ALL Unit 2 HW PDF Practice Sets

Change in Arithmetic & 2.1 Geometric Sequences Practice Set 1 Problems 1 − 4, determine whether or not the sequence is arithmetic. If it is, find the common difference. 1. 1,1,2,3,5,8, … 2. −83, −74, −65, −56, …...

Change in Arithmetic & 2.1 Geometric Sequences Practice Set 1 Problems 1 − 4, determine whether or not the sequence is arithmetic. If it is, find the common difference. 1. 1,1,2,3,5,8, … 2. −83, −74, −65, −56, … 5 −1 1 1 1 1 3. 4, , 1, ,… 4. , , , ,… 2 4 8 16 2 2 Problems 5 - 8, write the first five terms of each sequence. 5. 𝑎𝑎𝑛𝑛 = 7 + 5𝑛𝑛 6. 𝑎𝑎𝑛𝑛 = 2𝑛𝑛−1 7. 𝑎𝑎𝑛𝑛 = 120 − 8𝑛𝑛 𝑛𝑛 8. 𝑎𝑎𝑛𝑛 = 𝑛𝑛+2 Problems 9 – 12, find the 28th term of each sequence. 9. 22,25,28,31, … 10. 5,3,1, −1, … 11. 1.7, 1.9, 2.1, 2.3, … 12. 23,16,9,2, … © 2023 Jean Adams Flamingo Math.com Problems 13 − 16, write the general rule for the sequence and find the 14th term. 13. 𝑎𝑎13 = 15, 𝑑𝑑 = 4 14. 𝑎𝑎13 = 15, 𝑑𝑑 = −7 15. 𝑎𝑎15 = 15, 𝑑𝑑 = −11 16. 𝑎𝑎15 = 15, 𝑑𝑑 = 19 Problems 17 − 21, solve. 17. You know your school bus stops at 2-minute intervals. The first stop of the route begins at 6:20 A.M. every day. If your stop is the 18th stop, when will the bus arrive for you? 18. Your grandparents gave you a $500 gift during the holidays, that you deposited in a savings account. You plan to add $25 to the gift every week to save for summer break, which begins in June. A. Write the amount in the account after each deposit as an arithmetic sequence, let 𝑎𝑎1 = 500. B. If there are 23 weeks before your vacation begins, How much money will you have in your account by summer? © 2023 Jean Adams Flamingo Math.com 19. How many logs will be in a pile of timbered trees if there are 30 logs in the bottom layer, 29 in the second, and so on until there is one in the top layer? 20. Each school year, Mu Alpha Theta plans to add 5 more members to their tutoring team for helping math students who struggle. This year there were 37 student tutors. A. Write a general formula for the number of tutors the club expects to add each year. B. Write the first five terms of the sequence. C. How many tutors will the club have in the 15th year if this rate of increase remains steady? 21. An object with negligible air resistance is dropped from a hot air balloon. The behavior of the free-fall is shown in the table below. If this arithmetic pattern continues, how many meters will the object fall in 10 seconds? Justify your answer with a general formula and algebraic support. Time (seconds) 1 2 3 4 Distance (meters) 15.2 25 34.8 44.6 © 2023 Jean Adams Flamingo Math.com  Change in Arithmetic & 2.1 Geometric Sequences Practice Set 2 Problems 1 − 4, is the given sequence geometric? If so, identify the common ratio and find the next two terms. 1 1 1 2 1. 1, 2 , 3 , 4 ,… 2. 18, −6, 2, − 3 ,… 3. 9, 0.9, 0.09, 0.009, … 4. 0.5, 0.25, 0.125, 0.0625, … Problems 5 – 8, write the explicit formula for each sequence. Then generate the first five terms. 5. 𝑎𝑎1 = 8, 𝑟𝑟 = −3 6. 𝑎𝑎1 = 0.0175, 𝑟𝑟 = 5 1 1 7. 𝑎𝑎1 = 13, 𝑟𝑟 = − 2 8. 𝑎𝑎1 = 6561, 𝑟𝑟 = 3 Problems 9 – 12, find the missing term of each geometric sequence. Mean 𝑎𝑎2 = 𝑎𝑎1 𝑎𝑎3 9. 637, , 13, … 10. … , 8, , 1458, … 3 27 11. 12, , 17.28 12. … , 5 , , 80 , … © 2023 Jean Adams Flamingo Math.com Problems 13 – 16, identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. 13. 1, −2,4, −8, … 14. 40,45,50,55, … 15. −4,12, −36,108, … 16. 11,12,14,17,22, … Problems 17 – 20, find the missing terms of each geometric sequence. 17. 2187, , , , 27, … 18. 12, , , , 60.75, … 1 8 19. 0.1024, , , , 4, … 20. , , , , 81 , … 2 Problems 21 – 24, for the geometric sequence 2,8,32,128, … find the indicated term. 21. 5th term 22. 13th term 23. 8th term 24. nth term © 2023 Jean Adams Flamingo Math.com Problems 25 – 26, write the explicit formula for the sequence. Then generate the first five terms. 1 25. 𝑎𝑎1 = 5, 𝑟𝑟 = −2 26. 𝑎𝑎1 = 729, 𝑟𝑟 = − 3 Problems 27 − 28, solve. 27. Jordan got some helium balloons from her best friend Madison on her birthday. The balloons lose one-fifth of the helium each day. Each balloon was filled to a volume of 2500 in3. A. Write the geometric sequence representing the amount of helium in the balloons for the first five days. B. What is the common ratio of the sequence? [Hint: Volume is in cubic inches.] C. How much helium will be in the balloons on the 10th day? 28. Find the perimeter, in meters, of the sixth rectangle in the pattern. 1m 2m 4m 1.25 m 2.5 m 5m © 2023 Jean Adams Flamingo Math.com  Problems 29-30, solve. 29. Caliber One Construction purchased a new service vehicle for $58,200. At the end of each year they will depreciate the value of the vehicle for tax purposes by 15%. A. Write a general formula for the value of the vehicle. B. What will be the value of the vehicle after 5 years? C. When will the vehicles value be less than $20,000? 30. Andrea deposited $5000 at the beginning of the year into a Mutual Fund account that pays 8.5% interest compounded monthly. The balance, A at the end of 5 years is given by 0.085 1 0.085 60 𝐴𝐴 = 5000 1 + 12 +... +5000 1 + 12 A. Write a general formula that represents Andrea’s balance at the end of each month based on this rate of return. B. How long will it take before the value of the account exceeds $25,000? C. What will be the value of the account in 30 years when Andrea plans to retire? © 2023 Jean Adams Flamingo Math.com  Change in Linear & 2.2 Exponential Functions Practice Set 1 Problems 1 − 2, use the graph of the sequence to complete the table. Find the difference and write an explicit rule for a linear function that corresponds to the nth term expression. 1. 𝒏𝒏 0 1 2 3 4 5 6 𝒂𝒂𝒏𝒏 6 2. 𝒏𝒏 0 1 2 3 4 5 6 𝒂𝒂𝒏𝒏 −3 Problems 3 − 4, complete the table and write an explicit rule that will model the sequence. 3. Given the arithmetic sequence where 𝑎𝑎3 = −7 and 𝑎𝑎9 = 17, state the domain and range of the sequence. Write a linear function that will contain the domain of the sequence. 𝒏𝒏 3 4 5 6 7 8 9 𝒂𝒂𝒏𝒏 −7 17 4. Given the arithmetic sequence where 𝑎𝑎1 = 8.5 and 𝑎𝑎6 = −9, state the domain and range of the sequence. Write a linear function that will contain the domain of the sequence. 𝒏𝒏 1 2 3 4 5 6 𝒂𝒂𝒏𝒏 8.5 −9 © 2023 Jean Adams Flamingo Math.com Problems 5 − 8, find the common difference and write the general rule for the sequence in function notation. Find the total number of terms in the sequence, 5. {8, 1, −6,... , −48} 6. {−8, −2.75, 2.5,... , 55} 7. {1.8, −1.1, −4,... , −64.9} 8. {4.1, 3.7, 3.3,... , −15.5} Problems 9 −12, solve. 9. The graph at right shows how the cost of a hiking expedition depends on the number of hikers. A. Represent the sequence in a table. 𝒏𝒏 1 2 3 4 𝒇𝒇(𝒏𝒏) B. Is the sequence arithmetic? Explain. C. Write an explicit rule 𝑓𝑓(𝑛𝑛), for the sequence of costs. Show your work to support your answer. D. What does n and 𝑓𝑓(𝑛𝑛) represent in context to the scenario? © 2023 Jean Adams Flamingo Math.com 10. On a warm summer day in Colorado, a parcel of air rises over the Rocky Mountains. It cools at the dry adiabatic lapse rate of 5.5℉ per 1000 feet until it reaches its dew point. If the ground temperature is 76℉, A. Write an explicit formula for the sequence of temperatures, {𝑇𝑇 𝑛𝑛 }, for the parcel of air that has risen n thousand feet. B. Use your function to find the temperature of the air if it has risen 12,000 feet. Support your answer with work. 11. Cameron borrowed money interest-free to pay for his boat repairs. He is repaying the loan to the marina in equal monthly payments. Payment # 𝒏𝒏 1 2 3 4 Loan Balance 𝒇𝒇(𝒏𝒏) 960 835 710 585 A. Write an explicit formula for the sequence of loan balances. B. How many months will it take for Cameron to pay off the loan? C. How much did Cameron borrow? How much is his final payment? 12. You have received a job offer with a starting salary of $45,000 per year with a guaranteed raise of $2500 per year. How many years will it take before your total salary received is more than $500,000? [Consider your salary after 2 years is $45,000+$47,500.] What will the total gross salary be at the end of this year? © 2023 Jean Adams Flamingo Math.com  Change in Linear & 2.2 Exponential Functions Practice Set 2 Problems 1 − 2, use the graph of the sequence to complete the table. Find the ratio and write an explicit rule for an exponential function that corresponds to the nth term expression. 1. 𝒏𝒏 1 2 3 4 5 6 32 𝒂𝒂𝒏𝒏 18 12 9 2. 𝒏𝒏 1 2 3 4 5 6 7 1 𝒂𝒂𝒏𝒏 1 27 3 Problems 3 − 4, write an explicit rule 𝑓𝑓 𝑛𝑛 = 𝑎𝑎𝑏𝑏 𝑛𝑛 that will model the sequence. 3. Given the geometric sequence where 𝑎𝑎2 = 4, 𝑟𝑟 = 3. Write an exponential function that will contain the domain of the sequence. 4. Given the geometric sequence where 1250, 250, 50, 10, … , 0.016. Write an exponential function that will contain the domain of the sequence. © 2023 Jean Adams Flamingo Math.com  Problems 5 − 8, find the ratio and the 𝑎𝑎0 term. Then, write the general rule, 𝑓𝑓 𝑛𝑛 = 𝑎𝑎𝑏𝑏 𝑛𝑛 for the sequence. Find the total number of terms in the sequence, 5. {−2, −8, −32,... , −8192} 6. {3125, 1250, 500,... , 0.8192} 3 3 7. 12, 3, 4 , 16 ,... 8. {42, 126, 378,... , 275562} Problems 9 −12, solve. 9. The graph at right represents a chocolate factory during Valentine week. Workers packed boxes of candy over the 7-day period. A. Represent the sequence in a table. 𝒏𝒏 1 2 3 4 5 6 7 𝒇𝒇(𝒏𝒏) 40 80 640 B. Is the sequence geometric? Explain. C. Write an explicit rule 𝑓𝑓(𝑛𝑛), for the sequence of candy boxes packed. Show your work to support your answer. D. What does n and 𝑓𝑓(𝑛𝑛) represent in context to the scenario? E. How many total boxes of candy were packed during this time period? © 2023 Jean Adams Flamingo Math.com  10. Anna dropped a marble into a cylinder of gelatin solution that was 24 inches deep. The marble began to sink slowly. In each second, the marble descended 15% lower than the previous second. The pattern for 𝑛𝑛 = 0,1,2 creates the sequence {24,20.4,17.34 … } A. Write a general formula 𝑓𝑓 𝑛𝑛 = 𝑎𝑎𝑏𝑏 𝑛𝑛 that gives the height of the marble from the bottom of the cylinder for time in the gelatin. B. Use the formula from part (a) to find the depth the marble sinks after 10 seconds. C. At what time will the marble be 2 inches from the bottom of the cylinder? 11. A culture contains 10,000 bacteria initially and increases by 20% every hour. A. Find a formula for the number 𝑁𝑁(𝑡𝑡) of bacteria present after t hours. B. How many bacteria are in the culture after 10 hours? 12. A rubber ball is dropped from a height of 25 feet. The ball bounces to 80% of its previous height with each bounce. A. Set up a model in the table below to represent each “bounce” starting with the height of the first bounce, listing 5 bounces. 𝒏𝒏 1 2 3 4 5 Height (feet) B. Write an exponential function 𝑓𝑓 𝑛𝑛 = 𝑎𝑎𝑏𝑏 𝑛𝑛 to represent the height of each bounce. C. What is the height of the bounce at 10 bounces? D. When will the ball bounce less than 1 foot high? © 2023 Jean Adams Flamingo Math.com  Exponential Functions 2.3 Practice Problems 1-4, use a calculator to evaluate the function at the indicate value of x. Round your result to the nearest hundredth. 1. 𝑓𝑓 𝑥𝑥 = 2. 7𝑥𝑥 at 𝑥𝑥 = 3.4 2. 𝑔𝑔 𝑥𝑥 = 4𝑥𝑥 at 𝑥𝑥 = 2 1 3. ℎ 𝑥𝑥 = 1. 3𝑥𝑥 at 𝑥𝑥 = 3 4. 𝑘𝑘 𝑥𝑥 = 6. 8−2𝑥𝑥 at 𝑥𝑥 = −𝜋𝜋 Problems 5-10, identify whether the statement represents an exponential function. Explain how you know. 1 5. A population of bacteria decreases by a factor of 4 every 12 hours. 6. The average yearly population increase of a flock of geese is 65. 7. Your personal trainer will discount each session by $7 for every new referral you bring. 8. The value of your stamp collection has increased by 4.5% annually over the last 5 years. 9. The height of a book dropped from a building is modeled by ℎ 𝑡𝑡 = −16𝑡𝑡 2 + 30 10. A certain radioactive isotope used in medicine has a half-life of 15 hours. © 2023 Jean Adams Flamingo Math.com Problems 11-14, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. 3 𝑥𝑥 11. 𝑦𝑦 = 250 1 − 𝑥𝑥 12. 𝑓𝑓 𝑥𝑥 = 18.4 1.025 𝑡𝑡 13. 𝑔𝑔 𝑡𝑡 = 215 0.95 14. ℎ 𝑥𝑥 = 7 1.025 1 𝑥𝑥 Problems 15-18, determine whether the table could represent a function that is linear, exponential or neither. If it is exponential, find a function that passes through the points. 15. 16. 𝒙𝒙 −1 0 1 2 3 𝒙𝒙 −1 0 1 2 3 1 𝒇𝒇(𝒙𝒙) − −1 −4 −16 −64 𝒈𝒈(𝒙𝒙) −7 −4 −1 2 5 4 17. 18. 𝒙𝒙 −2 −1 0 1 2 𝒙𝒙 −1 0 1 2 3 𝒉𝒉(𝒙𝒙) 12 15 16 15 12 𝒌𝒌(𝒙𝒙) 3.5 7 14 28 56 Problems 19-20, use the table of values to determine whether 𝑓𝑓 𝑥𝑥 is an exponential function. Explain your reasoning and state the equation 𝑓𝑓(𝑥𝑥). 19. 𝑔𝑔 𝑥𝑥 = 𝑓𝑓 𝑥𝑥 − 3 𝒙𝒙 −2 −1 0 1 2 𝒇𝒇(𝒙𝒙) 𝒈𝒈(𝒙𝒙) −2.75 −2.5 −2 −1 1 © 2023 Jean Adams Flamingo Math.com 20. ℎ 𝑥𝑥 = 𝑓𝑓 𝑥𝑥 + 2 𝒙𝒙 −3 −2 −1 0 1 𝒇𝒇(𝒙𝒙) 𝒉𝒉(𝒙𝒙) 66 18 6 3 2.25 Problems 21 − 22, graph the function, label intercept(s), asymptotes, and analyze completely. 21. 𝑓𝑓 𝑥𝑥 = 4𝑒𝑒 𝑥𝑥 1 𝑥𝑥 22. 𝑔𝑔 𝑥𝑥 = 3 2 Domain: Range: Domain: Range: Continuity: Increase / Decrease Continuity: Increase / Decrease Extrema: Asymptotes: Extrema: Asymptotes: End-behavior: End-behavior: lim 𝑓𝑓 𝑥𝑥 = lim 𝑓𝑓 𝑥𝑥 = lim 𝑔𝑔 𝑥𝑥 = lim 𝑔𝑔 𝑥𝑥 = 𝑥𝑥→−∞ 𝑥𝑥→∞ 𝑥𝑥→−∞ 𝑥𝑥→∞ © 2023 Jean Adams Flamingo Math.com  Problems 23 – 24, use the table and given information to answer the questions. 23. An exponential function of the form 𝐹𝐹 𝑥𝑥 = 𝑎𝑎 𝑏𝑏 𝑥𝑥 + 𝑐𝑐 is shown in the table below. 𝒙𝒙 −8 −6 −3 −1 1 3 6 𝑭𝑭(𝒙𝒙) 4.9997 4.9973 4.9259 4.3333 −1 −49 −1453 A. Describe the end behavior of the function B. What is the equation of the horizontal using limit notation. asymptote? C. What conclusion can be made about the value D. What is the value of c ? Give a reason. of a? Why? 24. An exponential function of the form 𝐺𝐺 𝑥𝑥 = 𝑎𝑎 𝑏𝑏 −𝑥𝑥 + 𝑐𝑐 is shown in the table below. 𝒙𝒙 −8 −6 −3 −1 1 3 8 𝑮𝑮(𝒙𝒙) 487 92 17 4 3.25 2.477 2.003 A. Describe the end behavior of the function B. What is the equation of the horizontal using limit notation. asymptote? C. What conclusion can be made about the value D. What is the value of c ? Give a reason. of a? Why? © 2023 Jean Adams Flamingo Math.com  Exponential Function 2.4 Manipulation Practice Set 1 Problems 1-4, Use a calculator to evaluate the function at the given value for x. Round your answer to three decimal places. 1. 𝑓𝑓 𝑥𝑥 = 4𝑥𝑥 ; 𝑥𝑥 = −𝜋𝜋 2. 𝑔𝑔 𝑥𝑥 = 5. 3𝑥𝑥 ; 𝑥𝑥 = 7.8 1 3. ℎ 𝑥𝑥 = 2. 6−3𝑥𝑥 ; 𝑥𝑥 = − 2 4. 𝑘𝑘 𝑥𝑥 = 1. 8− 𝑥𝑥 ; 𝑥𝑥 = 4 Problems 5-8, Use the graph of f to describe the transformation that yields the graph of g. 5. 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = 3𝑥𝑥 + 4 6. 𝑓𝑓 𝑥𝑥 = −4𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = −4𝑥𝑥−3 1 𝑥𝑥 1 − 𝑥𝑥+1 8. 𝑓𝑓 𝑥𝑥 = 0. 2𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = −0. 2𝑥𝑥 − 5 7. 𝑓𝑓 𝑥𝑥 = 2 , 𝑔𝑔 𝑥𝑥 = 2 Problems 9-12, Graph the exponential function by hand, identify asymptotes and intercepts. Determine whether the function is increasing or decreasing, concave up or concave down. 3 𝑥𝑥 10. ℎ 𝑥𝑥 = −2𝑥𝑥 + 3 9. 𝑔𝑔 𝑥𝑥 = 2 © 2023 Jean Adams Flamingo Math.com  3 −𝑥𝑥 12. ℎ 𝑥𝑥 = −2−𝑥𝑥 + 4 11. 𝑔𝑔 𝑥𝑥 = 2 Problems 13-14, Given the function 𝑓𝑓(𝑥𝑥), A. Show that the function 𝑓𝑓 𝑥𝑥 = 𝑏𝑏 𝑥𝑥+𝑘𝑘 can be algebraically rewritten in the form of ℎ 𝑥𝑥 = 𝑎𝑎𝑏𝑏 𝑥𝑥 , where 𝑎𝑎 = 𝑏𝑏 𝑘𝑘. B. State the domain and range of the function. C. Graph each function on the grid provided. D. State the y-intercept and asymptotes. E. Describe the transformation from each parent function. 13. 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥+3 1 − 𝑥𝑥+2 14. 𝑓𝑓 𝑥𝑥 = − 3 © 2023 Jean Adams Flamingo Math.com  Exponential Function 2.4 Manipulation Practice Set 2 Problems 1-4, Graph the function 𝑓𝑓(𝑥𝑥), then use the graph to obtain the graph of 𝑔𝑔(𝑥𝑥). Label the asymptote and y-intercept of each graph. Describe the domain and range of 𝑔𝑔(𝑥𝑥). 1 𝑥𝑥 1 𝑥𝑥−3 1. 𝑓𝑓 𝑥𝑥 = ; 𝑔𝑔 𝑥𝑥 = −2 2. 𝑓𝑓 𝑥𝑥 = −3−𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = −32−𝑥𝑥 + 5 2 2 3. 𝑓𝑓 𝑥𝑥 = 2−𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = 2−2𝑥𝑥 − 3 4. 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥 , 𝑔𝑔 𝑥𝑥 = −2𝑥𝑥 + 3 © 2023 Jean Adams Flamingo Math.com Problems 5-8, Write an equation for the transformation of the graph of 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 then explain the transformation for each graph. 5. 𝑔𝑔 𝑥𝑥 = 𝑓𝑓(𝑥𝑥 + 5) 6. ℎ 𝑥𝑥 = 𝑓𝑓 𝑥𝑥 −6 7. 𝑘𝑘 𝑥𝑥 = 𝑓𝑓(−2𝑥𝑥) 8. 𝑝𝑝 𝑥𝑥 = −3 𝑓𝑓 𝑥𝑥 − 2 + 1 Problems 9-10, Given the function 𝑓𝑓(𝑥𝑥), A. Show that the function 𝑓𝑓 𝑥𝑥 = 𝑏𝑏 (𝑘𝑘𝑘𝑘) can be algebraically rewritten in the form of 𝑥𝑥 ℎ 𝑥𝑥 = 𝑏𝑏 𝑘𝑘 B. State the domain and range of the function. C. Graph each function on the grid provided. D. State the y-intercept and asymptotes. E. Describe the transformation from each parent function. 9. 𝑓𝑓 𝑥𝑥 = 2−3𝑥𝑥 1 −2𝑥𝑥 10. 𝑓𝑓 𝑥𝑥 = 4 © 2023 Jean Adams Flamingo Math.com  Exponential Function 2.5 Context and Data Modeling Practice Set 1 Problems 1 – 4, without graphing, determine whether the function represents exponential growth or exponential decay. Find the y-intercept. 1. 𝑦𝑦 = 158(1.34)𝑥𝑥 1 𝑥𝑥 2. 𝑓𝑓(𝑥𝑥) = 0.6 5 5 𝑥𝑥 4. 𝑓𝑓 𝑥𝑥 = 3−𝑥𝑥 3. 𝑔𝑔(𝑥𝑥) = 7 3 Problems 5 − 7, The tables below give selected values for exponential functions. Find an exponential function to model the data. 5. 𝒙𝒙 0 1 2 3 𝒇𝒇(𝒙𝒙) 3 6 12 24 6. 𝒙𝒙 0 1 2 3 𝒈𝒈(𝒙𝒙) 8 4 2 1 7. 𝒙𝒙 0 1 2 3 4 4 𝒉𝒉(𝒙𝒙) 12 4 3 9 © 2023 Jean Adams Flamingo Math.com Problems 8 −10, Given the ordered pairs and the form 𝑦𝑦 = 𝑎𝑎𝑏𝑏 𝑥𝑥 + 𝑑𝑑 for the model of an exponential function, write an equation to model each function. 8. 1,16 , 5,1 2 9. −1,18 , 3, 9 4 10. −2, 3 , 10,173 Problems 11 − 15, write an exponential function that models each scenario. Then use it to solve the question. 11. A new sports car sells for $65,000. It depreciates 22% every year. What is the value of the car after 4 years? 12. You have $15,000 to invest in a retirement account. The fund increases at a rate of 10.8% per year. What will be the value of the investment at the end of 30 years? © 2023 Jean Adams Flamingo Math.com  13. A population count of 1246 black bears were found in a region of North Carolina. The population has been increasing at a rate of 2.25% per year. If this growth rate continues, how many bears will populate the region in 5 years? 14. In a coastal region along the Gulf of Mexico, the sea turtle population has been declining by a rate of 0.75% per year due to development. The last count found 318 turtles. How many turtles will there be in 25 years if the population continues to decline at this rate? 15. Carbon-14 is used to date artifacts found in archaeological digs. It has a half-life of 5730 years. Find the amount of Carbon-14 remaining in a 30-mg sample after 20,000 years. Problems 16-18, For each table, Find the constant value that can be added/subtracted to the output values to reveal a proportional growth pattern. Then, write an equation to model the function. 16. 𝑥𝑥 0 1 2 3 4 𝑓𝑓(𝑥𝑥) 5 13 37 109 215 © 2023 Jean Adams Flamingo Math.com  17. 𝑥𝑥 0 1 2 3 4 𝑔𝑔(𝑥𝑥) 4 11 25 53 109 18. 𝑥𝑥 0 1 2 3 4 13 11 ℎ(𝑥𝑥) 59 29 14 2 4 Problems 19−20, write an exponential function that describes each graph. 19. 20. © 2023 Jean Adams Flamingo Math.com  Exponential Function 2.5 Context and Data Modeling Practice Set 2 Problems 1-2, use a graphing calculator to find the exponential equation that includes the points given below. Then, state the y-intercept. 1. (2, 25.8) and (5, 192.4) 2. −3, 372.05 and (4, 8.27) Problems 3-5, Solve. 3. Bacteria is being grown in a petri dish. The number of bacteria B after t hours can be modeled by 𝐵𝐵 𝑡𝑡 = 150𝑒𝑒 0.585𝑡𝑡. A. What is the initial number of bacteria present? B. How many bacteria will be in the petri dish after 12 hours? 4. An archaeologist found that an artifact contained 18 grams of radioactive material. The decay model 𝐴𝐴 𝑡𝑡 = 18𝑒𝑒 −0.000122𝑡𝑡 shows the amount of the radioactive matter, in grams, after t years. How many grams will be present after 5500 years? © 2023 Jean Adams Flamingo Math.com  5. Claire is planning to attend the University of Florida. She needs to save $9000 in four years. How much would Claire need to put into her money market account that has been earning 9.75% annual interest, compounded continuously? 6. Jay had a collection of X-Box games that he sold on E-Bay for $1200. He deposited half the money in an investment account that earned 6.75% interest, compounded continuously. How much money will Jay have in the account at the end of 8 years if he doesn’t withdraw funds? 7. Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 250,000. © 2023 Jean Adams Flamingo Math.com 8. The number N of bacteria in a culture is modeled by 𝑁𝑁 𝑡𝑡 = 250𝑒𝑒 𝑘𝑘𝑘𝑘 , where t is the time in hours. If N = 340 when 𝑡𝑡 = 4, find the time required for the population to double in size. 9. A zircon sample contains 2000 atoms of the radioactive element 235U. Given that 235U has a half-life of 700 million years, how long would it take to decay to 200 atoms? 10. A chemist has a 100-gram sample of a radioactive substance. The table below shows the amount of radioactive material at the end of the week for a 6-week period. Radioactive Substance A. Use a graphing utility to fit an exponential function to the Weight data. Write the equation for the data. Week (in grams) 0 100.0 B. When will there be 10-grams of material left? 1 90.4 2 76.8 3 69.7 4 58.2 C. How much radioactive material will be left after 52 weeks? 5 49.7 6 44.2 © 2023 Jean Adams Flamingo Math.com  11. The table below shows the degrees above room temperature for a cup of coffee after t minutes of cooling. Time (min) 0 5 10 15 20 25 30 35 40 45 50 55 60 ℉ Above Room 136 103 77 56 45 31 23 18 12 10 8 6 4 Temperature A. Graph the data and find an exponential function that models the data. B. What is the average rate of change in ℉/𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚, at which the cup of coffee is cooling? C. Explain why the function appears to level off. 12. The table shows the values of an investment after the given number of years of interest that is compounded yearly. 𝐴𝐴 𝑡𝑡 = 𝐴𝐴0 1 + 𝑟𝑟 𝑡𝑡 Years 0 1 2 3 4 5 Value $1500.00 $1620.00 $1749.60 $1889.57 $2040.73 $2203.99 A. Write an exponential function that models the investment. B. What is the rate of interest? C. To the nearest year, when will the investment be worth $7500.00? © 2023 Jean Adams Flamingo Math.com  Free Response Question 13. The table of values for an exponential function can be modeled by 𝐺𝐺 𝑥𝑥 = 𝑎𝑎 𝑏𝑏 𝑥𝑥 + 𝑐𝑐 is given below. 𝒙𝒙 −8 −6 −3 0 1 7 9 𝑮𝑮(𝒙𝒙) −374 −100 −15 −4.5 −3.75 −3.011 −3.004 A. State the end behavior for the function. B. Determine the domain and range of 𝐺𝐺(𝑥𝑥) C. What is the horizontal asymptote? Explain your reasoning. D. Use the table of values to find the value of 𝑎𝑎 and an equation for the function 𝐺𝐺(𝑥𝑥). Show your work. E. Describe and analyze the behavior of the graph of the function: Growth or decay? Positive or negative? Increasing or decreasing? Concave up or concave down? © 2023 Jean Adams Flamingo Math.com  Competing Function 2.6 Model Validation Practice Set 1 Problems 1-3, for each scenario given, find the residuals. 1. Sarah is selling bottled water on a hot summer day at a tennis tournament. She records each day’s high temperature and the number of bottled waters sold for the 7-day tournament. She found her data to be linear in nature, so she found a least squares regression equation to predict the sales of water, from the daily high temperature, in degrees Fahrenheit: 𝑦𝑦 = 0.75𝑥𝑥 − 32. What is the residual if a day had a high temperature of 93 degrees and Sarah sold 35 bottles of water? 2. Jackson is raising chickens and has recorded the width of each egg, in millimeters, compared to the mass, in grams of the baby chick when hatched. The data fit a linear model that could predict the mass of a baby chick from the egg’s width: 𝑦𝑦 = 1.8𝑥𝑥 − 45. What is the residual of a 23-gram chick that hatches from an egg with a 37 mm width? 3. The gasoline consumption rate in miles per gallons of 15 vehicles of various weights (pounds) showed data that fit an exponential regression model: 𝑦𝑦 = 91.7 0.99956 𝑥𝑥. Find the residual of a vehicle that weighs 2865 pounds with an average of 28 miles per gallon. Problems 4-7, find a regression equation for each scenario. Complete the exercises. 4. Mr. O’s Physics students recorded a lab experiment for Hooke’s Law by measuring the compressed length y (in cm) of a spring when a force of F pounds is applied. Recall Hooke’s Law states that the distance a spring is compressed varies directly as the force on the spring. Force, F (lbs.) 0 2 4 6 8 10 12 Length, y (cm) 0 1.12 2.26 3.55 4.6 5.82 7.2 A. Use the data in the table to create a scatter plot. Then find a regression equation. B. Use the model to interpret the meaning of the slope. © 2023 Jean Adams Flamingo Math.com  5. A local print shop sells yard signs for realtors in bundles of 50 signs. The profit margins for each bundle are shown below. Assume the behavior is a quadratic regression for a up to 800 signs. A. Find a quadratic regression model for data in the table below. Use the model to complete the table. Quadratic Regression Model: _________________________________________________________ B. Find the residual plot for the data. Based on the residual plot, could the 𝒙𝒙 𝒚𝒚 𝒚𝒚 − 𝒚𝒚 𝒚𝒚 (Signs) (Profit, $) (residuals) quadratic function be used to make predictions of profit for signs to the 50 15 nearest dollar? Explain why or why not. 100 36 150 110 200 125 6. The average hourly earnings, in US dollars, of US production workers for 2005 − 2013 are shown in the table below, where t = 1 corresponds to the year 2005. Year 1 2 3 4 5 6 7 8 9 Average Hourly 15.91 16.42 17.09 17.75 18.41 18.89 19.31 19.57 19.94 Earnings Source: FRED Economic Data, Federal Reserve Bank of St. Louis A. Find a linear regression equation. Round to the nearest 0.001. B. Find a quadratic regression equation. Round to the nearest 0.001. C. Find the hourly earnings predicted for 2023 for both models. D. Use your answer from part (C) and the residual plots from both curves to decide which regression equation models the data best. Explain. © 2023 Jean Adams Flamingo Math.com 7. The US bottled water producer’s revenue, in billions of US dollars, Revenue, from 2005 to 2012 is shown in the table at right. Use a graphing Year in billions utility to write a regression equation. The time, in years, is given by t. USD 2005 $10.01 A. Create a scatter plot to determine the type of regression model that best fits the data set. Find the equation. 2006 $10.86 2007 $11.55 ________________________________________________________________________ 2008 $11.18 2009 $10.60 B. Explain the meaning of the y-intercept in context. 2010 $10.69 2011 $11.07 2012 $11.82 C. Using the residual plot for the regression model, do the Source: Beverage Marketing residuals follow a definite pattern or are they randomly Corp. scattered? Explain. D. Find a linear model for the data. ________________________________________________________________________ E. Based on your findings, which model best fits (makes sense) of the data in this scenario? Explain why. © 2023 Jean Adams Flamingo Math.com  Competing Function 2.6 Model Validation Practice Set 2 Problems 1-2, Interpret the slope and y-intercept of the regression line. 1. Your precalculus class completed a survey for your teacher at the end of Unit 1 Test. The scatter plot and 90 trend line show the relationship between how 80 70 many hours students spent studying, x, and their Test score 60 score on the test, y. 50 A. The line fitted to the data has a slope of 14. 40 What is the best interpretation of this slope? 30 20 10 B. The y-intercept is 36. What is the best 1 2 3 4 5 interpretation of this value? Time spent studying (hours) C. What would you score if your study time was 5 hours? Is this reasonable for the context? 2. Casey collected data on how long he spent on his 9 phone compared to how much battery life was 8 Battery Life (hours) remaining (in hours) throughout the day. The 7 scatter plot and trend line show this relationship. 6 3 5 A. The line fitted to the data has a slope of − 4. 4 What is the best interpretation of this slope? 3 2 1 1 2 3 4 5 6 7 8 9 B. The y-intercept is 9. What is the best Time spent on phone (hours) interpretation of this value? C. Use the trend line to predict the battery life after 15 hours of phone use. Does the prediction seem reasonable in the context of the problem? Explain. © 2023 Jean Adams Flamingo Math.com Problems 3-4, Answer the following by analyzing the data given. 3. The scatterplot displays a set of data along with its least-squares regression line. Consider removing the outlier (2, −7) and calculating a new regression line. Which statement would be true? Explain. A. The slope of the regression line would increase. B. The y-intercept of the regression line would increase. 4. The table gives the weight of students, y, in pounds, 𝒙𝒙 𝒚𝒚 𝒙𝒙 𝒚𝒚 versus their height, x, in inches. Consider the points (inches) (pounds) (inches) (pounds) may be modeled by linear, exponential or power regression models shown in the figures below. 60 95 66.5 140 62 108 67 137 A. Use the endpoint behavior for each regression equation model to decide which of the three 63 105 67 128 are more reasonable. Explain why or why 63.5 132 68 153 not for each regression shown below. 64 138 68 164 Linear model: 𝑦𝑦 = 7.08𝑥𝑥 − 330 64 127 68.3 160 Exponential model: 𝑦𝑦 = 3.91 1.0552 𝑥𝑥 64.2 123 68.5 158 Power model: 𝑦𝑦 = 0.0000537𝑥𝑥 3.51962 65 118 69 160 66 145 70 155 66 129 70 173 © 2023 Jean Adams Flamingo Math.com 5. Your agriculture club plants a different bean variety each week, for 11 weeks. Two weeks after the last been was planted, the plants have heights shown in the table and in the graph. The behavior of the data has an upward trend, can you determine the best-fitting function model? Time 2 3 4 5 6 7 8 9 10 11 12 (weeks) Height 3 5 4 7 6 11 10 15 19 22 24 (inches) A. Use the data set to find a linear regression and an exponential regression. Linear model : _____________________________________________________ Exponential model: _______________________________________________ B. Analyze the residual plot for each regression equation. Explain your findings. C. Which regression equation best fits the data? 6. The population for a small town in the land of Leibniz is shown below. Create a scatter plot and find a quadratic, cubic and exponential regression. Then explain which model best represents the values. Time, yrs. 1995 2000 2005 2010 2015 Population, in thousands 8.21 8.63 8.49 8.84 8.92 A. Quadratic Model: _____________________________________________________ Cubic Model: ___________________________________________________________ Exponential Model: ____________________________________________________ B. The best model © 2023 Jean Adams Flamingo Math.com 7. A jet-ski’s average speed, y, (in knots) for several engine speeds, x, in hundreds of revolutions per minute (RPM’s) are shown below. Engine speed, x 8 11 14 17 19 22.5 Jet ski speed, y 8.97 14.18 19.62 26.05 31.29 43.13 A. Use the data to determine a polynomial model of best fit. How did you determine your model? B. Use your model to estimate the average speed for the jet ski when the engine speed is 2500 RPM’s. C. What engine speed will produce a jet-ski speed of 20 knots? 8. Vehicles are tracked at a busy intersection for a 24-hour period and reported in the table below, where t = 0 represents midnight. Hours 1 3 6 10 14 18 21 Vehicles 17 4 22 45 90 78 42 A. Find a regression equation model that fits the data. B. Based on your model, how many vehicles will pass through at midnight? C. Use your model to find the number of vehicles that pass through the intersection at 5 pm each day. © 2023 Jean Adams Flamingo Math.com 9. The table below shows hours Shelby worked, x and tips received, y for a week. Hours worked, x 4 5.5 4.75 3.5 7 6.5 3.5 Total Tips, y $38.50 $54.30 $148.75 $35.90 $115.65 $87.40 $34.85 A. Use the data to determine a regression model of best fit. B. Find a linear regression model for the data, what do you notice about 𝑦𝑦 (1)? C. Shelby said that her father gave her a $100 tip on the day she worked 4.75 hours. If you remove this $100 tip, 4.75, $48.75 how does the regression equation change? What does the adjusted equation show you related to the data points? D. Can you calculate a better linear model by adjusting the y-intercept to (0, 0) using 6.5, 87.40 ? Explain. 10. The table shows a lemonade stand data set based on temperature and revenue. Temperature (degrees) 82 77 85 87 81 88 Revenue $50 $32 $58 $72 $47 $92 A. Find a linear regression model for the data. B. Suppose the temperature is 80 degrees and the revenue for the day was $53, what is the error in your model? C. If a rainstorm was the result of the data point (77, 32) and you remove this point, how does your new regression model change the accuracy of the observed data? © 2023 Jean Adams Flamingo Math.com  Composition of 2.7 Functions Practice Set 1 Problems # 1 - 4, Find (A) 𝑓𝑓 ∘ 𝑔𝑔 , (B) 𝑔𝑔 ∘ 𝑓𝑓, and (C) 𝑓𝑓 ∘ 𝑓𝑓. State the domain of each. 1. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 + 1 ; 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2 − 1 2. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 + 1 ; 𝑔𝑔 𝑥𝑥 = 2 − 𝑥𝑥 3. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 2 ; 𝑔𝑔 𝑥𝑥 = 2 4. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 − 2 ; 𝑔𝑔 𝑥𝑥 = 1 − 2𝑥𝑥 𝑥𝑥 Problems #5 - 8, Use the functions below to find the equation for the indicated operations. 𝑥𝑥−3 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 2𝑥𝑥 − 3 𝑔𝑔 𝑥𝑥 = ℎ 𝑥𝑥 = 3𝑥𝑥 − 4 2𝑥𝑥 + 3 5. 𝑓𝑓 ℎ 𝑥𝑥 6. 𝑔𝑔 ℎ 𝑥𝑥 7. ℎ(𝑓𝑓(𝑥𝑥) 8. 𝑓𝑓 𝑔𝑔 𝑥𝑥 © 2023 Jean Adams Flamingo Math.com Problems #9 - 12, For each function h, find functions f and g so that ℎ 𝑥𝑥 = (𝑓𝑓 ∘ 𝑔𝑔)(𝑥𝑥). 3 9. ℎ 𝑥𝑥 = 3𝑥𝑥 − 2 10. ℎ 𝑥𝑥 = 𝑥𝑥 5 − 3 11. ℎ 𝑥𝑥 = 𝑥𝑥 4 − 2𝑥𝑥 + 3 12. ℎ 𝑥𝑥 = 𝑥𝑥 2 + 1 2 Problems #13 -20, Use the graphs of f , g, and h to evaluate the following. 13. (𝑓𝑓 ∘ 𝑔𝑔)(4) 14. 𝑓𝑓 𝑔𝑔 3 15. 𝑔𝑔 ℎ −2 16. ℎ ∘ ℎ (2) 17. (ℎ ∘ 𝑓𝑓)(−3) 18. 𝑔𝑔 𝑔𝑔 4 19. 𝑔𝑔(ℎ 𝑓𝑓 2 20. 𝑔𝑔(𝑓𝑓 ℎ 1 © 2023 Jean Adams Flamingo Math.com Problems 21-24, Interpret the meaning of each statement in context to the scenario. 21. A municipality determines that the tax revenue, R, in millions of dollars collected on p thousand people is represented by the function 𝑅𝑅(𝑝𝑝). The population of the city, in thousands, can be predicted by the function 𝑝𝑝(𝑡𝑡), where 𝑡𝑡 is measured in years after 2018. Explain the meaning of the statement: 𝑅𝑅 𝑝𝑝 5 = 15.025, in context to the scenario. 22. Action Outdoor Sports builds kayaks. The function 𝐶𝐶(𝑛𝑛) represents the cost in dollars to produce n kayaks. The number of kayaks that will be demanded when the price is p dollars is represented by the function 𝐷𝐷(𝑝𝑝). Explain the meaning of the statement: 𝐶𝐶 𝐷𝐷 250 = 9873, in context to the scenario. 23. The function 𝐴𝐴(𝑑𝑑) gives the pain level on a scale of 0 to 10 that a patient will experience with d milligrams of a drug in their system. The milligrams of the drug remaining in patient’s system after t minutes is modeled by 𝑚𝑚(𝑡𝑡). Explain the meaning of the statement: 𝐴𝐴 𝑚𝑚 8 = 4, in context to the scenario. 24. The function 𝑟𝑟(𝑥𝑥) represents the radius r, in inches, of a circle that forms x seconds after you toss a stone into a pool of water. The function 𝐴𝐴(𝑥𝑥) represents the area A of a circle with radius x. Explain the meaning of the statement: 𝐴𝐴 𝑟𝑟 5 = 8754, in context to the scenario. © 2023 Jean Adams Flamingo Math.com  Composition of 2.7 Functions Practice Set 2 Problems #1 - 8, The table of values for three functions, 𝑓𝑓, 𝑔𝑔, and ℎ are given below. Find the values for each composite function operation below. 𝒙𝒙 −3 −2 −1 0 1 2 3 𝒇𝒇(𝒙𝒙) −2 1 6 −2 0 5 −3 𝒈𝒈(𝒙𝒙) 0 3 2 −5 4 3 2 𝒉𝒉(𝒙𝒙) −1 0.5 1 2 −3 0 1 1. 𝑔𝑔 𝑓𝑓 1 2. 𝑔𝑔 ℎ 1 3. ℎ 𝑔𝑔 −2 4. 𝑔𝑔 ℎ 𝑔𝑔 −3 5. ℎ 𝑓𝑓 𝑔𝑔 −3 6. 𝑓𝑓 ℎ ℎ 1 7. If 𝑘𝑘 𝑥𝑥 = 2 𝑥𝑥 − 3 − 5𝑥𝑥, for what value(s) 8. If 𝑘𝑘 𝑥𝑥 = 3𝑥𝑥 − 4 , for what value(s) of x is of 𝑥𝑥 is 𝑘𝑘 𝑥𝑥 = 𝑓𝑓 ℎ 𝑓𝑓 3 𝑘𝑘 𝑥𝑥 = 𝑓𝑓 𝑔𝑔 −1 © 2023 Jean Adams Flamingo Math.com Problems 9 – 12, use the functions below to find an equation for the indicated composite function. Explain the transformation of the function 𝑓𝑓(𝑥𝑥) when composed with 𝑔𝑔 𝑥𝑥. 9. Find 𝑓𝑓 𝑔𝑔 𝑥𝑥 given: 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥 ; 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 − 3 10. Find 𝑓𝑓 𝑔𝑔 𝑥𝑥 given: 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 + 3 ; 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 + 4 11. Find 𝑓𝑓 𝑔𝑔 𝑥𝑥 given: 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 5𝑥𝑥 + 6 ; 12. Find 𝑓𝑓 𝑔𝑔 𝑥𝑥 given: 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 4; 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 + 3 𝑔𝑔 𝑥𝑥 = 2𝑥𝑥 13. Complete the table of values for 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) A. Find the values for ℎ 𝑥𝑥 = 𝑓𝑓 𝑔𝑔 𝑥𝑥. 𝒙𝒙 𝒇𝒇(𝒙𝒙) 𝒈𝒈(𝒙𝒙) 𝒉𝒉(𝒙𝒙) −1 0 1 2 3 4 B. Graph the points 𝑥𝑥, ℎ 𝑥𝑥 on the blank grid. © 2023 Jean Adams Flamingo Math.com Problems #14 – 20: Answer each question. 14. A rock is thrown into a calm lake, causing ripples to form in concentric circles. The radius, in feet, of the outer ripple is , 𝑟𝑟 𝑡𝑡 = 0.8𝑡𝑡 , where t is the number of seconds after the rock hit the water. The area of the circle is given by the function 𝐴𝐴 𝑟𝑟 = 𝜋𝜋𝑟𝑟 2. A. Find 𝐴𝐴 ∘ 𝑟𝑟 𝑡𝑡 B. What is the area of the outer ring after 6 seconds? C. Explain the meaning in the context of the scenario. 15. Your teacher offers to give the whole class a bonus if everyone passes the next math test. The teacher says she will give everyone a 10-point bonus and increase everyone’s grade by 9% of their score. A. You earned a 75 on the test. Would you rather have the 10-point bonus first and then the 9% increase, or the 9% increase first and then the 10-point bonus? B. Is this the best plan for all students? Explain your reasoning. © 2023 Jean Adams Flamingo Math.com  16. Your favorite restaurant is offering a 15% discount on their entrée menu items. You have a coupon valued at $5. A. Write functions to model the discount and to model applying the coupon. B. Use a composition of your two functions to model how much you pay for a dinner entrée if the cashier applies the discount first and then the coupon. C. Write a composition to represent applying the coupon first and then the discount. D. How much more is an entrée if the cashier applies the coupon first? 17. Suppose the function 𝑓𝑓 𝑥𝑥 = 0.0085𝑥𝑥 represents the number of US dollars equivalent to 𝑥𝑥 Japanese yen and the function 𝑔𝑔 𝑥𝑥 = 18.27𝑥𝑥 represents the number of Mexican pesos equivalent to 𝑥𝑥 US dollars. A. Write a composite function that represents the number of Mexican pesos equivalent to 𝑥𝑥 Japenese yen. B. Find the value in Mexican pesos of an item that costs 750 Japanese yen. 18. A pebble is tossed into a pool of water and the circular ripples grow outward. The radius 𝑟𝑟 in inches of the circle, at 𝑥𝑥 seconds can be found as 𝑟𝑟 𝑥𝑥 = 12.5𝑥𝑥. The function 𝑎𝑎 𝑥𝑥 = 𝜋𝜋𝑥𝑥 2 describes the area of the circle with radius x. A. Find 𝐴𝐴 ∘ 𝑟𝑟 𝑥𝑥 when 𝑥𝑥 = 3. B. Explain the meaning in context of the problem. © 2023 Jean Adams Flamingo Math.com 19. Angioplasty is a cardiac procedure where doctors insert a catheter into a heart vein and then inflate a small, spherical balloon on the tip of the catheter. If the balloon is inflated at a constant rate of 40 cubic millimeters per second, A. Find the volume after t seconds. 4 B. When the volume is 𝑉𝑉 = 3 𝜋𝜋𝑟𝑟 3 , what is the radius r? C. Write an equation that expresses the radius as a function of time. D. Find the radius of the balloon after 3 seconds. 20. A circular puddle of liquid is evaporating and slowly shrinking in size. After t minutes, the radius 18 of the surface of the puddle measures 𝑟𝑟 𝑡𝑡 = 2𝑡𝑡 + 3 inches. The area A of the surface of the puddle is given by 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2. Express the area as a function of time by finding 𝐴𝐴 ∘ 𝑟𝑟 𝑡𝑡 and compute the area of the surface of the puddle at 𝑡𝑡 = 12 minutes. © 2023 Jean Adams Flamingo Math.com  Inverse Functions 2.8 Practice Set 1 Problems 1 − 2, find the inverse of each relation. Graph the given relation and its inverse. 1. 2. 𝒙𝒙 𝒚𝒚 𝒙𝒙 𝒚𝒚 𝒙𝒙 𝒚𝒚 𝒙𝒙 𝒚𝒚 0 −3 −6 3 1 −1 −1 2 2 1 2 1 3 3 3 0 Problems 3 – 6, find the inverse of each function, if it exists. 3. 𝑦𝑦 = 2𝑥𝑥 − 5 4. 𝑦𝑦 = 3 − 2𝑥𝑥 2 2 5. 𝑦𝑦 = 𝑥𝑥 − 2 6. 𝑦𝑦 = 𝑥𝑥 − 1 © 2023 Jean Adams Flamingo Math.com Problems 7 – 10, for each function 𝑓𝑓, find 𝑓𝑓 −1 , if it exists. State the domain and range of 𝑓𝑓 and 𝑓𝑓 −1. 2 7. 𝑓𝑓 𝑥𝑥 = − 3 𝑥𝑥 + 4 8. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 + 1 9. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 + 5 10. 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 + 5 Problems 11 − 12, Let 𝑓𝑓 𝑥𝑥 = 5𝑥𝑥 + 8. Find each value. Hint: Rewrite 𝑓𝑓 𝑥𝑥 as 𝑦𝑦 and switch 𝑥𝑥 and 𝑦𝑦. 11. 𝑓𝑓 ∘ 𝑓𝑓 −1 (4) 12. 𝑓𝑓 −1 ∘ 𝑓𝑓 (−3) © 2023 Jean Adams Flamingo Math.com Problems #13-16: Sketch the graph of the inverse function for each function shown. State the domain for each. 13. 14. 15. 16. Problems #17- 20: Use a graphing utility to graph each function and determine whether the function has an inverse that is also a function. 1 2 𝑥𝑥−2 17. 𝑓𝑓 𝑥𝑥 = 2 𝑥𝑥 + 3 −4 18. 𝑔𝑔 𝑥𝑥 = 3𝑥𝑥 2 3 19. ℎ 𝑥𝑥 = −𝑥𝑥 9 − 𝑥𝑥 2 20. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 − 4 © 2023 Jean Adams Flamingo Math.com Problems #21- 24, Restrict the domain of the function 𝑓𝑓 so that the function is invertible and has an inverse function. Then find the inverse function 𝑓𝑓 −1. State the domain and ranges of 𝑓𝑓 and 𝑓𝑓 −1. 2 21. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 − 3 22. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 − 3 23. 𝑓𝑓 𝑥𝑥 = 3 − 2𝑥𝑥 2 24. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 + 3 − 2 Problems #25 - 30, Use the graphs of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) to evaluate the expressions. 25. 𝑓𝑓 −1 (0) 26. 𝑔𝑔−1 (0) 27. 𝑓𝑓 ∘ 𝑔𝑔 1 28. 𝑔𝑔 𝑓𝑓 −2 29. 𝑓𝑓 −1 𝑔𝑔 −4 30. 𝑓𝑓 −1 ∘ 𝑔𝑔−1 −1 © 2023 Jean Adams Flamingo Math.com  Inverse Functions 2.8 Practice Set 2 Problems 1-4, Explain the meaning of each statement in context to the scenario. Let 𝐻𝐻 𝑡𝑡 be the height, in feet, of an object that is launched into the air from the ground where t is the time, in seconds. 1. 𝐻𝐻 2 = 65 2. 𝐻𝐻 3 = 42 3. 𝐻𝐻 −1 8 = 3.25 4. 𝐻𝐻 −1 0 = 4 Problems 5- 8, 𝑓𝑓 and 𝑔𝑔 are defined by the given tables. Use the tables to evaluate the following. 5. 𝑓𝑓 −1 (2) 6. 𝑔𝑔−1 5 𝒙𝒙 𝒇𝒇(𝒙𝒙) 𝒙𝒙 𝒈𝒈(𝒙𝒙) −1 2 −1 −5 0 −1 1 −3 7. 𝑓𝑓 −1 𝑔𝑔 4 8. 𝑔𝑔−1 (𝑓𝑓 0 ) 1 −3 4 −1 3 −4 8 5 Problems 9 – 15, Solve. 9. Find the inverse of the linear function 𝑓𝑓 𝑥𝑥 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏; 𝑚𝑚 ≠ 0. © 2023 Jean Adams Flamingo Math.com  4 10. The formula for the volume of a sphere is 𝑉𝑉 = 3 𝜋𝜋𝑟𝑟 3 A. Find the inverse of the formula, if it exists. B. Use the inverse to find the radius of a sphere that has a volume of 28,000 ft3. 𝑟𝑟 2 11. The function 𝑑𝑑 = is a model for the distance 𝑑𝑑 in feet that a car with locked brakes skids 24 when coming to a complete stop from a speed of 𝑟𝑟 mi/hr. Find the inverse of the function. Use the inverse to estimate the speed of a car that made skid marks 136 feet long. 12. Mr. Swenson used the formula 𝑦𝑦 = 140𝑥𝑥 + 175 to predict a student’s score on a state End-of-Course exam using the student’s 11-grade GPA number 𝑥𝑥. A. Find the inverse of the formula. B. Using the inverse, what GPA does a student need to get a passing score of 525 on the End-of-Course exam? © 2023 Jean Adams Flamingo Math.com 13. The circumference of a child’s head C, is related to the height of the child H, by the equation 𝐻𝐻 𝐶𝐶 = 2.15𝐶𝐶 − 10.53 where both C and H are in inches. A. Express the head circumference 𝐶𝐶, as a function of height 𝐻𝐻. B. What is the predicted circumference of a child’s head who measures 32 inches tall? 14. In June 2015, you are planning a trip to Japan and find the exchange rate for converting US dollars (x) to yen (y) was 𝑦𝑦 = 123.65𝑥𝑥. A. How many yen could you get for $100 U.S.? B. What is the inverse function, and what conversion does it represent? C. While in Toyko you have lunch in an outside café at a cost of 2000 yen. How much was that in U.S. dollars? 15. The consumption of bottled water in the US, in gallons per person, from 2005 to 2011 is shown in the table below. The time, in years, is given by t, where t = 0 corresponds to 2005. 𝒕𝒕 0 1 2 3 4 5 6 𝒇𝒇 (𝒕𝒕) 25.4 27.6 29.0 28.5 26.4 28.3 29.2 A. What does the inverse represent? B. Does 𝑓𝑓 −1 exist? © 2023 Jean Adams Flamingo Math.com Problems 16 - 17, restrict the domain of the function f so that the function is invertible and has an inverse function. Then find the inverse function 𝑓𝑓 −1. State the domains and ranges of 𝑓𝑓 and 𝑓𝑓 −1. 16. 𝑓𝑓 𝑥𝑥 = 3 − 𝑥𝑥 2 2 17. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 + 3 Problems 18 – 20, Consider each situation and determine if the statement represents an invertible function. If so, write a statement that describes the inverse function. Explain why if not possible. 18. The number of miles, n of a race car driver has completed in a race in terms of the time, t in hours. 19. The depth of the tide at a fishing pier, d in terms of the time, t over a 24-hour period. 20. The population, p of Miami, Florida in terms of the year t from 2020 to 2023. © 2023 Jean Adams Flamingo Math.com  Logarithmic Expressions 2.9 Practice Set 1 Problems 1 - 6, express in exponential form. 1. log 6 216 = 3 2. log 3 81 = 4 1 1 3. log 3 9 = −2 4. log 64 8 = 2 5. log 7 = 0.845 6. log 1 32 = −5 2 Problems 7- 12, express in logarithmic form. 7. 27 = 128 2 8. 8− 3 = 1 4 1 9. 81 2 = 9 10. 10−3 = 0.001 11. 𝑏𝑏 𝑘𝑘 = 𝑁𝑁 1 −4 12. 2 = 16 Problems 13 – 18, evaluate each expression without a calculator. 1 13. log 5 125 14. log 3 27 1 15. log 16. log 5 57 100 17. 4log4 6 18. log 25 5 © 2023 Jean Adams Flamingo Math.com Problems 19 − 24, Use a calculator to evaluate to four decimal places. 19. log 4 18 20. log 1 23 2 2𝟏𝟏. log 𝜋𝜋 57 22. log 0.8 16 23. log 136 24. log 9 (0.85) Problems 25-30, solve. 25. The amount of energy released from one earthquake was 8,000 times greater than the amount of energy released from another. The equation 10𝑥𝑥 = 8000 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes? 26. One formula for the decibel level D of sound is 𝐷𝐷 = 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 − 𝐿𝐿𝐿𝐿𝐿𝐿𝐿0−16. In this formula, I represents the intensity level in watts per square meter. Find D if I = 10-12. 27. A road construction crew is blasting through a mountain side to create a new highway. Safety regulations require loudness of the sound in watts to remain below 110 dB. If a blasting charge has an intensity of 1.65 × 10−2 W/m2 , is this in violation of safety standards? 𝐼𝐼 Use: 𝐷𝐷 = 10 𝑙𝑙𝑙𝑙𝑙𝑙 𝐼𝐼 where 𝐼𝐼0 = 10−12 represents a barely audible sound. 0 © 2023 Jean Adams Flamingo Math.com 28. The Richter scale model for measuring magnitude R of an earthquake is modeled by the 𝑎𝑎 equation 𝑅𝑅 = log 𝑇𝑇 + 𝐵𝐵, where a is the amplitude in micrometers, T is the period in seconds, and B represents the dampening effect (weakening) of the wave due to the distance from the epicenter of the quake. A. Find the magnitude R of a quake where 𝑎𝑎 = 325, 𝑇𝑇 = 4 and 𝐵𝐵 = 3.25 B. Find the magnitude R of a quake where 𝑎𝑎 = 230, 𝑇𝑇 = 2 and 𝐵𝐵 = 4.5 29. Venus and Jupiter are the two brightest planets in our solar system. When the planets are closest Venus will glow with a magnitude of –4.3 and Jupiter will shine at –2.1 on astronomers’ magnitude scale (where lower numbers mean brighter objects). How many times brighter is 𝑏𝑏 Venus than Jupiter? Use: 𝑚𝑚2 − 𝑚𝑚1 = −2.5 𝑙𝑙𝑙𝑙𝑙𝑙 2 𝑏𝑏 1 𝐼𝐼1 30. The intensity levels 𝐼𝐼 of two earthquakes can be compared by the formula log = 𝑀𝑀1 − 𝑀𝑀2 , where 𝐼𝐼2 𝑀𝑀 is the magnitude on the Richter Scale. In August 2009, Honshu, Japan registered a 6.1 magnitude quake. In March 2011 a 9.0 quake hit at a nearby area in Fukushima, Japan. How many times greater was the intensity of the 2011 earthquake? Round to the nearest hundredth. © 2023 Jean Adams Flamingo Math.com  Inverses of 2.10 Exponential Functions Practice Set 1 Problems 1 – 2, In the tables shown below, the value for b is the same positive constant. Table I contains outputs of the function 𝑓𝑓 𝑥𝑥 = 𝑏𝑏 𝑥𝑥 for some x values. Table II contains outputs of the function 𝑔𝑔 𝑥𝑥 = log 𝑏𝑏 𝑥𝑥 for some x values. Find the values for a, b, c, and d without using a calculator. 1. Table I Table II 𝒙𝒙 0.631 1.262 1.465 1.631 𝒚𝒚 𝑎𝑎 3 4 2𝑐𝑐 4𝑑𝑑 𝒃𝒃𝒙𝒙 2 4 5 6 log 𝒃𝒃 𝒚𝒚 0 1 1.262 1.465 1.631 2. Table I Table II 𝒙𝒙 0.792 1.161 1.292 1.5 𝒚𝒚 𝑎𝑎 4 4𝑐𝑐 7 12𝑑𝑑 𝒃𝒃𝒙𝒙 3 5 6 8 log 𝒃𝒃 𝒚𝒚 0 1 1.292 1.404 1.5 Problems 3 – 6, Find the inverse of each function. 3. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑥𝑥 4. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙 𝑥𝑥 + 2 © 2023 Jean Adams Flamingo Math.com 5. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙2 2𝑥𝑥 6. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙 1 𝑥𝑥 2 Problems 7 – 8, Use the given function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) to: A. Find the domain of 𝑓𝑓 B. Graph 𝑓𝑓. C. From the graph, determine the range and any asymptotes of 𝑓𝑓. D. Find 𝑓𝑓 −1 , the inverse of 𝑓𝑓. E. Graph 𝑓𝑓 −1. F. Find the domain and range and any asymptotes of 𝑓𝑓 −1. 7. 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥−3 Domain Range Asymptotes 8. 𝑓𝑓(𝑥𝑥) = log 2 2𝑥𝑥 + 1 Domain Range Asymptotes © 2023 Jean Adams Flamingo Math.com Problems 9 – 11, Graph and analyze completely. 9. Graph 𝑦𝑦 = log 1 𝑥𝑥 + 2. Analyze the function where 0 < 𝑏𝑏 < 1. 2 Domain: Range: lim 𝑓𝑓(𝑥𝑥) 𝑥𝑥→∞ lim 𝑓𝑓(𝑥𝑥) 𝑥𝑥→0+ 10. Graph 𝑦𝑦 = log 3 (𝑥𝑥 − 2). Analyze the function where 𝑏𝑏 > 1. Domain: Range: lim 𝑓𝑓(𝑥𝑥) 𝑥𝑥→∞ lim 𝑓𝑓(𝑥𝑥) 𝑥𝑥→2+ 11, Graph the following scenario as an inverse function. A certain type of bacteria increases according to the model 𝑃𝑃 𝑡𝑡 = 80 2 0.25𝑡𝑡 where 𝑡𝑡 is measured in hours. A. Use the table of value for 𝑃𝑃(𝑡𝑡) to graph a logarithmic model that will display the population of the bacteria as the input with the time as the output. B. Estimate how many hours before the bacteria reaches a population of 1000. 𝒕𝒕 𝑷𝑷(𝒕𝒕) 𝑷𝑷 𝒕𝒕 , 𝒕𝒕 0 4 8 12 16 © 2023 Jean Adams Flamingo Math.com  Inverses of 2.10 Exponential Functions Practice Set 2 Problems 1 − 4, Describe how the graph of each function compares with the graph of the parent function 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 𝑥𝑥. Write the (𝑥𝑥, 𝑦𝑦) rule. 1. 𝑦𝑦 = −1 𝑙𝑙𝑙𝑙𝑙𝑙4 (𝑥𝑥) − 3 2. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙5 − 𝑥𝑥 − 1 + 2 3. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙6 𝑥𝑥 + 3 − 5 4. 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑥𝑥 + 2 + 4 Problems 5 – 8, Use a graphing utility to graph each function. Then, analyze. 5. 𝑓𝑓 𝑥𝑥 = log(𝑥𝑥 − 3) Domain: Left-end behavior: Range: Asymptote: Right-end behavior: Intercept(s): Concavity: 6. ℎ 𝑥𝑥 = 2 log 𝑥𝑥 − 3 Domain: Left-end behavior: Range: Asymptote: Right-end behavior: Intercept(s): Concavity: © 2023 Jean Adams Flamingo Math.com 7. 𝑔𝑔 𝑥𝑥 = ln(4 − 𝑥𝑥) Domain: Left-end behavior: Range: Asymptote: Right-end behavior: Intercept(s): Concavity: 8. 𝑓𝑓 𝑥𝑥 = − ln 𝑥𝑥 + 3 − 4 Domain: Left-end behavior: Range: Asymptote: Right-end behavior: Intercept(s): Concavity: Problems 9 – 12, In the table given, determine whether the function is exponential behavior or logarithmic behavior, explain how you know for each question. 9. 𝒙𝒙 1 2 3 4 5 10. 𝒙𝒙 2 4 8 16 32 𝒇𝒇(𝒙𝒙) −1 1 5 13 29 𝒈𝒈(𝒙𝒙) 1 2 3 4 5 11. 𝒙𝒙 5 25 125 625 3125 12. 𝒙𝒙 1 2 3 4 5 𝒉𝒉(𝒙𝒙) 1 2 3 4 5 𝒌𝒌(𝒙𝒙) 1 0.5 0.25 0.125 0.0625 © 2023 Jean Adams Flamingo Math.com  Free Response Question 13. Given the function 𝑓𝑓 𝑥𝑥 = ln 𝑥𝑥 for all 𝑥𝑥 > 0, and 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2 − 9 for all real 𝑥𝑥. A. Find the composition 𝐻𝐻 𝑥𝑥 = 𝑓𝑓 𝑔𝑔 𝑥𝑥. B. State the domain and range of 𝐻𝐻(𝑥𝑥). C. Describe the symmetry of the graph of 𝐻𝐻(𝑥𝑥). Show work to support your answer. D. Write the formula for 𝐻𝐻 −1 (𝑥𝑥) for all 𝑥𝑥 > 3. E. Graph 𝐻𝐻 𝑥𝑥 for 𝑥𝑥 > 0 and 𝐻𝐻 −1 (𝑥𝑥) © 2023 Jean Adams Flamingo Math.com  Free Response Question 14. The table below is a representation of values for the function 𝑓𝑓 𝑥𝑥 = 𝑒𝑒 𝑥𝑥−3 + 2. 𝒙𝒙 −6 −1 0 1 4 7 𝒇𝒇(𝒙𝒙) 2.0001 2.018 2.049 2.135 4.718 56.598 A. What is the equation of the horizontal asymptote for 𝑓𝑓(𝑥𝑥). Justify your answer. B. Does the graph of 𝑓𝑓 𝑥𝑥 lie above or below the horizontal asymptote. Explain your reasoning. C. Let 𝑔𝑔 𝑥𝑥 represent the inverse of 𝑓𝑓(𝑥𝑥). Find the equation of the inverse function 𝑓𝑓 −1 𝑥𝑥 = 𝑔𝑔(𝑥𝑥). Show your work to support your answer. D. What is the vertical asymptote of 𝑦𝑦 = 𝑔𝑔(𝑥𝑥)? Justify your answer. E. Sketch both functions on the grid provided. State the domain and range of both 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥). © 2023 Jean Adams Flamingo Math.com  Logarithmic Functions 2.11 Practice Set 1 Problems 1-8, the graph of a logarithmic function is given. Match each graph to one of the following functions. Find domain and range for each function. [A] 𝑦𝑦 = − log 3 (2 − 𝑥𝑥) [B] 𝑦𝑦 = −2 log 3 (𝑥𝑥) [C] 𝑦𝑦 = 2 − log 3 (𝑥𝑥) [D] 𝑦𝑦 = log 3 (2 − 𝑥𝑥) [E] 𝒚𝒚 = log 𝟑𝟑 (𝒙𝒙) [F] 𝑦𝑦 = log 3 (−𝑥𝑥) [G] 𝑦𝑦 = log 3 (𝑥𝑥 − 2) [H] 𝑦𝑦 = − log 3 (3 − 𝑥𝑥) 1. 2. 3. 4. 5. 6. 7. 8. © 2023 Jean Adams Flamingo Math.com Problems 9 – 14, find the domain of each logarithmic function. 2 9. 𝑓𝑓 𝑥𝑥 = log 2 (𝑥𝑥 + 6) 10. 𝑓𝑓 𝑥𝑥 = ln 𝑥𝑥 − 3 11. 𝑓𝑓 𝑥𝑥 = log(5 − 𝑥𝑥) 12. 𝑓𝑓 𝑥𝑥 = log 5 (4 − 𝑥𝑥) 2 13. 𝑓𝑓 𝑥𝑥 = ln 𝑥𝑥 + 2 14. 𝑓𝑓 𝑥𝑥 = log 2 𝑥𝑥−2 𝑥𝑥+1 Problems 15 – 18, given the function, determine the asymptotes and the end behaviors, use limit notation. 15. 𝑓𝑓 𝑥𝑥 = ln(𝑥𝑥 + 3) 16. 𝑔𝑔 𝑥𝑥 = log 1 (𝑥𝑥 − 2) 2 17. ℎ 𝑥𝑥 = log 2 𝑥𝑥 + 1 18. 𝑘𝑘 𝑥𝑥 = − log 3 (−𝑥𝑥) © 2023 Jean Adams Flamingo Math.com Problems 19 – 20, answer each question. 19. Given 𝐹𝐹 𝑥𝑥 = log 3 2𝑥𝑥 − 1 − 4 A. Is the graph left or right of the vertical E. If 𝐹𝐹 𝑥𝑥 = −2, what is x? What point is on the asymptote? graph of F? B. Is 𝐹𝐹 increasing or decreasing? Why? F. What is the zero of F? C. What is the domain of F? G. lim 𝐹𝐹 𝑥𝑥 𝑥𝑥→∞ D. What is 𝐹𝐹 122 ? What point is on the graph H. What is the concavity of F? why? of F? 20. Given 𝐺𝐺 𝑥𝑥 = 4 − 2 log 2 3 − 𝑥𝑥 A. Is the graph left or right of the vertical E. If 𝐺𝐺 𝑥𝑥 = −4, what is x? What point is on the asymptote? graph of G? B. Is 𝐺𝐺 increasing or decreasing? Why? F. What is the zero of G? C. What is the domain of G? G. lim 𝐺𝐺 𝑥𝑥 𝑥𝑥→−∞ D. What is 𝐺𝐺 −125 ? What point is on the graph H. What is the concavity of G? why? of G? © 2023 Jean Adams Flamingo Math.com Problems 21 – 22, Use the table of values below to determine whether the function is increasing or decreasing, and the concavity of the function. Make a sketch. Can you determine the domain of the function? 21. 𝒙𝒙 −500 −6 −2 0 1 1.99 𝒇𝒇(𝒙𝒙) −8.972 −3 −2 −1 0 6.644 22. 𝒙𝒙 −1.99 −1 1 7 25 500 𝒈𝒈(𝒙𝒙) −4.192 0 1 2 3 5.660 Problem 23, complete the table and interpret your results. Then, use a calculator to graph the function on the grid provided. 23. The time 𝑡𝑡 in hours for a culture of bacteria to double if it is increasing at a continuous ln 2 rate of 𝑟𝑟 is given by 𝑡𝑡 = 𝑟𝑟. 𝒓𝒓 0.005 0.010 0.015 0.020 0.025 0.030 𝒕𝒕 © 2023 Jean Adams Flamingo Math.com  Logarithmic Function 2.12 Manipulation Practice Set 1 Problems 1 − 6, Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms. 1. log 32𝑥𝑥 2. ln 5 𝑦𝑦 3. ln 𝑒𝑒 4 𝑦𝑦 6 4. log(1000𝑥𝑥 4 ) 𝑥𝑥 3 6. log 2 𝑥𝑥 2 𝑦𝑦 3 5. ln 𝑦𝑦 Problems 7 − 12, Assuming x , y, and z are positive, use properties of logarithms to write the expression as a single logarithm. 7. log 2 64 − log 2 4 8. ln 𝑥𝑥 + 5 + 2 ln 𝑥𝑥 9. 5 ln 𝑥𝑥 + 3 ln 𝑦𝑦 − 4 ln 𝑧𝑧 10. 1 log 𝑥𝑥 − 2 log 𝑦𝑦 3 11. 1 2 log 𝑥𝑥 + 1 − log 𝑥𝑥 − log(𝑥𝑥 − 3) 12. 3 ln 𝑥𝑥 + ln 𝑥𝑥 − 2 − 2 ln(𝑥𝑥 2 − 4) 2 © 2023 Jean Adams Flamingo Math.com Problems 13 − 16, Use the change of base formula to write as base 10 or base 𝑒𝑒. Use a calculator to evaluate to three decimal places. 13. log 4 18 14. log 1 32 2 15. log 𝜋𝜋 58 16. log 0.8 16 Problems 17- 18, Use the product property to rewrite the function. Write the 𝑥𝑥, 𝑦𝑦 rule. 17. 𝑔𝑔 𝑥𝑥 = log 3 (9𝑥𝑥) A. Show this is a vertical translation of the function 𝑓𝑓(𝑥𝑥) = log 3 𝑥𝑥 B. Graph both 𝑓𝑓 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) on the grid at right. C. Describe the change in the output values over equal-length input-value intervals. 18. ℎ 𝑥𝑥 = log 1 4𝑥𝑥 2 A. Show this is a vertical translation of the function 𝑓𝑓(𝑥𝑥) = log 1 𝑥𝑥 2 B. Graph both 𝑓𝑓 𝑥𝑥 and ℎ(𝑥𝑥) on the grid at right. C. Describe the change in the output values over equal-length input-value intervals. © 2023 Jean Adams Flamingo Math.com  Logarithmic Function 2.12 Manipulation Practice Set 2 Problems 1- 2, Use the power property to rewrite the function. Write the 𝑥𝑥, 𝑦𝑦 rule. 1. Given 𝑔𝑔(𝑥𝑥) = log 3 𝑥𝑥 A. Use the power property to show this is a vertical dilation of the function 𝑓𝑓(𝑥𝑥) = log 3 𝑥𝑥. B. Graph both 𝑓𝑓 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) on the grid at right. C. Describe the change in the output values over equal-length input-value intervals. 2. ℎ 𝑥𝑥 = log 1 𝑥𝑥 2 for 𝑥𝑥 > 0 2 A. Use the power property to show this is a vertical dilation of the function 𝑓𝑓(𝑥𝑥) = log 1 𝑥𝑥. 2 B. Graph both 𝑓𝑓 𝑥𝑥 and ℎ(𝑥𝑥) on the grid at right. C. Describe the change in the output values over equal-length input-value intervals. © 2023 Jean Adams Flamingo Math.com Problems 3- 4, Use the power property to rewrite the function. Write the 𝑥𝑥, 𝑦𝑦 rule. 3. 𝑔𝑔(𝑥𝑥) = log 2 (𝑥𝑥 + 3) ln 𝑥𝑥 A. Use the change of base property log 𝑏𝑏 𝑥𝑥 = ln 𝑏𝑏 where 𝑎𝑎 > 0 and 𝑎𝑎 ≠ 1 to show this is a horizontal shift of the function 𝑓𝑓(𝑥𝑥) = log 2 𝑥𝑥. B. Graph both 𝑓𝑓 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) on the grid at right. C. Describe end behavior using limits. D. Describe the change in the output values over equal-length input-value intervals. 4. ℎ 𝑥𝑥 = log 5 5 − 𝑥𝑥 ln 𝑥𝑥 A. Use the change of base property log 𝑏𝑏 𝑥𝑥 = ln 𝑏𝑏 where 𝑎𝑎 > 0 and 𝑎𝑎 ≠ 1 to show this is a horizontal shift of the function 𝑓𝑓(𝑥𝑥) = log 5 (−𝑥𝑥). B. Graph both 𝑓𝑓 𝑥𝑥 and ℎ(𝑥𝑥) on the grid at right. C. Describe end behavior using limits. D. Describe the change in the output values over equal-length input-value intervals. © 2023 Jean Adams Flamingo Math.com  Problems 5 - 6, Graph and analyze each function. Write the 𝑥𝑥, 𝑦𝑦 rule. 5. 𝑓𝑓 𝑥𝑥 = −2 ln(5 − 𝑥𝑥) A. Find the domain of 𝑓𝑓 B. Graph 𝑓𝑓 C. State the range and asymptotes. D. State any intercepts. E. Function behavior and concavity. F. Use limits to state end behavior. 2 6. 𝑔𝑔 𝑥𝑥 = ln 𝑥𝑥 + 4 for 𝑥𝑥 > −4 A. Find the domain of 𝑔𝑔 B. Graph 𝑔𝑔 C. State the range and asymptotes. D. State any intercepts. E. Function behavior and concavity. F. Use limits to state end behavior. © 2023 Jean Adams Flamingo Math.com  Exponential & Logarithmic 2.13 Equations and Inequalities Practice Set 1 Problems 1 − 6, Solve for 𝑥𝑥 and find the exact solution algebraically. 1. 3𝑥𝑥 = 81 2. 5𝑥𝑥 = 625 1 1 𝑥𝑥 4. 5 10𝑥𝑥 = 175 3. 2 = 64 1 𝑥𝑥⁄ 3 2 𝑥𝑥 32 5. 32 =2 6. 3 = 243 4 Problems 7 – 12, Solve the logarithmic equation algebraically. Be sure to check for extraneous solutions. 7. log 𝑥𝑥 625 = 4 8. 5 ln 2𝑥𝑥 = 20 9. ln 𝑥𝑥 + ln 𝑥𝑥 − 4 = ln 5 10. log 3 9 + 2 = 2 log 3 (𝑥𝑥 − 1) © 2023 Jean Adams Flamingo Math.com 11. log 2 𝑥𝑥 + 8 + log 2 𝑥𝑥 + 7 = 1 12. ln 2𝑥𝑥 − 1 = 3 Problems 13 – 16, Solve the exponential equation algebraically. Approximate the result to three decimal places. Verify your answer using a graphing utility. 13. 100𝑒𝑒 −2𝑥𝑥 = 75 14. 52𝑥𝑥−1 = 2𝑥𝑥+3 15. 5 36−𝑥𝑥 = 40 16. 𝑒𝑒 𝑥𝑥+3 = 𝜋𝜋 𝑥𝑥 Problems 17- 22, solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. 17. ln 𝑥𝑥 + 2 = 1 18. 5 log 4 0.5𝑥𝑥 = 12 © 2023 Jean Adams Flamingo Math.com 19. 3 + 2 ln 𝑥𝑥 = 9 20. log 4 𝑥𝑥 + 2 − log 4 𝑥𝑥 − 1 = 1 21. ln 𝑥𝑥 − 8 = 5 22. log 6 𝑥𝑥 + log 6 𝑥𝑥 + 5 = 2 © 2023 Jean Adams Flamingo Math.com  Exponential & Logarithmic 2.13 Equations and Inequalities Practice Set 2 Problems 1-4, Solve each quadratic type exponential equation. Support your answer graphically. 1. 𝑒𝑒 2𝑥𝑥 − 𝑒𝑒 𝑥𝑥 = 42 2. 32𝑥𝑥 + 3𝑥𝑥 = 6 3. 22𝑥𝑥+1 + 5 2𝑥𝑥 = 3 4. 9𝑥𝑥 − 3𝑥𝑥+2 + 18 = 0 © 2023 Jean Adams Flamingo Math.com Problems 5-8, Solve each quadratic type logarithmic equation. Support your answer graphically. 5. log 2 𝑥𝑥 + 4 + log 2 𝑥𝑥 + 3 = 1 6. log 2𝑥𝑥 2 + 5𝑥𝑥 − 2 = 1 7. 𝑥𝑥1+log 𝑥𝑥 =100 8. ln 𝑥𝑥 2 + 15 = ln 𝑥𝑥 + ln(8) © 2023 Jean Adams Flamingo Math.com Problems 9-10, complete the table to find an interval containing the solution of the equation. Then use a graphing utility to graph both sides of the equation to estimate the solution. Round your answer to three decimal places. 1 9. 3 2 𝑥𝑥 = 24 𝒙𝒙 5.5 5.6 5.7 5.8 5.9 30.5𝑥𝑥 10. 2 ln 5𝑥𝑥 = 6 𝒙𝒙 4 5 6 7 8 2 ln(5𝑥𝑥) Problems 11-15, use the properties of logarithms to express 𝑦𝑦 as a function of 𝑥𝑥. The constant 𝐶𝐶 is a positive number. 11. ln 𝑦𝑦 = 2 ln 𝑥𝑥 + ln 𝐶𝐶 12. ln 𝑦𝑦 = ln 𝐶𝐶 + 4𝑥𝑥 13. ln 𝑦𝑦 + 3 = 4𝑥𝑥 + ln 𝐶𝐶 14. ln 𝑦𝑦 − 5 = −2𝑥𝑥 + ln 𝐶𝐶 15. 2 ln 𝑦𝑦 = 3 ln 𝑥𝑥 + 1 + ln 𝐶𝐶 © 2023 Jean Adams Flamingo Math.com Problems 16 – 18, Solve. 16. The demand equation for a paddleboard is 𝑃𝑃 𝑥𝑥 = 850 − 0.5 𝑒𝑒 0.002𝑥𝑥. A. Find the demand x for a paddleboard with a price of 𝑃𝑃 = $575. B. What price would the manufacturer set for a demand of 𝑥𝑥 = 2500 paddleboards? 17. The sales of a cell phone screen protector (in thousands of units) after 𝑥𝑥 hundred dollars is spent on advertising are modeled by 𝑆𝑆 = 20(1 − 𝑒𝑒 𝑘𝑘𝑘𝑘 ). When $1500 is spent on advertising and 5000 units are sold: A. Find the value of k and write the model. B. Estimate the number of units that will be sold if the advertising budget is raised to $3500. 18. A widget manufacturer found that the maximum number of widgets a worker can create in a day is 32. There is a learning curve associated with building up to this maximum production rate for new employees. The learning curve model for the number N of widgets built per day after a new employee has worked t days is 𝑁𝑁 = 32(1 − 𝑒𝑒 𝑘𝑘𝑘𝑘 ). After 15 days on the job a new employee builds 14 widgets. A. Find the value of k and write the learning curve model. B. How many days will it take until the employee is building 28 widgets each day? © 2023 Jean Adams Flamingo Math.com  Exponential & Logarithmic 2.13 Equations and Inequalities Practice Set 3 Problems 1-8, Find the solution set of the inequality. Check for extraneous solutions and verify your answer with a graphing utility. 1. 22𝑥𝑥−1 > 8 2. 915−𝑥𝑥 > 27𝑥𝑥 3. 3𝑥𝑥+1 ≤ 81 4. 25𝑥𝑥−1 ≥ 5𝑥𝑥 2 𝑥𝑥 27 6. 24𝑥𝑥+1 ≤ 64 5. 3 ≥ 8 7. 𝑒𝑒 3𝑥𝑥 < 𝑒𝑒 2−𝑥𝑥 8. 𝑒𝑒 𝑥𝑥 < 𝑒𝑒 3𝑥𝑥+8 © 2023 Jean Adams Flamingo Math.com Problems 9-16, Find the solution set of the inequality. Check for extraneous solutions and verify your answer with a graphing utility. 9. log 2 𝑥𝑥 − 2 ≤ 4 10. log 5 4𝑥𝑥 − 8 ≥ log 5 (2𝑥𝑥 + 4) 11. log 2 2𝑥𝑥 + 1 < 3 12. log 1 2𝑥𝑥 − 8 < log 1 𝑥𝑥 − 5 2 2 13. log 𝑥𝑥 + log 𝑥𝑥 + 6 ≥ log(5𝑥𝑥 + 12) 14. log 1 𝑥𝑥 + 3 ≤ log 1 2𝑥𝑥 − 4 2 2 15. log 1 𝑥𝑥 − 3 < log 1 2𝑥𝑥 − 5 16. log 3 𝑥𝑥 + 3 ≤ log 3 2𝑥𝑥 − 6 2 2 © 2023 Jean Adams Flamingo Math.com  Logarithmic Functio