Do work on a separate sheet of paper. Do work on a separate sheet of paper.

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1 Algebra 2H- WS 0.X Recursion- Loans Do work on a separate sheet of paper.. Find the first month s interest on a $32,000 loan at an annual interest rate of a. 4.9% b. 5.9% c. 6.9% d. 7.9% 2. Write a recursive formula for each financial situation. a. You borrow $0,000 at an annual interest rate of 0%, compounded monthly, and each payment is $300. b. You buy $7000 worth of furniture on a credit card with an annual interest rate of 8.75%, compounded monthly. You plan to pay $250 each month. 3. Cici purchased $2000 worth of merchandise with her credit card this past month. Then she was unexpectedly laid off from her job. She decided to make no more purchases with the card and to make only the minimum payment of $40 each month. Her annual interest rate is 8%, compounded monthly. a. Find the balance on the credit card over the next six months. b. When will Cici pay off the total balance on her credit card? c. What is the total amount paid for the $2000 worth of merchandise? 4. Megan Flanigan is a loan officer with L. B.Mortgage Company. She offers a loan of $60,000 to a borrower at 9.6% annual interest, compounded monthly. a. What should she tell the borrower the monthly payment will be if the loan must be paid off in 25 years? (Note: This does require guess and check.) b. Make a graph that shows the unpaid balance over time. 5. Is this sequence arithmetic, geometric, shifted geometric, or something else?,,,,, Consider the geometric sequence generated by u0 4 and un 0.7un where n a. What is the long-run value? b. What is the long-run value if the common ratio is changed to.3? c. What is the long-run value if the common ratio is changed to? Algebra 2H- WS 0.X Recursion- Investments Do work on a separate sheet of paper.. Assume the sequence generated by u0 450 and un.039un 50 where n represents a financial situation and n is measured in years. a. Is this a loan or an investment? Explain your reasoning. b. What is the principal? c. What is the deposit or payment amount? d. What is the annual interest rate? e. What is the frequency with which interest is compounded? 2. Write a recursive formula for each financial situation. a. You invest $8000 at 6%, compounded quarterly, and you deposit $500 every three months. b. You enroll in an investment plan that deducts $00 from your monthly paycheck and deposits it into an account with an annual interest rate of 7%, compounded monthly. 3. Find the balance after 5 years if $500 is deposited into an account with an annual interest rate of 3.25%, compounded monthly. 4. Consider a $000 investment at an annual interest rate of 6.5%, compounded quarterly. Find the balance after a. 0 years b. 20 years c. 30 years 5. Find the balance of a $000 investment, after 0 years, at an annual interest rate of 6.5% when compounded a. Annually b. Monthly c. Daily (In financial practice, daily means 360 times per year, not 365.) d. After the same length of time, how will the balances compare from investments that are compounded annually, monthly, or daily? 6. Beau and Shaleah each get a $000 bonus at work and decide to invest it. Beau puts his money into an account that earns an annual interest rate of 6.5%, compounded yearly. He also decides to deposit $200 each year. Shaleah finds an account that earns 6.5%, compounded monthly, and decides to deposit $00 each month. a. Compare the amounts of money that Beau and Shaleah deposit each year. Describe any differences or similarities. b. Compare the balances of Beau s and Shaleah s accounts over several years. Describe any differences or similarities.

2 Algebra 2H- WS 0.X Recursion- Review Do work on a separate sheet of paper.. Consider this sequence: 256, 92, 44, 08,... a. Is this sequence arithmetic or geometric? b. Write a recursive formula that generates the sequence. Use u for the starting term. c. What is the 8 th term? d. Which term is the first to have a value less than 20? e. Find u Consider this sequence: 3, 7,, 5,... a. Is this sequence arithmetic or geometric? b. Write a recursive formula that generates the sequence. Use u for the starting term. c. What is the 28 th term? d. Which term has the value 59? e. Find u A large barrel contains 2.4 gal of oil 8 min after its drain is opened. How many gallons of oil were in the barrel to start, if it drains at a rate of 4.2 gal/min? 6. The enrollment at a college is currently From now on, the board of administrators estimates that each year the school will graduate 24% of its students and admit 250 new students. What will the enrollment be during the sixth year? What will the enrollment be in the long run? Sketch a graph of the enrollment over 5 years. 7. You deposit $500 into a bank account that has an annual interest rate of 5.5%, compounded quarterly. a. How much money will you have after 5 yr if you never deposit more money? b. How much money will you have after 5 yr if you deposit an additional $50 every 3 mo after the initial $500? 8. This table gives the consumer price index (CPI) for medical care from 970 to Use a graph to find a sequence model that approximately fits these data. U.S. Consumer Price Index for Medical Care YEAR CPI List the first five terms of each sequence. For each set of terms, what minimum and maximum values of n and u n would you use on the axes to make a good graph? u 3 u 2 a. b. u u.5 where n 2 u 3u 2 where n 2 n n 4. Match each recursive formula with the graph of the same sequence. Give your reason for each choice. u0 5 u0 5 a. b. un un where n un 0.5un where n u0 5 u0 5 c. d. u 0.5 u where n u u where n n n n n n n

3 Algebra 2H- WS 04.X Properties of Logarithms Do work on a separate sheet of paper.. Change the form of each expression below using properties of logarithms or exponents. a. logs logt b. log h k c. log g b d. 2. Determine whether each equation is true or false. If false, rewrite one side of the equation to make it true. a. log3 log7 log2 b. log5 log3 log8 c. log6 4log2 d. log5 log2 log2.5 e. log9 log3 log6 f. n k m 7 log 7 log 2 g. log35 5log7 h. j. log64.5log6 log log4 4 i. log3 3 log log Draw the graph of a function whose inverse is not a function. Carefully describe what must be true about the graph of a function if its inverse is not a function. 4. Find an equation to fit the set of data. x y Write each expression in terms of logm and logn logm 6. log 9. N 2 3 logmn log M N 3 2 log M 0. logm N Write each expression as a single logarithm.. log2 3log7 2. log36 2log35 log log47 log logx logx log52 log5 49 log log6t 8log6u 5log6v Write the following as a single logarithm and give its numerical value. 7. log336 log log25 log8 9. log52 log560 Let x log3 2 and y log3 0. Write each expression in terms of x and y. 20. log log3 22. log

4 Algebra 2H- WS 04.X Properties of Logarithms 2 Do work on a separate sheet of paper.. The half-life of carbon-4, which is used in dating archaeological finds, is 5730 yrs. a. Assume that 00% of the carbon-4 is present at time 0 yr. Write the equation that expresses the percentage of carbon-4 remaining as a function of time. b. Suppose some bone fragments have 25% of their carbon-4 remaining. What is the approximate age of the bones? c. In the movie Raiders of the Lost Ark (98), a piece of the Ark of the Covenant found by Indiana Jones contained 62.45% of its carbon-4.what year would this indicate that the ark was constructed in? d. Coal is formed from trees that lived about 00 million years ago. Could carbon-4 dating be used to determine the age of a lump of coal? Explain your answer. 2. Use the properties of logarithms and exponents to solve these equations. a. 5. x 247 b x 30 x x c. d. 6. Prove that these statements of equality are true. Take the logarithm of both sides, then use the properties of logarithms to re-express each side until you have two identical expressions. np n p a b. 0 0 d e 0 7. Solve each equation. Check your answers by substituting your answer for x. a x b x c x d x de e x f x 8. Suppose you invest $3000 at 6.75% annual interest compounded monthly. How long will it take to triple your money? 9. Find an exponential function that passes through (4, 8) and (0, 44). 0. Solve each equation. Round to the nearest hundredth. 5 a. x 348 b. x c x 3. Carbon- decays at a rate of 3.5% per minute. Assume that 00% is present at time 0 min. a. What percentage remains after min? b. Write the equation that expresses the percentage of carbon- remaining as a function of time. c. What is the half-life of carbon-? d. Explain why carbon- is not used for dating archaeological finds. 4. Describe how each function has been transformed from the parent function y or y log x. Then graph the function. a. y 4 32 x x b. y 2log 3 2 x 5. A driver charges $4 per hour plus $20 for chauffeuring if a client books directly with her. If a client books her through an agency, the agency charges 5% of what the driver charges plus $25. a. What percentage remains after min? Write a function to model the cost of hiring the driver directly. Identify the domain and range. b. Write a function to model what the agency charges. Identify the domain and range. c. Give a single function that you can use to calculate the cost of using an agency to hire the driver for h hours.

5 Algebra 2H- WS 04.X Real World Logs Do all work neatly, showing all work, on a separate sheet of paper.. Hazel wants to invest $2000 for her newly born grandson. At what interest rate must she invest to end up with $0,000 on the lad's 2 st birthday? Assume continuous compounding and round your answer to the nearest hundredth of a percent. 2. Hazel has a granddaughter who is 7 years old. She wants to present the lass with a gift of $0,000 on the child's 2 st birthday. a. If the highest interest rate she can find is % compounded quarterly, how much must she invest now? b. Answer the same question for an interest rate of % compounded continuously. 3. It is commonly said that the human population of the world is doubling every 35 rt years. Find the value for r in the growth formula y Pe where t is measured in years. Round your answer to the nearest thousandth. 4. A piece of machinery is worth $35,000 depreciates 20% per year by the fixed rate method. After how many years will the value have depreciated to $5000? Round your answer to the nearest tenth of a year. 5. In 970, a grand piano cost $6300. In 984, the same model cost $,00. Assuming a steady rate of increase, what was the yearly rate of inflation? Round your answer to the nearest tenth of a percent. 6. In living matter, the proportion of Carbon-4 remains constant. When the matter dies, the Carbon-4 decays radioactively so that only half of the Carbon-4 atoms are left after 5570 years. rt a. Find the constant r for the decay formula: y Pe b. Charcoal from an ancient fire was found to have one-fifteenth of the Carbon-4 that a living sample of the same size has. How old was the charcoal, to the nearest 0 years. 7. What interest rate compounded annually is equivalent to 2% compounded continuously? n% compounded continuously? 8. A toxic chemical has been mistakenly introduced into a lake which is a city's source of drinking water. The level of toxicity is 2 times the safe level. Through natural flushing action of the lake, the level of toxicity will be reduced by 30% each week. In how many weeks will it be safe to use the water again? Numbers 9 and 0 are an Algebra review and do not require exponents. 9. Find two numbers whose difference is 3 and whose product is a minimum. 0. Two candles are the same length. One burns up in 8 hours and the other in 2 hours. If they are both lit at the same time, how long is it before one is twice as long as the other? Algebra 2H- WS 04.X Logarithmic Word Problems Show a complete calculator set-up for each of the following.. One thousand dollars is invested at 2% interest compounded annually. Determine how much the investment is worth after: a) year b) 3 years 2. One thousand dollars is invested at 2% annual interest for three years. Determine how much the investment is worth if the interest is compounded: a) semi-annually b) quarterly c) daily 3. The value of a $2,500 used car decreases 20% per year. Find its value after: a) year b) 3 years 4. The value of a $3500 sailboat depreciates 0% per year. Find its value after: a) year b) 0 years 5. How long will it take you to double your money if you invest $000 at 8% compounded annually? 6. How long will it take you to triple your money if you invest $4000 at 6% compounded annually? 7. A gold coin appreciated in value from $00 to $238 in eight years. Find the average annual rate of appreciation. 8. Ten years ago Michael paid $250 for a rare 823 stamp. Its current value is $000. Find the average annual rate of growth. 9. A used car valued at $2,000 decreased in value to $4900 in 5 years. Find the annual rate of depreciation. 0. A certain radioactive element decays over time according to the equation t 300 y A 2 where A= the number of grams present initially and t=time in years. If 000 grams were present initially a) How many grams are present after 900 years? b) How long will it take for there to be 00 grams remaining?. Bacteria in a culture are growing exponentially with time according to the table shown. a) Write an equation to model the number or bacteria present at any time t. b) How may bacteria a there after 8 hours? c) How long will it take for there to be 00,000 bacteria present? ANSWERS a) $20.00 b) $ a) $ b) $ c) $ a) $0,000 3b) $6400 4a) $350 4b) $ ) 9.0 years 6) 8.9 years 7).4% 8) 4.9% 9) 6.4% Bacteria Growth Hour Bacteria a) 25 grams 0b) years a) y 60(2) t b) 5,360 bacteria c) 0.7 hours

6 Algebra 2H- WS Sequences and Series Arithmetic Sequences and Series ' ' ! Fill in the blanks above and to the right with the letter that represents the correct answer. A = 3 B = 40 C = 2 D = -38 E = -59 F = 20,00 G = 2.5 H = 7 I = 9000 J = 3200 K = 52 L = 203 M = 2250 N = O = 8,500 P = -25 Q = 4 R = 3 S = -4 T = 8 U = -57 V = -3 W = -203 X = 400 Y = -67 Z = -2 Show all work with numerical substitutions on a separate sheet of paper.. In 5, 3,,..., find a In, 3,,..., find a In 2, 0, 2,..., find a In 5,, 4,..., find a In, 2, 3, 4, 5,..., find S In 4, 2, 0,..., find S a 3, d 2, find a a 8, a5 4, find a a7 3, d 4, find a. 0. a0 2, S0 20, find d.. Find the arithmetic mean of 70 and If six arithmetic means are inserted between -4 and 45, what is the value of the first one inserted? 3. If seven arithmetic means are inserted between -8 and 38, what is the value of the 7 th one inserted? 4. The first year a politician is in office she saves $500. Each succeeding year, she saves $300 more than the year before. How much has she saved (in total) after her 0 th year in office? 5. A secretary is employed at a starting salary of $6000 and promised annual raises of $600. What will his annual salary be for the 6 th year of employment? 6. Find the arithmetic mean of -38 and a5 6, a8 3, find a. 8. a4, a2 7, find a 33.

7 Algebra 2H- WS Sequences and Series Infinite Geometric Series. Consider the repeating decimal or 0.4 a. Express this decimal as the sum of terms of an infinite geometric series. b. Identify the first term and the ratio. c. Use the formula you learned in this lesson to express the sum as a ratio of integers. 2. Repeat the three parts of Exercise with the repeating decimal or An infinite geometric sequence contains the consecutive terms 28, 32, 8, and 2. The sum of the series is 43, What is the first term? 7. A computer software company decides to set aside $00,000 to develop a new video game. It estimates that development will cost $955 the first week and that expenses will increase by $65 each week.. a. After 25 weeks, how much of the development budget will be left? b. How long can the company keep the development phase going before the budget will not support another week of expenses? 8. Suppose square ABCD with side length 8 in. is cut out of paper. Another square, EFGH, is placed with its corners at the midpoints of ABCD. A third square is placed with its corners at midpoints of EFGH, and so on. a. What is the perimeter of the tenth square? b. What is the area of the tenth square? c. If the pattern were repeated infinitely many times, what would be the sum of the perimeters of the squares? d. What would be the sum of the areas? 4. Consider the sequence u n n a. List the first ten terms, u to u 0. 0 b. Find the sum n n c. Make a graph of partial sums for n n d. Find the sum n 5. A ball is dropped from an initial height of 00 cm. The rebound heights to the nearest centimeter are 80, 64, 5, 4, and so on. What is the total distance the ball will travel, both up and down? 6. A sporting event is to be held at the Superdome in New Orleans, Louisiana, which holds about 95,000 people. Suppose 50,000 visitors arrive in New Orleans and spend $500 each. In the month after the event, the people in New Orleans spend 60% of the income from the visitors. The next month, 60% is spent again, and so on. a. What is the initial amount the visitors spent? b. In the long run, how much money does this sporting event seem to add to the New Orleans economy? c. The ratio of the long-run amount to the initial amount is called the economic multiplier. What is the economic multiplier in this example? d. If the initial amount spent by visitors is $0,000,000 and the economic multiplier is.8, what percentage of the initial amount is spent again and again in the local economy?

8 Algebra 2H- WS Sequences and Series Geometric Sequences and Series 0 3) a 5, r, find a. 4) S 4 40, r 3, find a ) Find the negative geometric mean between 2 and 27. 6) Find the positive geometric between 2 and ) If 3 geometric means are inserted between and 4, what is 64 the value of the middle one inserted. 2 8) a 500, a 5, and r 0, find r ) a 63, a 5, and r 0, find a ) a 300, r 0.3,, find S 4. Connect each problem with its answer. Show all work, including formulas with substitutions, neatly on a separate sheet of paper. ) In 896, 448, 224,... find a 6. 2) In 4, 4, 4,... find a ) In 3 3 3,,, ) In 7 5,5 6,5 5,... 7) In , 23, 23,... find a 6. 4) In 29, 29, 29,... find a 3. find a 0. 6) In 208, 04, 52,... find a 5. find a 2. 8) In 3, 6, 2,... find S 6. 9) In,,,... find S 7. 0) a 8, r, find a ) a, a 6 64, find a 4. 2) a 3 2, a 6 324, find a. 6 For problems 2-25, supply the next term. 2) 25, 5,, 22) 4, 4, 4, ) 32, 6, 8, 24) 0240,280, 60, 25),,, ) You deposit cent in an antique bank on March 3 rd. Every day thereafter you double the amount deposited. How many cents are deposited on March 0 th? 27) Referring to problem 26, how does the March 0 th deposit compare to the March 2 th deposit? 28) You have $480 hidden under the mattress. On June 3 th you spend half the amount and decide to spend half the remaining amount each day thereafter. How many dollars do you spend on June 7 th? 29) Referring to problem 28, take your June 9 th "mattress allotment" and add it to the $5.25 already in your wallet. Your wallet now contains how many dollars? 30) What is a 204 in 43, 43, 43,...

9 Algebra 2H- WS Sequences and Series Review. Consider the geometric sequence 256, 92, 44, 08,... a. What is the 8 th term? b. Which term is the first one smaller than 20? c. Find u 7. d. Find S 7. u Find each partial sum of this sequence: u 0.6u where n 2 n n a. S 5 b. S 5 c. S Identify the first term and the common ratio or common difference of each series. Then find the partial sum. a b n n c. d. n n 8. An Indian folktale, recounted by Arab historian and geographer Ahmad al- Yaqubi in the 9th century, begins, It is related by the wise men of India that when Husiya, the daughter of Balhait, was queen..., and goes on to tell how the game of chess was invented. The queen was so delighted with the game that she told the inventor, Ask what you will. The inventor asked for one grain of wheat on the first square of the chessboard, two grains on the second, four grains on the third, and so on, so that each square contained twice the number of grains as on the square before. (There are 64 squares on a chessboard.) a. How many grains are needed i. For the 8th square? ii. For the 64th square? iii. For the first row? iv. To fill the board b. In sigma notation, write the series you used to fill the board. 4. Find the missing value in each set of numbers. u 3, r 2, S a. 0 b. u r S 4, 0.6, c. u r S5 d. u r S8,.4, ,, Find the nearest integer value of n if n is approximately Suppose you begin a job with an annual salary of $7,500. Each year, you can expect a 4.2% raise. a. What is your salary in the tenth year after you start the job? b. What is the total amount you earn in ten years? c. How long must you work at this job before your total earnings exceed $ million? 7. As a contest winner, you are given the choice of two prizes. The first choice awards $000 the first hour, $2000 the second hour, $3000 the third hour, and so on. For one entire year, you will be given $000 more each hour than you were given during the previous hour. The second choice awards the first week, 2 the second week, 4 the third week, and so on. For one entire year, you will be given double the amount you received during the previous week. Which of the two prizes will be more profitable, and by how much?

10 Algebra 2H- WS 08.X Random Variables and Expected Value Do all work neatly, showing all work, on a separate sheet of paper.. Previous versions of the SAT would penalize students for incorrect answers. Each question has five choices (A E). You decide to roll a six-sided die and mark the answer according to the number on the die and leave the answer blank if the die is a 6. Each question is scored one point for a right answer, minus one-quarter point for a wrong answer, and no points for a question left blank. a. What is the expected value for each question? b. What is the expected value for a 30-question test? 2. Current versions of the SAT do not penalize students for an incorrect answer. There are still five answer choices per question with the option to leave an answer blank. A correct answer is worth one-point, an incorrect answer is worth zero-points, and leaving a question blank is worth zero-points. a. What is the expected value for each question? b. What is the expected value for a 30-question test? 3. In a concert hall, 6% of seats are in section A, 24% are in section B, 32% are in section C, and 28% are in section D. Section A seats sell for $35, section B for $30, section C for $25, and section D for $5. You see a ticket stuck high in a tree. a. What is the expected value of the ticket? b. The markings on the ticket look like either section A or C. If this is true, then what is the probability that the ticket is for section C? c. If the ticket is for section A or C, then what is the expected value of the ticket? 4. The tree diagram at right represents a game played by two players. Find a value of x that gives approximately the same expected value for each player. 5. Two varieties of flu spread through a school one winter. The probability that a student gets both varieties is.8. The probability that a student gets neither variety is.42. Having one variety of flu does not make a student more or less likely to get the other variety. What is the probability that a student gets exactly one of the flu varieties? 6. This table gives counts of different types of paperclips in Maricela s paperclip holder. Create a Venn diagram of the probabilities of picking each kind of clip if one is selected at random. Use metal, oval, and small as the categories for the three circles on your diagram. Algebra 2H- WS 08.X Permutations and Combinations Do all work neatly, showing all work, on a separate sheet of paper.. Evaluate the factorial expressions. (Some answers will be in terms of n) 2! 7! n! n! 20! n! a. b. c. d. e. f.! 6! n! n! 8! n 2! 2. Consider making a four-digit I.D. number using the digits 3, 5, 8, and 0. a. How many I.D. numbers can be formed using each digit once? b. How many can be formed using each digit once and not using 0 first? c. How many can be formed if repetition is allowed and any digit can be first? d. How many can be formed if repetition is allowed but 0 is not used first? 3. A combination lock has four dials. On each dial are the digits 0 to 9. a. Suppose you forget the correct combination to open the lock. How many combinations do you have to try? If it takes 0 s to enter each combination, how long will it take you to try every possibility? b. Suppose you replace your lock with one that has five dials, each with the digits 0 to 9. How many combinations are possible? If it still takes 0 s to enter each combination, how long will it take to try every possibility? c. For a lock to be secure, it has to be difficult for someone to guess the correct combination. How many times as secure as a 4-dial lock is a 5- dial lock? 4. Suppose each student in a school is assigned one locker. How many ways can three new students be assigned to five available lockers? 5. You have purchased four tickets to a charity raffle. Only 50 tickets were sold. Three tickets will be drawn, and first, second, and third prizes will be awarded a. What is the probability that you win the first prize (and no other prize)? b. What is the probability that you win both the first and second prizes, but not the third prize? c. What is the probability that you win the second or third prize? d. If the first, second, and third prizes are gift certificates for $25, $0, and $5, respectively, what is the expected value of your winnings? 6. Suppose you need to answer any four of seven essay questions on a history test and you can answer them in any order. a. How many different question combinations are possible? b. What is the probability that you include Essay Question 5 if you randomly select your combination? 7. Typically, 2 jurors and 2 alternates are chosen from a pool of 30 prospective jurors. If a juror is unable to serve, then the first alternate will replace that juror. The second alternate will be called on if another juror is dismissed. In how many ways can 2 jurors and first and second alternates be chosen from 30 people?

11 Algebra 2H- WS 08.X Binomial Expansion Do all work neatly, showing all work, on a separate sheet of paper.. Given the expression x y 47, find the terms below. a. st term b. th term c. 4 st term d. 47 th term 2. Expand the binomial expression: 2x Expand the binomial expression: 3x 4 4. Algebra 2H- WS 08.X Binomial Experiments Do all work neatly, showing all work, on a separate sheet of paper.. If the probability of success for each trial is.25 and all trials are independent, then... a. What is the probability of failure for a single trial? b. What is the probability of two successes in two trials? c. What is the probability of n successes in n trials? d. What is the probability that there will be some combination of two successes and three failures in five trials? 2. Suppose that the probability of success is.62. What is the probability that there are 35 successes in 50 trials? 3. Dr. Miller is using a method of treatment that is 97% effective. a. What is the probability that there will be no failure in 30 treatments? b. What is the probability that there will be fewer than 3 failures in 30 treatments? 4. Suppose that a blue-footed booby has a 47% chance of surviving from egg to adulthood. For a nest of four eggs... a. What is the probability that all four birds will hatch and survive to adulthood? b. What is the probability that none of the four birds will hatch and survive to adulthood? c. How many birds would you expect to survive? 5. Suppose that 350 points are randomly selected within the rectangle at right and 56 of them fall within the closed curve. What is an estimate of the area within the curve?

12 Investigation # Name The Sine Function: Amplitude Period In this lesson you will learn how A affects the graph of y Asinx. Use a graphing calculator to graph each of the following functions. All work will be done in degrees, so you must set the mode setting on your calculator to degrees. The suggested window settings are Xmin = -360, Xmax = 360, Xscl = 90, Ymin = -4, Ymax = 4, Yscl =. Equation A Sketch Max Min Amp. Is the graph inc. or dec. from 0 to 90? y sinx 2. Use your graphs to answer the following questions about y Asin( x). a. As A increases, does the graph become steeper or flatter? b. Does the sign of A affect the value of the maximum, minimum, or amplitude? If so, how? c. How do the graphs of y Asin( x) and y Asin( x) differ? d. Are the graphs of y Asin( x) and y Asin( x) symmetric? If so, are they symmetric about the x-axis or the y-axis? y 2sinx e. If A 2.5, will the maximum be greater or less than the graph with A? f. Suppose you want the maximum value of y Asin( x) to be.5 and the graph to be increasing from 0 to 90. What value of A would you choose? Check your answer on the calculator. y 0.5sinx g. Suppose you want the minimum value of y Asin( x) to be between -.25 and -.50 and the graph to be decreasing from 0 to 90. What value of A would you choose? Check your answer on the calculator. y 2sinx 3. Write formulas for the maximum value, minimum value, and amplitude in terms of the constant A in the equation y Asin( x). Remember that A can be either positive or negative. Maximum: y 3.5sinx Minimum: Amplitude: Explain how the constant A affects the graph of y Asin( x).

13 Investigation #2 The Sine Function: Vertical Shift In this lesson you will learn how A and D affect the graph of y Asin x D.. Use a graphing calculator to graph each of the following functions. All work will be done in degrees, so you must set the mode setting on your calculator to degrees. The suggested window settings are Xmin = -360, Xmax = 360, Xscl = 90, Ymin = -4, Ymax = 4, Yscl =. 2. Use your graphs to answer the following questions about y Asin( x) D a. If the constant D is positive, does the graph shift up or down? b. If the constant D is negative, does the graph shift up or down? c. Write a sentence describing what happens to the graph if we add a non-zero constant D to the equation y Asin( x). Equation A D Sketch Max Min x y sin 0 Max Min 2 d. If you want the graph of y Asin( x) to shift.5 units above the x-axis, what value of D should you choose? e. If a function is periodic, like the sine function, then one-half the sum of the maximum value plus the minimum value is the center line of the function. What does the center line of y Asin( x) D tell you about the graph? x y sin 2 f. The graph of y sin( x) 2 has a new center line because it has been shifted up from the x-axis. What is the equation of the new center line? g. Explain how the constants A and D affect the shape and location of the graph of y Asin( x) D. x y sin 3 3. State formulas for the vertical shift, center line, maximum value and minimum value in terms of the constants A and D in the equation y Asin( x) D. Vertical Shift: Center Line: Maximum: Minimum: x y 2sin 2 x y sin 4. Write equations of the form y Asin( x) D for the maximum, minimum, and vertical shift values given below. The first entry has been completed for you. Maximum Minimum Amplitude Vertical Shift Equation 3 2 y sin( x) Write an equation whose graph is a sine curve between the graphs of the equations y sin( x) 2 and y sin( x) 0.5. Verify your answer using the calculator. 6. Explain how the constant D affects the graph of y Asin( x) D.

14 Investigation #3 The Sine Function: Phase Shift In this lesson you will learn how C affects the graph of y sinx C. Phase shift tells how far (in degrees) the graph has moved in the horizontal direction.. Use a graphing calculator to graph each of the following functions. All work will be done in degrees, so you must set the mode setting on your calculator to degrees. The suggested window settings are Xmin = -360, Xmax = 360, Xscl = 90, Ymin = -4, Ymax = 4, Yscl =. The first has been done for you. Equation C Sketch Phase Shift y sin x 45 x-intercepts between 0 and 360 Equation C Sketch Phase Shift y sinx x-intercepts between 0 and 360 y sin x 90 0 none 0, 80, 360 y sin x Compared to y sin( x), have the graphs in Exercise been shifted horizontally or vertically? 3. In what direction does the graph shift when C > 0? 4. In what direction does the graph shift when C < 0? 5. Explain why the x-intercepts of y sinx and y sinx 45 are different. y sin x Explain the difference between the phase shifts in the graphs y sin( x 45) and y sin( x 45). 7. What is the formula for the phase shift in terms of C? y sin x Use what you know about phase shift and period to explain why y sin( x 360) has the same graph as y sin( x). 9. Write an equation of the form y sin( x C) that has x-intercepts at 60 and 240. Check your y sin x 270 answer using the calculator. 0. Give a value of C in the equation y sin( x C) that would produce a graph between y sin( x 20) and y sin( x 45). Check your answer using the calculator.. Explain how the constant C affects the graph of y sin( x C)

15 Investigation #4 The Sine Function: Period y sin 8x In this lesson you will learn how B affects the graph of y sin( Bx). Recall that the period of a sine graph is the length along the x- axis of one complete cycle.. Use a graphing calculator to graph each of the following functions. Our work will be done in degrees, so you must set the calculator mode to DEG. The suggested window settings are Xmin = -360, Xmax = 360, Xscl = 90, Ymin = -4, Ymax = 4, Yscl =. The first one has been done for you. Equation B Sketch Number of Cycles in 360 y sin x Period Use the results of Exercise to answer the following questions: a. If B =, the period of y sin( Bx) is 360. As B gets larger than, what happens to the period? b. As B gets smaller than (but still greater than 0) what happens to the period of the graph? y sin 2x c. How does the number of cycles in 360 of a sine graph compare to the constant B? d. Give a formula for the period of the sine function in terms of B. (Your formula must work for each graph in Exercise ) 3. If the graph of a sine wave shows 0 complete cycles in 360, what is its period? y sin x 2 y sin 4x 4. Write an equation of the form y sin( Bx) for each of the following periods. a. Period = 80 Equation: b. Period = 20 Equation: c. Period = 60 Equation: 5. Write an equation of the form y Asin( Bx) D whose graph is: a. A sine curve with amplitude 2 and period 80. b. A sine curve with vertical shift -2 and period 90. y sin x 4 c. A sine curve with amplitude.5, vertical shift 0.5 and period Explain how the constant B affects the graph of y sin( Bx).

16 Investigation #5 The General Sine Function y AsinB x C D. In this lesson you will learn how A, B, C, and D affect the graph of. Use a graphing calculator to graph each of the following functions. Our work will be done in degrees, so you must set the calculator mode to DEG. The suggested window settings are Xmin = -360, Xmax = 360, Xscl = 90, Ymin = -4, Ymax = 4, Yscl =. y 2sin 2 x 90 A = B = C = D = Be sure to identify the values of A, B, C, and D for each graph. Equation Sketch Max & Min y sinx Amp Vert Shift Per. Phase Shift 2. In the equation y x The value.5 determines the The value 2 determines the.5sin , A = B = The value 0.5 determines the The value 45 determines the C = D = y sin x 90 A = B = C = D = 3. Examine the graph below and find the values of A, B, C, and D that would generate it. Check your answer on the calculator. Amplitude 2 Vertical Shift Period Phase Shift Equation -2 y 2sin x Complete the table below. The first one has been done for you. A = B = C = D = y 2sin( x 90) Max Min Vertical Period Phase A B C D Equation Shift Shift y = 2sin(x) A = B = C = D = 5. In your own words, explain how the constants A, B, C, and D affect the graph of the equation y AsinBx C D. y 2sin( x 90) A = B = C = D =

17 Algebra 2H- WS 07.X Right Triangle Trig- WS 230 Algebra 2H- WS 07.X Right Triangle Trig- WS 229

18 Algebra 2H- WS 07.X TBD Algebra 2H- WS 0.X Mixed Trig Review DO ALL WORK NEATLY ON A SEPARATE SHEET OF PAPER Draw an angle with the given measure in standard position ) 20 2) 305 3) 580 4) 35 5) 450 6) 560 Find one angle with positive measurement and one angle with negative measurement coterminal with each given angle. 7) 65 8) 80 9) 285 0) 0 ) 37 2) 93 3) Find the degree measure of the angle through which the hour hand on the clock rotates from 5:00am to 0:00am. Find the exact values of sin, cos, and tan if the terminal side of in standard position contains the given point. 4) 6, 8 5) 20, 2 6) 2, 5 Find the reference angle for the angle with the given measure. 7) 236 8) 97 9) 20 Find the exact value of each trigonometric function. 20) tan35 2) sin20 22) sin90 Suppose is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the functions listed. 2 23) Given tan in Quadrant IV, find sin and cos. 5 24) Given 2 sin in Quadrant II, find cos and tan. 3

19 Algebra 2 CP- Section 7.3. Defining the Circular Functions sin cos sin cos sin cos sin cos