmin = Ymin= Xmax= Ymax = b) Estimate population after 35 years. 66

65 7.4 Characteristics of Logarithmic Functions with Base 10 and Base e Logarithmic Function A log function is a function of the form, where 0, 1 and 0 Investigate the Characteristics of the Graphs of Log Function Example 1. Graph each log function. Determine the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range. a) log as log x f(x) number of y-intercepts: x-intercept: domain: range: end behaviour: b) 2log as 2log x g(x) number of y-intercepts: x-intercept: domain: range: end behaviour: 67

66 c) 5log as 5log x h(x) number of y-intercepts: x-intercept: domain: range: end behaviour: d) 5log as 5log x i(x) number of y-intercepts: x-intercept: domain: range: end behaviour: All logarithmic functions of the form log or ln have these unique characteristics: - If a > 0, the function increases. - If a < 0, the function decreases. 68

67 Connect the Characteristics of an Increasing Log Function to Its Equation and Graph Example 2. Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the function 15log. x-int: domain: # of y-int: range: end behaviour Connect the Characteristics of an Decreasing Natural Log Function to Its Equation and Graph Example 3. Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the function 4ln x-int: domain: # of y-int: range: end behaviour Try. Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the following functions. 12ln 5log x-int: domain: x-int: domain: # of y-int: range: # of y-int: range: end behaviour 69end behaviour

68 Match Equations of Exponential and Log Functions with Their Graphs Example 4. Which exponential function matches each graph below? Provide your reasoning. 70

69 7.5 Modelling Data Using Logarithmic Functions You can graph the scatter plot and interpolate using Technology (TI 83). Step 1. Enter the data Press STAT key Select EDIT Clear any numbers that are written in L1, L2 Under Column L1, enter the data (x-values) Under Column L2, enter the data (y-values) Step 2. Choose window Press WINDOW and adjust Xmin, Xmax, Ymin, Ymax Graph Step 3. Obtain the function Make sure Plot1 is highlighted to see the scatter plot L1 & L2 deleted: STAT Select 5 SetUpEditor Enter Press STAT key Select CALC Select #9 LnReg Press Y= key, clear Y 1 = VARS Select #5 Statistics Select EQ Select #1 RegEQ Use Log Regression to Solve a Problem Graphically and Algebraically Example 1. The decay of radioactive elements can sometimes be used to date events from the earth's past. In a living organism, the ratio of radioactive carbon, carbon-14, to ordinary carbon remains fairly constant. However, when the organism dies, no new carbon is ingested and the proportion of carbon -14 decreases as it decays. The table below shows data for five recently discovered fossils. % carbon-14 (x) Age in years (y) a) Determine if the data can be represented by an log model. Xmin = Ymin= Xmax= Ymax = b) Use the natural log regression feature of a calculator (LnReg) to determine a function that models the data. Use integer values for a and b. c) A bone fragment was discovered. If the carbon dating test indicated that approximately 20.3% of carbon-14 was left, estimate the age of the bone fragment to the nearest 1000 years. 71

70 Example 2. The number of years, y, that it takes for an investment of $1000 to increase in value to x dollars can be modelled by a log function. The table shows the value of Scott's investment over a period of 10 years. Value of Investment (x) Number of Years (y) a) Use the natural log regression feature of a calculator (LnReg) to determine a function that models the data. Round a and b to three decimal places. b) Estimate the number of years if the value of investment is $ Try. Martin is a fruit grower. He has planted and tracked the growth of a new variety of cherry tree he is considering planting on 10 acres of his farm. Age of Tree Height Age of Tree Height (years) (feet) (years) (feet) Xmin = Ymin= Xmax= Ymax = a) Determine the equation of the log regression function that models the tree's growth. b) Determine the height of a tree of this variety when it is 15 years old. c) Determine the age of a tree of this variety when it is 12 feet tall. 72

71 8.1 Understanding Angles Measure Angles Using Protractor Example 1. Measure the angle using the protractor. In math, the symbol " " following a number means the unit of angular measure in degrees. If there is no unit after the number, or there is the abbreviation "rad", or the word radians, then the unit is radians. Radian Measure Radian measure is an alternative way to express the size of an angle. 180 is equal to π radians and 360 is equal to 2π radians. Converting Between Degrees and Radians π radians = 180 Degrees to Radians: multiply the angle measure by Radians to Degrees: multiply the angle measure by 73

72 Convert Degrees to Radians as an Exact Value Example 2. Sketch and convert 60 and 450 to a radian measure as an exact value. Try. Express each angle measure in radians. Leave answers in terms of π. a) 45 b) 75 c) 210 Example 3. Sketch and estimate the value of 135 in radian measure, to one decimal place. Try. Estimate the value of each angle in radian measure, to one decimal place. a) 240 b) 450 c) 690 Convert Radians to Degrees Example 4. Convert 3.14 radians to a degree measure to three decimal places. Try. Estimate, to the nearest degree, the measure of each angle in degrees. a) 2.4 rad b) rad c) rad 74

73 8.2 Exploring Graphs of Periodic Functions Periodic Function A periodic function is a function whose graph repeats in regular intervals or cycles. Period is the length of interval of the domain to complete one cycle. Midline A midline is the horizontal line halfway between the maximum and minimum values of a periodic function. Amplitude The amplitude is the distance from the midline to either the maximum or minimum value of a periodic function; the amplitude is always expressed as a positive number. 75

74 Investigate the Characteristics of the Graphs of Sine and Cosine Functions Example 1. Complete the table below and sketch the function sin for 0 720, Period: Domain: Midline: Amplitude: Range: y-intercept: x-intercept: What is the maximum value of sin? For what value of x does a maximum occur? What is the minimum value of sin? For what value of x does a minimum occur? 76

75 Try. Complete the table below and sketch the function cos for 0 720, Period: Domain: Midline: Amplitude: Range: y-intercept: x-intercept: What is the maximum value of cos? For what value of x does a maximum occur? What is the minimum value of cos? For what value of x does a minimum occur? 77

76 Set calculator to RADIAN mode when dealing with angle measure in radians. Example 2. Complete the table below and sketch the function cos for 0 4, Period: Domain: Midline: Amplitude: Range: y-intercept: x-intercept: What is the maximum value of cos? For what value of x does a maximum occur? What is the minimum value of cos? For what value of x does a minimum occur? 78

77 Try. Complete the table below and sketch the function sin for 0 4, Period: Domain: Midline: Amplitude: Range: y-intercept: x-intercept: What is the maximum value of sin? For what value of x does a maximum occur? What is the minimum value of sin? For what value of x does a minimum occur? 79

78 8.3 The Graphs of Sinusoidal Functions The period is the horizontal distance between consecutive maximum values or consecutive minimum values. It is also twice the horizontal distance between a maximum value and the next minimum value. The amplitude of a periodic function is defined as half the distance between the maximum and minimum values of the function. Amplitude = Midline can be determined by Midline = Describe the Graph of a Sinusoidal Function in Degree Measure Example 1. The graph of a sinusoidal function is shown. Describe this graph by determining its maximum, minimum, range, the equation of its midline, its amplitude, and its period. Maximum: Minimum: Range: Midline: Amplitude Period: 80

79 Describe the Graph of a Sinusoidal Function in Radian Measure Example 2. The graph of a sinusoidal function is shown. Describe this graph by determining its maximum, minimum, range, the equation of its midline, its amplitude, and its period. Maximum: Minimum: Range: Midline: Amplitude Period: Try. Determine the range, amplitude, period, and equation of the midline of this sinusoidal function. Maximum: Minimum: Range: Midline: Amplitude Period: 81

80 Connect a Sinusoidal Function to Oscillating Motion Example 3. Olivia was swinging back and forth in front of a motion detector. Her distance from the detector, in terms of time, can be modelled by the graph shown. a) What is the equation of the midline? What does it represent in this situation? b) What is the amplitude of the function? c) What is the period of the function? What does it represent in this situation? d) How close did Olivia get to the motion detector? e) At t = 7s, would it be safe to run between Olivia and the motion detector? Try. For a physics project, Morgan and Lily had to graph and analyze an example of simple harmonic motion. Morgan swung on a swing, and Lily used a motion detector to measure Morgan's height above the ground over time, as she swung back and forth. The girls then graphed their data as shown. At the end of each cycle, the swing returned to its initial position, which resulted in a sinusoidal graph. a) Determine the range, amplitude, equation of the midline, and period of the graph. Range: Period: Midline: Amplitude: b) Determine Morgan's height above the ground at 4 s. 82

81 8.4 The Equations of Sinusoidal Functions A sinusoidal function has the form of sin or cos Explore the Characteristics of a Sine or Cosine Function Base on Its Equation Example 1. Graph, in radian mode, y = sin x. Then graph the following functions on the same axes. i) 3sin ii) 1.5sin iii) sin How does changing the value of a affect the graph of y = sin x? Example 2. Graph, in radian mode, y = sin x. Then graph the following functions on the same axes. How does changing the value of b affect the graph of y = sin x? 83

82 Example 3. Graph, in degree mode, y = sin x. Then graph the following functions on the same axes. How does changing the value of c affect the graph of y = sin x? sin x + 1 means sin (x) + 1. Must provide the end round bracket when entering to graphing calculator. Example 4. Graph, in degree mode, y = sin x. Then graph the following functions on the same axes. How does changing the value of d affect the graph of y = sin x? 84

83 Determine the Characteristics of a Cosine Function Based on Its Equation Example 5. Consider the function 2cos4 1 for 0 360,. Describe the graph of the function by stating the amplitude, equation of the midline, range, and period, as well as the relevant horizontal translation of cos. amplitude: period: midline: horizontal translation: range: Try. Consider the function 5cos 3 for 0 360,. Describe the graph of the function by stating the amplitude, equation of the midline, range, and period, as well as the relevant horizontal translation of y = cos x. amplitude: period: midline: horizontal translation: range: Determine the Characteristics of a Sine Function Based on Its Equation 85

84 Match Equations to Graphs Example 7. Match each graph with the corresponding equation below. Try. Match each graph with the corresponding equation below. 86

85 8.5 Modelling Data with Sinusoidal Functions You can graph the scatter plot and interpolate using Technology (TI 83). Step 1. Enter the data Press STAT key Select EDIT Clear any numbers that are written in L1, L2 Under Column L1, enter the data (x-values) Under Column L2, enter the data (y-values) Step 2. Choose window Press WINDOW and adjust Xmin, Xmax, Ymin, Ymax Graph Step 3. Obtain the function Make sure Plot1 is highlighted to see the scatter plot L1 & L2 deleted: STAT Select 5 SetUpEditor Enter Press STAT key Select CALC Select C SinReg Press Y= key, clear Y 1 = VARS Select #5 Statistics Select EQ Select #1 RegEQ Solve Problems Using a Sinusoidal Model Example 1. The average high temperature for each month in Vancouver, British Columbia, is shown in the following table. Month 1 represents January and Month 12 represents December. a) Determine the equation of the sinusoidal regression function that models the relationship between the month of the year and average high temperature. b) Determine the average high temperature in Vancouver in July. c) Estimate when the average high temperature in Vancouver will be greater than 60 F. 87

86 Example 2. The table below gives the height of a rider on a Ferris wheel from the ground at different times. Jordan got on the Ferris wheel and rode it for four consecutive rotations. His friend Scottie was in a building directly across from the Ferris wheel, at a height of 400 ft. When was Jordan level with Scottie? Try. The diameter of the Ferris wheel was approximately 76 metres and the maximum height of the wheel was approximately 80 metres. a) Use sinusoidal regression to determine the equation. Round the parameters to the nearest hundredths if required. b) How high, to the nearest metre, is the cart 5 minutes after the wheel starts rotating? c) How many seconds after the wheel starts rotating does the cart first reach 10 metres from the ground. Answer to the nearest second. 88

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89 3.1 Types of Sets and Set Notation In math, a set is defined as a collection of distinguishable objects. A set is usually denoted by a capital letter and a description inside curly brackets { }. For example, the set of whole numbers is W = {0, 1, 2, 3,... }. Each object in a set is referred to as an element. For example, object 3 is an element of set W. There are three general ways of defining the contents of a set. The curly brackets { } together is read as "the set of". 1. Describing the set C in words. C = {natural numbers less than 7} 2. Listing the elements of the set C. C = {1, 2, 3, 4, 5, 6} 3. Using set-builder notation to describe the set C. C = { x x < 7, x N } represents, and is read as, "such that". represents, and is read as, "is a member of". N represents the natural numbers. The number of elements in set C is written as n C( ). Therefore, n C( ) = 6 in the above example. Example 1: Sets and Set Notation Consider the following two sets. set P = { Whole numbers less than or equal to 3 } set Q = { even whole numbers less than 10 } a) List the elements of set P and Q. b) Write set P using set-builder notation. c) Write set Q using set-builder notation. d) Complete the following: i) n (P) ii) n (Q) 91

90 Universal Set A set of all the elements under consideration for a particular context (also called the sample space). We can define the universal set depending on this situation we are dealing with. An universal set should cover all elements in the context of the question. Example 2. Sharon is defining set using non-negative single digit numbers. She defined the following sets. set E = { even numbers } set L = { whole numbers less than 7 } set O = { odd numbers less than 7 } a) List the elements of each set. b) Choose an universal set U which will contain all elements in sets E, L and O. Subset A subset is a set that contains some or all or possible none of the elements from a previously defined set. All sets we deal with in a problem must be subsets of the universal set and every set is a subset of itself. The symbol is used to represent this relationship between sets. For example, set E = {even whole numbers} is a subset of U = {whole numbers}, since all the elements in set E belongs to set U. This is denoted by E U. If a set Y is not the subset of a set X, then it is denoted by Y X. If set Z is a subset of set Y, which is a subset of set X, this relationship can be denoted by Z Y X. 92

91 Complement of a Set The complement of a set C is the set of all element in the universal set that are not in set C. The complement of a set C is denoted by C'. The complement of set E = { even whole numbers } in the universal set U = {whole numbers less than or equal to 10} is written as E' = { 1, 3, 5, 7, 9 } Example 3. Consider the following two sets. set A = { natural numbers less than 20 that are divisible by 3 } set B = { natural numbers less than 20 that are divisible by 5 } a) State the universal set for this example. Label this set U. b) List the elements of the following sets: i) A = ii) B = iii) A' = c) Write a description of the elements of the complement of set B in words. d) Set C = {natural numbers less than 20 that are multiples of 6 }. State whether the following are true or false. i) C A ii) C B Disjoint Sets Disjoint sets are two or more sets have no elements in common; for example, the set of even numbers and the set of odd numbers are disjoint. Example 4: Determine the Disjoint Sets If P = { 2, 4, 8, 16 }, Q = { factors of 6 }, and R = { 4, 8, 12, 16 }, which pairs of sets are disjoint? 93

92 Empty Set The empty set is a set that contains no elements. The empty set is denoted by or { }. set X = { integers that are both even and odd } = or { }. Example 5: Determine the Empty Set List the element of following set: W = { prime numbers between 24 and 28 }. Example 6: Sort Numbers Using Set Notation and a Venn Diagram Consider the multiples of 5 and 10, from 1 to 500. a) State the universal set for this example. Label this set U. b) Define a subset F, for the multiples of 5, from 1 to 500, using set builder notation. c) Define a subset T, for the multiples of 10, from 1 to 500, using set builder notation. d) Describe the relationships among U, F and T and represent the sets and subsets in a Venn diagram. 94

93 Example 7: Determine the Number of Elements in Sets A triangular number, such as 1, 3, 6, or 10, can be represented as a triangular array. a) Determine a pattern you can use to determine any triangular number. f t b) Define the universal set for this example. Label this set U. c) Define each subset and determine the number of elements i) even and triangular ii) odd and triangular 95

94 Example 8: Solve Problem Using a Venn Diagram Becky recorded the possible sums that can occur when you roll two four-sided dice in an outcome table. Blue Die Red Die a) Define the universal set for this example. Label this set U. b) Define and display the following sets in one Venn diagram: rolls that produce a sum less than 5 rolls that produce a sum greater than 5 c) Record the number of elements in each set. 96

95 3.2 Exploring Relationships between Sets Example 1: Explore Relationships between Sets There are 65 Grade 12 students. Of these students, 23 play volleyball and 26 play basketball. There are 31 students play neither sport. The following Venn diagram represents the sets of students. a) Determine the number of students who play both volleyball and basketball, denoted by n ( V B ). b) Determine the number of students who play volleyball only, denoted by n ( V \ B ). c) Determine the number of students who play basketball only, denoted by n ( B \ V ). Try. Jordan asked 40 students at his school cafeteria what they bought for lunch. He recorded his results in the table below. How many students bought a beverage and soup? Purchase Number of Students beverage 34 soup 18 no beverage or soup 5 97

96 Example 2. In a survey of 400 households, 285 had personal Video recorders (PVRs) and 320 had multi-function printers (MFPs). Determine the number of households in the survey who had both an MFP and PVR, denoted by n ( P M ). Sets that are not disjoint share common elements. Each area of a Venn diagram represents something different. When two non-disjoint sets are represented in a Venn diagram, you can count the elements in both sets by counting the elements in each region of the diagram just once. Each element in a universal set appears only once in a Venn diagram. If an element occurs in more than one set, it is placed in the area of the Venn diagram where the sets overlap. 98

97 3.3 Intersection and Union of Two Sets Suppose the universal set U is defined as non-negative single digit numbers, set E is even numbers, set L is whole numbers less than 7, and set O is odd numbers less than 7. Intersection of Sets The intersections of set A and set B is called A and B and is denoted by A B. It is the set of elements that are members of both set A and B. For example, the intersection of sets of E and L is defined as E L = { 0, 2, 4, 6 } In set notation, A B is read as intersection of A and B. It denotes the elements that are common to A and B. The intersection is the region where the two sets overlap in the Venn diagram on the left. Union of Sets The union of set A and set B is called A or B and is denoted by A B. It is the set of elements that are members of either set A or B, or both (inclusive use of "or"). For example, the union of sets of E or L is defined as E L = { 0, 1, 2, 3, 4, 5, 6, 8 } A B is read as union of A and B. It denotes all elements that belong to at least one of A or B. The union is the red region in the Venn diagram on the left. 99

98 Example 1: Determine the Union and Intersection of Disjoint Sets If you draw a card at random from a standard deck of cards, you will draw a card from one of four suits: clubs (C), spades (S), hearts (H), or diamonds (D). a) Describe sets C, S, H, and D, and the universal set U for this situation. b) Determine n(c ), n(s ), n(h ), n(d ), and n(u ). c) Describe the union of S and H. Determine n ( S H ). d) Describe the intersection of S and H. Determine n ( S H ). e) Determine whether the events that are described by sets S and H are mutually exclusive, and whether sets S and H are disjoint. f ) Describe the complement of S H. 100

99 Example 2: Determine the Number of Elements n a Set Using a Formula The athletics department at a large high school offers 16 different sports: badminton, basketball, cross-country running, curling, football, golf, hockey, lacrosse, rugby, cross-country skiing, soccer, softball, tennis, ultimate, volleyball, wrestling. Determine the number of sports that require following types of equipment: U = {sports offered by the athletics department} B = {sports that use a ball} I = {sports that use an implement} a) a ball and an implement, denoted by n ( B I ). b) only a ball, denoted by n ( B \ I ). c) an implement but not a ball, denoted by n ( I \ B ). d) either a ball or an implement, denoted by n ( I B ). e) neither a ball nor an implement, denoted by n ( ( B I )' ). 101

100 102

101 Example 3: Determine the Number of Elements in a Set by Reasoning Jamaal surveyed 34 people at his gym. He learned that 16 people do weight training three times a week, 21 people do cardio training three times a week, and 6 people train fewer than three times a week. How can Jamaal interpret his results? Example 4. Morgan surveyed the 30 students in her mathematics class about their eating habits. 18 of these students eat breakfast. 5 of the 18 students also eat a healthy lunch. 3 students do not eat breakfast and do not eat a healthy lunch. How many students eat a healthy lunch? Tyler solved this problem, as shown below, but made an error. What error did Tyler make? Determine the correct solution. 103

102 3.4 Applications of Set Theory Example 1: Use Sets to Model and Solve Problems Rachel surveyed Grade 12 students about how they communicated with friends over the previous week. 66% called on a cellphone. 76% texted. 34% used a social networking site. 56% called on a cellphone and texted. 18% called on a cellphone and used a social networking site. 19% texted and used a social networking site. 12% used all three forms of communication. a) Define an universal set and its subsets. b) What percent of students used at least one of these three forms of communication? 104

103 Example 2: Solve a Puzzle Using the Principle of Exclusion and Inclusion Use the following clues to answer the questions below: 28 children have a dog, a cat, or a bird. 13 children have a dog. 13 children have a cat. 13 children have a bird. 4 children have only a dog and a cat. 3 children have only a dog and a bird. 2 children have only a cat and a bird. No child has two of each type of pet. a) How many children have a cat, a dog, and a bird? b) How many children have only one pet? 105

104 Try. Shannon s high school starts a campaign to encourage students to use green transportation for travelling to and from school. At the end of the first semester, Shannon s class surveys the 750 students in the school to see if the campaign is working. They obtain these results: 370 students use public transit. 100 students cycle and use public transit. 80 students walk and use public transit. 35 students walk and cycle. 20 students walk, cycle, and use public transit. 445 students cycle or use public transit. 265 students walk or cycle. a) Define an universal set and its subsets. b) How many students use green transportation for travelling to and from school? 106

105 Example 4: Use Sets to Win a Game Star is playing a game that involves sets. She is using the nine cards shown, which have three different attributes: shape, colour, and number of shapes. There are three shapes (triangle, square, and circle), three colours (red, blue, and green), and three numbers of shapes (one, two, and three). To win, Star must create four sets, using three cards in each set from the nine cards shown. Each card may be used more than once in a set. What sets can Star make to win the game? 107

106 3.5 Conditional Statements and Their Converse A conditional statement is a statement that is written using if and then. A conditional statement has two parts to it: a hypothesis, p, (the part following if ) and a conclusion, q, (the part following "then"). A conditional statement, sometimes called an "if-then" proposition, may be true or false. For example, "if it is Monday, then it is a school day" or "if it is raining outside, then we practice indoors". A conditional statement has a converse, which may or may not be true. This occurs when the hypothesis and the conclusion are interchanged. For example, "if it is a school day, then it is Monday" or "if we practice indoors, then it is raining outside". Example 1: Write and Verify a Conditional Statement Write the following statement as a conditional statement "if-then" and identify the hypothesis and the conclusion. "An acute-angled triangle has three angles that are each less than 90 " Is the conditional statement true? Example 2: Determine if the Converse of a Conditional Statement is True Write the converse of the conditional statement in example 1. Is the converse true? Example 3. Is the converse of the following statement true or false? "If a number is divisible by 4, then it is an even number." Try. Consider the following conditional statement: "If x > 3, then x 2 > 9. a) If the conditional statement true? If not, give a counterexample. b) Write the converse. Is it true? If not, give a counterexample. 108

107 If a conditional statement and its converse are both true, you can combine them to create a biconditional statement using the phrase if and only if. For example, If a number is even, then it is divisible by 2 is true. The converse, If a number is divisible by 2, then it is even, is also true. The biconditional statement is A number is even if and only if it is divisible by 2. Example 4. A conditional statement and its converse are given. "If a triangle is obtuse-angled, then the triangle has one angle between 90 and 180." "If a triangle has one angle between 90 and 180, then the triangle is obtuse-angled." Verify both statements and write a bi-conditional statement. Try. Write a conditional statement and its converse for each of the following. Examine both statements and then write a bi-conditional statement, if necessary. a) A polygon with 6 sides is a hexagon. b) A quadratic function is a polynomial function of degree

108 Try. Reid stated the following biconditional statement: A quadrilateral is a square if and only if all of its sides are equal. Is Reid s biconditional statement true? Explain. 110

109 3.6 The Inverse and the Contrapositive of Conditional Statements An inverse is a statement that is formed by negating both the hypothesis and the conclusion of a conditional statement. For example, for the statement If a number is even, then it is divisible by 2, the inverse is If a number is not even, then it is not divisible by 2. You form the inverse of a conditional statement by negating the hypothesis and the conclusion. A contrapositive is a statement that is formed by negating both the hypothesis and the conclusion of the converse of a conditional statement; for example, for the statement If a number is even, then it is divisible by 2, the contrapositive is If a number is not divisible by 2, then it is not even. You form the contrapositive of a conditional statement by exchanging and negating the hypothesis and the conclusion. Conditional Statement If p then q Converse If q then p Inverse If p then q Contrapositive If q then p Example 1: Verify the Inverse and Contrapositive of a Conditional Statement Consider the following conditional statement: If today is February 29, then this year is a leap year. a) Verify the statement, or disprove it with a counterexample. b) Verify the inverse, or disprove it with a counterexample. c) Verify the contrapositive, or disprove it with a counterexample. 111

110 Try. Consider the following conditional statement: "If Peter lives in BC, then Peter lives in Vancouver." (In Canada only) a) Is the conditional statement true? b) Write the converse. Is it true? c) Write the inverse. Is it true? d) Write the contrapositive. Is it true? Example 2. Consider the following "if-then" proposition: "If z > 2, then z 2 > 4. a) Is the original proposition true? If not, give a counterexample. b) Write the converse. Is it true? If not, give a counterexample. c) Write the inverse. Is it true? If not, give a counterexample. d) Write the contrapositive. Is it true? If not, give a counterexample. 112

111 Example 3: Examine the Relationships between a Conditional Statement and Its Contrapositive Consider the following conditional statement: If a number is a multiple of 10, then it is a multiple of 5. a) Write the contrapositive of this statement. b) Verify that the conditional and contrapositive statements are both true. Example 4: Examine the Relationships between the Converse and Inverse of a Conditional Statement Arizona is studying the colour wheel in art class. She observes the following: If a colour is red, yellow, or blue, then it is a primary colour. a) Write the converse of this statement. b) Write the inverse of this statement. c) Verify that the converse and the inverse are both true. d) Is Arizona s statement biconditional? Explain. If a conditional statement is true, then its contrapositive is true, and vice versa. If the inverse of a conditional statement is true, then the converse of the statement is also true, and vice versa. 113

112 4.1 Counting Principles Select a Strategy to Solve a Counting Problem Example 1: Hannah plays on her school soccer team. The soccer uniform has: three different sweaters: red, white, and black, and three different shorts: red, white, and black. How many different variations of the soccer uniform can the coach choose from for each game? Make a tree diagram. Try. A toy manufacturer makes a wooden toy in three parts. Determine how many different coloured toys can be produced? Make a tree diagram. Part 1: the top part may be coloured red or blue Part 2: the middle part may be orange, white or black Part 3: the bottom part may be yellow or green Fundamental Counting Principle If there are a ways to perform one task and b ways to perform another, then there are a b ways of performing both. Consider a task made up of several stages. The fundamental counting principle states that if the number of choices for the first stage is a, the number of choices for the second stage is b, the number of choices for the third stage is c, etc... then the number of ways in which the task can be completed is a b c

113 Solve a Counting Problem by Extending the Fundamental Counting Principle Example 2: A luggage lock opens with the correct three-digit code. Each wheel rotates through the digits 0 to 9. a) How many different three-digit codes are possible? b) Suppose each digit can be used only once in a code. How many different codes are possible when repetition is not allowed? Example 3. A manufacturer uses a six-character serial number for a line of products. The first and second characters are upper-case letters (A to Z). The third, fourth, and fifth characters are digits (0 to 9). There are only three choices for the last position: A, B, and X. a) How many different serial numbers are possible, if repetition of characters is allowed? b) How many different serial numbers are possible, if no repetition is allowed? Try. A vehicle license plate consists of 3 digits followed by 3 letters. How many license plates are possible if a) there are no restrictions on the letters or digits used? b) no letter can be repeated? c) the first digit must not be zero and no digits can be repeated? 115

114 Example 4. How many ways are there of getting from A to C in each diagram, passing through each point at most once? a) b) Try. How many ways are there of getting from A to C in the diagram, passing through each point at most once? 116

115 The Fundamental Counting Principle applies when tasks are related by the word AND. It does not apply when the tasks are related by the word OR. In the case of OR situation, n ( A B ) = n ( A ) + n ( B ) if A and B are mutually exclusive (disjoint sets). n ( C D ) = n ( C ) + n ( D ) n ( C D ) if C and D are not mutually exclusive (not disjoint). Solve a Counting Problem when the Fundamental Counting Principle Does Not Apply Example 5: A standard deck of cards contains 52 cards as shown. Count the number of possibilities of drawing a single card and getting: a) either a red face card OR an ace n ( RF A ) b) either a club OR a two n ( C 2 ) 117

116 4.2 Introducing Permutations and Factorial Notation Permutation is an arrangement of distinguishable objects in a definite order. For example, the objects a and b have two permutations, ab and ba. Solve a Counting Problem Where Order Matters Example 1: Determine the number of arrangements that 4 children can form while lining up to go to the washroom. Example 2. When you press the shuffle button on an i-pod, it plays a list of the songs (all songs will be played only once). If the i-pod has 6 songs on it, how many playlists of the songs are possible? Try. How many different ways can 5 different books, Math, Chemistry, Physics, English and Biology be arranged on a shelf? Factorial Notation! 1! = 2! = 3! = 4! = 5! = A concise representation of the product of consecutive descending natural numbers: n! = (n +1)! = (n 1)! = 6! = 118

117 In the expression n!, the variable n is defined only for values that belong to the set of whole numbers; that is, n { 0, 1, 2, 3,...}. Please note that 0! is defined to be 1. Example 3: Evaluate Numerical Expressions Involving Factorial Notation a) 10! b) 12! 9!3! Simplify an Algebraic Expression Involving Factorial Notation Example 4: Simplify each expression, where n N. a) ( n + 3 )( n + 2 )! b) ( n + 1 )! ( n 1 )! Example 5. Write each expression without using the factorial symbol. a) ( n + 2 )! b) ( n 3 )! n! n! Try. Evaluate each without using the factorial buttom on the calculator. a) 43! b) 37! 40! 33!4! 119

118 Solve an Equation Involving Factorial Notation Example 5: Solve for n. a)!! 90, where n I b)!! 42, where n I Try. Solve!! 12, where n I 120

119 4.3 Permutations When All Objects Are Distinguishable Solve a Permutation Problem Where Only Some of the Objects are Used in Each Arrangement Example 1: How many permutations can be formed using all the letters of the word CLARINET? Example 2. How many 3-letter arrangements can be made from the letters of the word CLARINET? The number of permutations of " n " different objects taken " r " at a time is npr!, 0. If all " n " objects are used, then is " n " equal to " r " and! the number of permutations is npr = n! Example 3. Matt has downloaded 10 new songs from an online music store. He wants to create a playlist using 6 of these songs arranged in any order. How many different 6-song playlists can be created from his new downloaded songs? Try. Calculate the number of ways that a committee of 3 people (president, vice-president and secretary) can be selected from a group of 20 people. 121

120 If order matters in a counting problem, then the problem involves permutations. To determine all possible permutations, use the formula for npn or npr, depending on whether all or some of the objects are used in each arrangement. Both of these formulas are based on the Fundamental Counting Principle. If a counting problem has one or more conditions that must be met, consider each case that each condition creates first, then add the number of ways each case can occur to determine the total number of outcomes. Solve a Permutation Problem Involving Cases Example 4: How many numbers (1-digit, 2-digit, 3-digit or 4-digit numbers) can be formed using the number cards 2, 3, 4, and 5 if none of them can be repeated? Try. An online store allows its users to register with passwords with a minimum of 5 characters and a maximum of 7 characters. The passwords can use any digits from 0 to 9 and/or any letters of the alphabet. The passwords are case sensitive. How many different passwords are possible if none of the characters can be repeated? 122

121 Solve a Permutation Problem with Conditions Example 5: How many arrangement of the following words can be made if all the vowels must be kept together? a) FATHER b) DAUGHTER c) EQUATION Example 6. In how many ways can four adults and five children be arranged in a single line a) if the adults are all together and the children are all together? b) if the adults are all together? The number of permutations that can be created from a set of " n " objects, using " r " objects in each arrangement, where repetition is allowed and r n, is n r. Example 7. A social insurance number (SIN) in Canada consists of a nine-digit number that uses the digits 0 to 9. Compare the number of possible social insurance numbers with and without repetitions. 123

122 4.4 Permutations When All Objects Are Identical Solve a Permutation Problem Where Objects are Alike Example 1. Three cans are to be put on a shelf. a) List all permutations. b) If the Red Bull is replaced by another Pepsi, list all permutations. The number of permutations of "n"objects, where "a" are identical, another "b" are identical, another "c" are identical, and so on, is!!!! Example 2. Beck bought a carton containing 6 mini boxes of cereal. There are 3 boxes of Cheerios, 2 boxes of Fruit Loops, and 1 box of Mini-Wheats. Over a six day period, Beck plans to eat the contents of one box of cereal each morning. How many different orders are possible? 124

123 Try. Naval signals are made by arranging coloured flags in a vertical line and the flags are then read from top to bottom. How many signals using six flags can be made if you have a) 3 red, 1 green, and 2 blue flags b) 2 red, 2 green, and 2 blue flags. Example 3. Determine the number of permutations of all the letters in each of the following words. a) OGOPOGO b) STATISTICIAN Try. Find the number of permutations of the letters of the word. a) VANCOUVER b) MATHEMATICAL Solve a Conditional Permutation Problem Involving Identical Objects Example 4. How many ways can the letters of word CANADA be arranged, if the first letter must be N and the last letter must be C? Try. Tina is playing with a tub of building blocks. The tub contains 3 red blocks, 5 blue blocks, 2 yellow blocks, and 4 green blocks. How many different ways can Tina stack the block in a single tower, if there must be a yellow block at the bottom of the tower and a yellow block at the top. 125

124 Solve a Permutation Problem Involving Routes Example 5. Julie's home is two blocks north and three blocks west of her school. How many routes can Julie take from home to school if she always travels either south or east? home school home School Try. On the following grid, how many different paths can A take to B, assuming one can only travel east and south? Explain. A B 126

125 Example 6. A supervisor of the city bus department is determining how many routes there are from the bus station to the concert hall. Determine the number of routes possible if the bus must always move closer to the concert hall. Bus Station Concert Hall Example 7. A taxi company is trying to find the quickest route during rush hour traffic from the train station to the football stadium. How many different routes must be considered if at each intersection the taxi must always move closer to the football stadium? Train Station Football Stadium Try. Find the number of pathways from A to B if paths must always move closer to B. a) b) A A B B 127

126 4.5 & 4.6 Exploring Combinations Explore Combinations Example 1. If 5 sprinters compete in a race, how many different ways can the medals for first, second, and third place, be awarded? Example 2. Five sprinters are to compete in a race. If the fastest 3 qualify for relay team, how many different relay teams can be formed. A permutation is an arrangement of elements in which the order of the arrangement is taken into account. A combination is a selection of element in which the order of selection is NOT taken into account. Solve a Simple Combination Problem Example 3. Three students from a class of 10 are to be chosen to go on a school trip. In how many ways can they be selected? Combinations of "n" different elements taken "r" at a time is ncr! can also be written as! where 0.!!!! Try. There are 16 students in a class. Determine the number of ways in which four students can be chosen to complete a survey. 128

127 Example 4. To win the LOTTO 649 a person must correctly select six numbers between 1 to 49, in no particular order. Jack selected the six numbers from the birth dates of his family, How many different selections of number could he have made? Try. A restaurant serves 10 flavours of ice cream. Danielle has ordered a large sundae with three scoops of ice cream. How many different ice-cream combinations does Danielle have to choose from, if she wants each scoop to be a different flavour? Solve a Combination Problem Using the Fundamental Counting Principle Example 5. The Athletic Council decides to form a sub-committee of 7 council members to look at how funds raised should be spent on sport activities in the school. There are a total of 15 athletic council members, 9 males and 6 females. The sub-committee must consist of exactly 3 females. Determine the number of ways of selecting the sub-committee. 129

128 Try. A basketball coach has 5 guards and 7 forwards on his basketball team. In how many different ways can he select a starting team of two guards and three forwards? Solve Combination Problems with "At Least" or "At Most" Cases Example 6. A planning committee is to be formed for a school-wide Earth Day program. There are 13 volunteers: 8 teachers and 5 students. How many ways can the principal choose a 4-person committee that has at least 3 teachers? Example 7. City Council decides to form a 5 people sub-committee to investigate transportation concerns. There are 8 males and 7 females. How many different ways can the sub-committee be formed consisting of at most one female member? Try. An all-night showing at a movie theatre is to consist of five movies. There are fourteen different movies available, ten disaster movies and four horror movies. How many possible schedule include at least four disaster movies? 130

129 4.7 Solving Counting Problems When solving counting problems, you need to determine if order plays a role in the situation. Once this is established, you can use the appropriate permutation or combination formula. Solve a Permutation Problem with Conditions Example 1. A piano teacher and her students are having a group photograph taken. There are three boys and five girls. The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many ways can the students and teacher sit in a row of nine chairs for this pose? Example 2. In how many ways can all of the letters of the word ORANGES be arranged if: a) there are no restrictions? b) the first letter must be an N? c) the vowels must be together in the order O, A, and E? Try. In how many of the arrangements of the letters of the word BRAINS are the vowels together? 131

130 Solve a Combination Problem Involving Multiple Choices Example 3. A standard deck of 52 playing cards consists of 4 suits (spades, hearts, diamonds, and clubs) of 13 cards each. a) How many different 5-card hands can be formed? b) How many different 5-card hands can be formed that consist of all hearts? c) How many different 5-card hands can be formed that consist of all face cards? d) How many different 5-card hands can be formed that consist of 3 hearts and 2 spades? e) How many different 5-card hands can be formed that consist of exactly 3 hearts? Try. A group of 4 journalists is to be chosen to cover a murder trial. There are 5 male and 7 female journalists available. How many possible groups can be formed consisting of exactly 2 men and 2 women? 132

131 Solve a Combination Problem Involving Cases Example 4. The Student Council decides to form a sub-committee of five council members to look at how funds raised should be spent on the students of the school. There are a total of 11 student council members, 5 males and 6 females. How many different ways can the sub-committee consist of at least three females? Example 5. Consider a standard deck of 52 cards. How many different five card hands can be formed containing a) at least 1 red card? b) at most 2 kings? Try. The Athletic Council decides to form a sub-committee of 6 council members to look at a new sports program. There are a total of 15 student council members, 6 females and 9 males. How many different ways can the sub-committee consist of at most one male? 133

132 5.1 Exploring Probability Probability theory deals with the mathematics of chance or prediction. Trial A trial is any operation whose outcome cannot be predicted with certainty. e.g. a coin is tossed, a die is rolled. Experiment An experiment consists of one or more trials. e.g. a coin is tossed, two dice are rolled, a coin is tossed and a die is rolled. Outcome An outcome is the result of the carrying out an experiment. e.g. H, 6 and 4, T and 4. Sample Space The sample space (S) of an experiment is the set of all possible outcomes. e.g. {H,T}, {(1,1),(1,2),...,(6,6)}, {(H,1),(H,2),...,(T,5),(T,6)} Event An event is a subset of the sample space. It consists of one or more of the possible outcomes of an experiment. e.g. {H}, {(1,4)}, {(T,2)} Probability To denote the probability of an event A, we write P (A), which is read as "the probability of A ". Probabilities are usually represented by decimals or fractions between 0 and 1, but sometimes percentages are used. For any event A, 0 P (A ) 1. If the event X does not include any of the outcomes in the sample space, then the event X is impossible and we write P (X ) = 0. e.g. P(rolling a 9 on a standard die ) = 0 If the event Y includes all the outcomes in the sample space, then the event Y is certain and P (Y ) = 1. e.g. P(rolling a natural number less than 7 on a standard die ) = 1 Complimentary Events The probability of an event and the probability of the complement will always add to 1 (100%). P (A ) + P (A ' ) = 1 134

133 The theoretical probability of an event A is determined from the formula # of outcomes favourable to total # of outcomes in the sample space Determine the Theoretical Probability Example 1. Four black counters and five white counters are placed in a bag. One of the counters is selected at random. a) State the probability that the counter selected is black ( B ). b) State the probability that the counter selected is not black ( B'). The experimental probability of an event A consisting of multiple trials is determined from the formula # of times event occurs total # of trials Determine the Experimental Probability Example 2. Simulate rolling a die 60 times and recording the number of times a 5 appears. From your simulation, what is the experimental probability of getting a 5? 135

134 Fair Game A game in which all the players are equally likely to win; for example, tossing a coin to get heads or tails is a fair game. Determine the Fair Game Example 3. Sasha and Mika have to invent a fair game for a class project. Sasha suggests this game: Two people play. Each player has a container. Both players put three identical slips of paper, numbered 1, 2, and 3, into their own container. For each turn, both players draw one slip of paper from their container. o Player 1 scores a point if the product of the two numbers drawn is less than their sum. o Player 2 scores a point if the product of the two numbers drawn is greater than their sum. Neither player gets a point if the product and sum are equal. After each turn, the players return their slip of paper to their container. A game consists of 10 turns. Is Sasha s game a fair game? 136

135 5.2 Probability and Odds Odds in Favour The ratio of the probability that an event will occur to the probability that the event will not occur, or the ratio of the number of favourable outcomes to the number of unfavourable outcomes. The odds in favour of event A occurring are given by the ratio P (A) : P (A' ). Odds Against The ratio of the probability that an event will not occur to the probability that the event will occur, or the ratio of the number of unfavourable outcomes to the number of favourable outcomes. The odds against event A occurring are given by the ratio P (A) : P (A '). Determine the Odds Using Sets Example 1. Bailey holds all the hearts from a standard deck of 52 playing cards. He asks Morgan to choose a single card without looking. a) Determine the odds in favour of Morgan choosing a face card. b) Determine the odds against Morgan drawing a face card. P (A ') is the probability of the complement of A, where P (A ') = 1 P (A ). Example 2. Research shows that the probability of an expectant mother, selected at random, having twins is 1/32. a) What are the odds in favour of an expectant mother having twins? b) What are the odds against an expectant mother having twins? 137

136 If the odds in favour of event A occurring are m : n:, then Determine the Probability from Odds Example 3. A computer randomly selects a university student s name from the university database to award a $100 gift certificate for the bookstore. The odds against the selected student being male are 57 : 43. Determine the probability that the randomly selected university student will be male. Make a Decision Based On Odds and Probability Example 4. A hockey game has ended in a tie after a 5 min overtime period, so the winner will be decided by a shootout. The coach must decide whether Ellen or Brittany should go first in the shootout. The coach would prefer to use her best scorer first, so she will base her decision on the players shootout records. Who should go first? Interpret Odds Against and Make a Decision Example 5. A group of Grade 12 students are holding a charity carnival to support a local animal shelter. The students have created a dice game that they call Bim and a card game that they call Zap. The odds against winning Bim are 5 : 2, and the odds against winning Zap are 7 : 3. Which game should Madison play? 138

137 5.3 Probabilities Using Counting Methods You may be able to use the Fundamental Counting Principle and techniques involving permutations (npr) and combinations (ncr) to solve probability problems with many possible outcomes. The context of each particular problem will determine which counting techniques you will use. Use permutations when order is important in the outcomes. Use combinations when order is not important in the outcomes. # of outcomes favourable to total # of outcomes in the sample space Solve a Probability Problem Using Counting Techniques Example 1. Two cards are drawn without replacement from a well shuffled deck of 52 cards. Calculate the probability that the two cards drawn are both aces, to the nearest tenth of a percent. Example 2. A bag of marbles contains 5 red, 3 green and 6 blue marbles. If a child grabs 3 marbles from the bag, determine the probability that exactly 2 are blue, to the nearest tenth of a percent. Try. Two seeds are chosen from a packet containing 10 seeds; 3 that will produce red flowers, 4 that will produce white flowers and 3 that will produce blue flowers. What is the probability that both seeds produce red flowers, to the nearest tenth of a percent? 139

138 Example 3. A game, called Euchre, is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but with only the ace (A), 9, 10, jack (J), queen (Q), king (K) for all four suits. Each player is dealt 5 cards. Determine the probability, to the nearest tenth of a percent, that a dealt hand will contain the following: a) A, K, Q, J of the same suit, plus 1 other card b) Five cards of the same colour c) Four cards of the same rank, plus 1 other card Example 4. Three prizes are awarded in a raffle during a halftime show at a school basketball game. Ben, Janelle, James and 17 other students each have one ticket. If the raffle has three identical prizes, determine the probability that Ben, Janelle, and James win the prizes. Try. Three people are randomly chosen from a fellowship of 14 people to be president, treasurer, and secretary, in any order. What is the probability that Frank, Sam and Aaron will be ones chosen? 140