Name. Unit 4B: Exponential Functions

Transcription

1 Name Unit 4B: Exponential Functions Math 1B Spring 2017

2 Table of Contents STANDARD 6-LINEAR vs EXPONENTIAL FUNCTIONS... 3 PRACTICE/CLOSURE... 4 STANDARD 7-CREATING EXPLICIT EQUATIONS COMPOUND INTEREST NOTES PRACTICE/CLOSURE PRACTICE/CLOSURE HALF-LIFE Guided Notes PRACTICE/CLOSURE PRACTICE/CLOSURE STANDARD 8-CONSTRUCT EXPONENTIAL GRAPHS Practice/Closure PRACTICE/CLOSURE STANDARD 9-EVALUATING EXPONENTIAL FUNCTIONS AND GRAPHS Exponential Decay: Depreciation Problems PRACTICE/CLOSURE STANDARD 10-INTERPRETING KEY PARTS OF EXPONENTIAL FUNCTIONS Practice problems: PRACTICE/CLOSURE Extensions

3 STANDARD 6-LINEAR vs EXPONENTIAL FUNCTIONS Linear vs. Exponential Word Problems At separate times in the course, you ve learned about linear functions and exponential functions, and done word problems involving each type of function. Today s assignment combines those two types of problems. In each problem, you ll need to make a choice of whether to use a linear function or an exponential function. Below is some advice that will help you decide. Linear Function f(x) = mx + b or f(x) = m(x x 1) + y 1 b is the starting value, m is the rate or the slope. m is positive for growth, negative for decay. Exponential Function f(x) = a b x a is the starting value, b is the base or the multiplier. b > 1 for growth, 0 < b < 1 for decay. See below for ways to find the base b. Choosing linear vs. exponential In growth and decay problems (that is, problems involving a quantity increasing or decreasing), here s how to decide whether to choose a linear function or an exponential function. If the growth or decay involves increasing or decreasing by a fixed number, use a linear function. The equation will look like: y = mx + b f(x) = (rate) x + (starting amount). If the growth or decay is expressed using multiplication (including words like doubling or halving ) use an exponential function. The equation will look like: f(x) = (starting amount) (base) x. 3

4 PRACTICE/CLOSURE 1. Decide whether the word problem represents a linear or exponential function. Circle either linear or exponential. Then, write the function formula. a. A library has 8000 books, and is adding 500 more books each year. Linear or exponential? y =. b. A gym s customers must pay $50 for a membership, plus $3 for each time they use the gym. Linear or exponential? y =. c. A bank account starts with $10. Every month, the amount of money in the account is tripled. Linear or exponential? y =. d. At the start of a carnival, you have 50 ride tickets. Each time you ride the roller coaster, you have to pay 6 tickets. Linear or exponential? y =. e. There are 20,000 owls in the wild. Every decade, the number of owls is halved. Linear or exponential? y =. 2. Decide whether the table represents a linear or exponential function. Circle either linear or exponential. Then, write the function formula. a. x y Linear or exponential? y =. b. x y Linear or exponential? y =. c. x y Linear or exponential? y =. 4

5 d. x y Linear or exponential? y = e. x y Linear or exponential? y = f. x y Linear or exponential? y = g. x y Linear or exponential? y = 5

6 n 1a. Complete the tables below, for the functions f n 2 and f n 2n n, f n on a coordinate plane for each of the formulas. and then graph the points y x 2 4 b. Describe the change in each sequence when n increases by 1 unit. 6

7 2. Two equipment rental companies have different penalty policies for returning a piece of equipment late: Company 1: On day 1, the penalty is $5. On day 2, the penalty is $10. On day 3, the penalty is $15. On day 4, the penalty is $20 and so on, increasing by $5 each day the equipment is late. Company 2: On day 1, the penalty is $0.01. On day 2, the penalty is $0.02. On day 3, the penalty is $0.04. On day 4, the penalty is $0.08 and so on, doubling in amount each additional day late. Jim rented a digger from Company 2 because he thought it had the better late return policy. The job he was doing with the digger took longer than he expected, but it did not concern him because the late penalty seemed so reasonable. When he returned the digger 15 days late, he was shocked by the penalty fee. a. What did he pay, and what would he have paid if he had used Company 1 instead? b. Describe how the amount of the late charge changes from any given day to the next successive day in both companies 1 and 2. c. How much would the late charge have been after 20 days under Company 2? 3. Solve the following system of equations graphically. 7

8 f g x x x x y x

9 4. Estimated Female Population of the United States Year Population in millions a. Use the statistical features of your calculator to construct a scatter plot and fit a linear function to the data. Calculate and interpret the correlation coefficient. b. Use the statistical features of your calculator to construct a scatter plot and fit an exponential function to the data. Calculate and interpret the correlation coefficient. c. Which regression provides the more appropriate model for the data set? Explain. d. Compare the growth rates of the two functions e. Using both functions project the population in the year

10 STANDARD 7-CREATING EXPLICIT EQUATIONS COMPOUND INTEREST NOTES When interest is computed on the principal and any previously earned interest, it is called Compound Interest When interest is calculated once each year, we say that it is compounded In many cases, interest is computed at more frequent intervals than that. It can be compounded semiannually ( a year), quarterly ( a year), monthly ( a year), or even daily ( day). Formula for Computing Compound Interest r A P 1 n where A is the future value (principal + interest) r is the yearly interest rate in decimal form n is the number of times per year the interest is compounded (yearly, semiannually, quarterly, monthly or daily) t is term of the investment in years nt Example: Find the interest on $7, compounded at 3% for 5 years. Compounded Work Amount Yearly Semiannually Quarterly Monthly Yearly 10

11 You have $5000 to invest and you want to know which of the following investment situations will give you the most money at the end of 8 years. The interest rate for all of the situations is 6%. *Make sure you put your exponents in parentheses! Show what you are plugging into the calculators! 1. Calculate the investment if it is compounded annually. 2. Calculate the investment if it is compounded semi-annually (twice a year). 3. Calculate the investment if it is compounded quarterly (four times a year). 4. Calculate the investment if it is compounded monthly (12 times a year). 5. Calculate the investment if it is compounded daily (365 times a year). 6. What did you discover? Which situation will give you the most? Which situation is the most realistic for banks? How does the situation change if you are now a credit card company? Why is it unrealistic to compound daily? Explain your reasoning. Compound Interest Foldable 11

12 PRACTICE/CLOSURE Solve each problem. Show all work. 1. How much money will you have in 8 years if you invest $4000 at 3 ½ % compounded quarterly? 2. What interest rate do you need for a $5000 investment to double in 10 years? 3. How much money do you need to invest at 2 ¾ % in order to have $12,000 after 7 years? 4. How much money will you have in 6 months if you invest $1000 at 3% compounded monthly? 5. How much interest will you earn in 8 years if you invest $7500 at 4 ¼ % compounded semiannually? 6. You are investing $1500 at 5.2% compounded daily. How much money will you have in 12 years? 7. How much money do you need to invest at 2.8% compounded daily in order to have $25,500 at the end of 8 years? 12

13 8. If you deposit $4500 at 5% annual interest compounded quarterly, how much money will be in the account after 10 years? 9. If you deposit $4000 into an account paying 9% annual interest compounded monthly, how long until there is $10000 in the account? 10. If you deposit $2500 into an account paying 11% annual interest compounded quarterly, ho w long until there is $4500 in the account? 11. How much money would you need to deposit today at 5% annual interest compounded mon thly to have $20000 in the account after 9 years? 12. If you deposit $6000 into an account paying 6.5% annual interest compounded quarterly how long until there is $12600 in the account? 13. If you deposit $5000 into an account paying 8.25% annual interest compounded semiannually, how long until there is $9350 in the account? 13

14 PRACTICE/CLOSURE 14

15 HALF-LIFE Guided Notes Key Points of Half-Life b = ½ The amount will never reach 0! If you are only taking away half each interval you are still left with half! Definition of Half-life The term half-life refers to the time it takes for half of the atoms in a radioactive substance to decay. t h as a whole represent the NUMBER of HALF-LIVES that exists!!!!!! Guided Practice 1. The half-life of carbon-11 is 20 minutes. Fill in the table to find the amount of carbon-11 left after 3 half-life cycles. Number of Half Life Cycles Amount remaining 15

16 2. The half-life of a radioactive substance is 2 days. There are currently 150 grams of this substance. a) Write an explicit equation. b) How many grams will remain in 8 days? 3. Hg-197 is used in kidney scans and it has a half-life of hours. a. Write the exponential decay function for a 12-mg sample. b. Find the amount remaining after 72 hours. Half-Life Foldable 16

17 Group Work 1) Arsenic-74 is used to locate brain tumors. It has a half-life of 17.5 days. Write an exponential function for a 90-gram sample. Use the function to find the amount remaining after 70 days. 2) Phospohorus-32 is used to study a plant s use of fertilizer. It has a half-life of 28 days. Write the exponential decay function of a 50-mg sample. Find the amount of phosphorus-32 remaining after 84 days. 3) Iodine-131 is used to find leaks in water pipes. It has a half-life of 8 days. Write the exponential decay function for a 200-mg sample. Find the amount of iodine-131 remaining after 24 days. 4) Hg-197 is used in kidney scans and it has a half-life of hours. Write the exponential decay function for a 12-mg sample. Find the amount remaining after 72 hours. 5) Sr-85 is used in bone scans and is has a half-life of 64.9 days. Write the exponential decay function for an 8-mg sample. Find the amount remaining after 100 days. 6) A decaying radioactive substance originally weights 500 grams and is reduced to grams in 68 days. Find the half-life of the substance. 17

18 PRACTICE/CLOSURE Scientists use half-life to determine the time it takes for any specified concentration of a substance in the body or in nature to decrease by half its original value. Half-Life: y = a( 1 2 )^t h or y = a(.5)^ t h Example 1. Iron-59 is used in medicine to diagnose blood circulation disorders. The half-life of Iron 5 is 44.5 days. How much of a mg sample will remain after days? Example 2. Iodine-131 has a half-life of 8 days. What fraction of the original sample would remain at the end of 32 days? Example 3: Titanimu-51 decays with a half-life of 6 minutes. You started with 200 mg. How much titanium would remain after one hour? Example 4. With a half life of 28.8 years, how long will it take 1g of strontium-90 to decay to 125mg? 18

19 5. The half-life of radium 226 is 1602 years. If you have 500 grams of radium today how many grams would have been present 9612 years ago? 6. A 1.000kg block of phosphorus-32, which has a half life of 14.3 days, is stored for days. At the end of this period, how much phosphorus-32 remains? 7. The half-life of chromium-51 is 28 days. If the sample contains 510 grams, how much chromium would remain after 1 year? 8. A 0.5g sample of radioactive Iodine-131 has a half life of 8.0 days. After 40 days, how much is left? 19

20 Half-Life Labs PRACTICE/CLOSURE Most things are composed of stable atoms. However, the atoms in radioactive substances are unstable and the break down in a process called radioactive decay. The rate of decay varies from substance to substance. The term half-life refers to the time it takes for half of the atoms in a radioactive substance to decay. For example, the half-life of carbon-11 is 20 minutes. This means that 2,000 carbon-11 atoms will be reduced to 1,000 carbon- 11 atoms in 20 minutes. Half-lives vary from a fraction of a second to billions of years. For example, the half-life of polonium-214 is seconds. The half-life of rubidium-87 is 49 billion years. In the problems below, write an exponential decay function in order to find the solution to each problem. (Use function notation) 1) Hg-197 is used in kidney scans and it has a half-life of 1 day. Write the exponential decay function for a 12-mg sample. Find the amount remaining after 6 days. 2) Sr-85 is used in bone scans and it has a half-life of 20 days. Write the exponential decay function for an 8-mg sample. Find the amount remaining after 100 days. 3) I-123 is used in thyroid scans and has a half-life of 5 hours. Write the exponential decay function for a 45-mg sample. Find the amount remaining after 35 hours. 4) An exponentially decaying radioactive ore originally weighs 28 grams and is reduced to 14 grams in 1,000 years. How much will be left in 3,000 years? Write an exponential decay function in order to find the solution. 5) Some radioactive ore which weighed 24 grams 200 years ago has been reduced to 12 grams today. How much will be left 400 years from now? Write an exponential decay function in order to find the solution. 20

21 STANDARD 8-CONSTRUCT EXPONENTIAL GRAPHS Practice/Closure 1) When dropped on to a hard surface, a brand new softball should rebound to about 2/5 the height from which it is dropped. a. If the softball is dropped 25 feet from a window onto concrete, what pattern of rebound heights can be expected? b. Make a table and plot of predicted rebound data for 5 bounces. Bounce Number Rebound Height (in feet) 25 c. What NOW-NEXT rule and y = rules give ways of predicting rebound height after any bounce? 2) Here are some data from bounce tests of a softball dropped from a height of 10 feet. Bounce Number Rebound Height (in feet) a. What do these data tell you about the quality of the tested softball? b. What are the first six bounce heights would you expect from this ball if it were dropped from 20 feet instead of 10 feet? 21

22 3) If a basketball is properly inflated, it should rebound to about ½ the height from which it is dropped. a. Make a table and plot showing the pattern to be expected in the first 5 bounces after a ball is dropped from a height of 10 feet. Bounce Number Rebound Height (in feet) 10 b. At which bounce will the ball first rebound less than 1 foot? Show how the answer to this question can be found in the table and on the graph. c. Write a rule using NOW-NEXT and a rule beginning y = that can be used to calculate the rebound height after many bounces. d. How will the data table, plot, and rules change for predicting rebound height if the ball is dropped from a height of 20 feet? Bounce Number Rebound Height (in feet) 20 NOW-NEXT Rule: Y = e. How will the data table and rules change for predicting rebound height if the ball is somewhat overinflated and rebounds to 3/5 of the height from which it is dropped? NOW-NEXT Rule: Y = Bounce Number Rebound Height (in feet) 20 22

23 PRACTICE/CLOSURE Other vehicles and equipment depreciate over time as well, such as trucks, boats, tractors, computer equipment, etc. Let us look at other items which depreciate over time. The cost of a new truck is $32,000. It depreciates at a rate of 15% per year. This means that it loses 15% of each value each year. Tasks: Draw the graph of the truck s value against time in year. Find the formula that gives the value of the truck in terms of time. Find the value of the truck when it is four years old. Estimate when the truck will be worth half of its value (about $16,000). Let s start by making a table of values. To fill in the values we start with 32,000 at time t = 0. Then we multiply the value of the car by 85% for each passing year. (Since the car looses 15% of its value, which means that it keeps 85% of its value). Remember that 85% means that we multiply by the decimal Number of Years Value of the Truck ,00 0 Graph the data on the coordinate grid below. Remember to label your axes. 23

24 Now let us write the equation for the data. Initial value: Percentage rate of depreciation: Equation: Use the equation to determine the value of the truck when it is 4 years old. Value of the 4 year old truck: Compare this value with the value in the data table. It should be the same value if your equation is correct. Use the table, graph, equation, or graphing calculator to estimate the time it will take for the truck to worth half of its initial value. Try a few more: 1) The cost of a new ATV (all-terrain vehicle) is $7200. It depreciates at 18% per year. Draw the graph of the vehicle s value against time in years. Find the formula that gives the value of the ATV in terms of time. Find the value of the ATV when it is ten year old. Number of Years Value of the ATV

25 2) A tool & die business purchased a piece of equipment of $250,000. The value of the equipment depreciates at a rate of 12% each year. a. Write an exponential decay equation for the value of equipment. b. What is the value of equipment after 5 years? c. Make a table and graph the model. Number of Years Value of the Equipment d. Estimate when the equipment will have a value of $70,

26 STANDARD 9-EVALUATING EXPONENTIAL FUNCTIONS AND GRAPHS Exponential Decay: Depreciation Problems Most cars lose value each year by a process known as depreciation. You may have heard before that a new car loses a large part of its value in the first 2 or 3 years and continues to lose its value, but more gradually, over time. That is because the car does not lose the same amount of value each year, it loses approximately the same percentage of its value each year. What kind of model would be useful for calculating the value of a car over time? Let us look at an example of depreciation: Suppose the value of car when new is $20,000 and it depreciates at a rate of 20% each year. What is the percentage rate of depreciation each year? The percentage rate of depreciation is 20%, which means that 80% of the value of the car remains every year. We can calculate this percentage rate by subtracting 20% from 100% in order to calculate the value remaining of 80% each year. What is the initial value of the car? $ What is the percentage rate? 100% - 20% = % each year. Write that percentage as a decimal. 26

27 Let us look at the depreciation data over a 5 year period of time (rounded to the nearest dollar). Number of Years Value of the Car , ,00 0 8,192 Graph the table on the graph below. Write an explicit equation for the data in order to calculate the value of deprecation for any year. Y = initial value (1 percentage rate of depreciation) time Y = 20,000(1-0.20) x Y = 20,000(0.80) x Use this equation to find the depreciated value of the car for year 8. Y = 20,000(0.80) 8 = When will the depreciated value of the car be worth $5000? Estimate this value to the nearest tenth of a year with your calculator by inputting the equation in your calculator in Y1 and 5000 in Y2 then find the point of intersection (2 nd TRACE, 5, ENTER 3 times). 27

28 YOUR TURN Matt bought a new car at a cost of $25,000. The car depreciates approximately 15% of its value each year. a.) What is the percentage rate of depreciation for the value of this car? (Remember that the percentage rate of depreciation is 0 < b < 1.) b.) Write an equation to model the decay value of this car. y = where y is the value of the car; x is the number of years since new purchase c.) What will the car be worth in 10 years? 28

29 PRACTICE/CLOSURE Use equations, graph, or tables to find the solutions to the problems below. 1) A computer valued at $6500 depreciates at the rate of 14.3% per year. Write a function that models the value of the computer. Find the value of the computer after three years. 2) A new truck that sells for $29,000 depreciates 12% each year. Write a function that models the value of the truck. Find the value of the truck after 7 years. 3) A new car that sells for $18,000 depreciates 25% each year. Write a function that models the value of the car. Find the value of the car after 4 years. 4) You purchased a car for $19,500. The car will depreciate at a rate of 12% each year. Write a formula to represent the value of the car after x number of years. Find the value of the car after 4 years. 5) Each graph below shows the expected decrease in a car s value over the next five years. Write a function to model each car s depreciation. Determine which car will be worth more after 10 years. 29

30 STANDARD 10-INTERPRETING KEY PARTS OF EXPONENTIAL FUNCTIONS Exponential Function Models Students will work in small groups to complete this assignment. Students should be communicating and collaborating on the material discussed in this assignment. Be ready for a whole group discussion at the conclusion of the assignment. Arithmetic sequences are modeled by polynomial functions: Linear: y = mx + b or quadratic: y = ax 2 + bx + c (coming soon) Geometric sequences are modeled by exponential functions: y = a b x where b = constant ratio and a is a constant Stages Members What is a practical domain and range for this function? What is the initial value? What is the Recursive Function? What is the Explicit Function? Is this problem Realistic? 30

31 Now assume that every person calls 5 people Stages Member What is a practical domain and range for this function? What is the initial value? What is the Recursive Function? What is the Explicit Function? Is this problem Realistic? What type of event would a function of this size be necessary? Why can I write 1 (1/2) x = 1 2 -x? Explain? 31

32 Most cars depreciate as they get older. Suppose that an automobile that originally costs $14,000 loses one-fifth of its value each year. What is the value after 2 years? After 2.5 years? The recursive function doing iteration will easily give us the answer for the first part of this question, but not the second. Fill in the table below and write the instructions that you would need to give another person to find the value of the car after x years. Original value After 1 year After 2 year After 2.5 year After 3 year After 4 year The general formula for an exponential function is: y = a b x Look at the method used and figure out what each of these variables represent in the formula for exponential growth or decay. a = b = x = What is the value of the car at 2.5 years? Why isn t the answer half-way between the value of y when x = 2 and x =3? Approximately when will the car be worth $7000? What is the half-life of the car? 32

33 Radioactivity is measured in units called rads. A sample of phosphorus-33 is found to give off 480 rads initially. The half-life of phosphorus-33 is 25 days. What continuous function represents the radioactivity measure of this sample? What ordered pair represents the initial? At period 25, the reading is 240. (25, 240) WHY????? Explain your work Create an exponential model using your calculator? Explain your work. Suppose you deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if the interest is compounded a) quarterly b) daily Suppose you want to have $2500 after 2 years. Find the amount you should deposit for each of the situations described. a) The account pays 2.25% annual interest compounded monthly. b) The account pays 5% annual interest compounded yearly. The amount g (in trillions of cubic feet) of natural gas consumed in the United States from 1940 to 1970 can be modeled by G = 2.9(1.07) t where t is the number of years since a) Identify the initial amount, the growth factor, and the annual percent increase. b) Estimate the natural gas consumption in

34 The student enrollment E of a high school was 1240 in 1990 and increased by 15% per year until Which exponential growth model shows the school's student enrollment in terms of t, the number of years since 1990? a) E = 15(1240) t b) E = 1240(1.15) t c) E = 1240(15) t d) E = 0.15(1240) t e) E = 1.15(1240) t Practice problems: 1) In 1965 the federal debt of the United States was $322.3 billion. During the next 30 years, the debt increased by about 10.2% each year. a. Write a model giving the amount D (in billions of dollars) of debt t years after b. About how much was the federal debt in 1980? c. Estimate the year when the federal debt was $2,120 billion. 2) You deposit $2500 in a bank that pays 4% interest compounded annually. Use the process below and a graphing calculator to determine the balance of your account each year. a. Find the balance after one year. b. What is the balance after five years? c. How would you enter the formula in part (a) if the interest is compounded quarterly? What would you have to do to find the balance after one year? d. Find the balance after 5 years if the interest is compounded quarterly. Compare this result with your answer to part (b). 34

35 PRACTICE/CLOSURE James invested $10,000 for his child s college fund. The annual interest rate is 8%. He doesn t deposit or withdrawal any money into the account. Let b represent the account balance and t the time in years. 1. Identify the independent & dependent variables. 2. What is the value of the initial balance? What percent represents the initial balance? 3. Consider that you are trying to find the next year s balance. a. What percent represents each of the previous year s balance? For example: When looking for year 2 s balance, what percentage of James balance does year 1 s account balance represent? b. Is the account s balance increasing or decreasing? What percent represents the change? c. Explain the relationship between the two percentages in parts a & b and the account s balance. 4. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. 5. Write a recursive rule for the given situation. 6. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 7. Write an explicit rule for the given situation. Support your findings with work or an explanation. 8. In how many years will the account s balance be $26,000? Show your work. 9. What will the account s balance be in 17 years? Show your work. 35

36 The recent ice storm knocked out your power. Duke Energy tells you that the temperature will increase by 4% each hour. The temperature inside your house is 54º F when the power comes back on. Let t represent the temperature and h represent the time in hours. Identify the independent & dependent variables. What is the value of the initial temperature? What percent represents the initial temperature? Consider that you are trying to find the temperature after the next hour. What percent represents each of the previous hour s temperature? For example: When looking for the 2 nd hour s temperature, what percentage of the house s temperature does the 1 st hour s temperature represent? Is the temperature increasing or decreasing? What percent represents the change? Explain the relationship between the two percentages in parts a & b and the temperature. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. Round to the nearest 10 th if necessary. Write a recursive rule for the given situation. Graph your data on graph paper. Be sure to label your graph and choose an appropriate scale. Is your graph continuous or discrete? Support your reasoning. Write an explicit rule for the given situation. Support your findings with work or an explanation. How long will it take for the temperature each 72 º F? Show your work. What will the temperature be after 2.5 hours? Show your work. 36

37 A culture of bacteria increases by 50% every half hour and begins with 375 bacteria. Let b represent the amount of bacteria and h represent the time in half hours. 1. Identify the independent & dependent variables. 2. What is the value of the initial bacteria amount? What percent represents the initial bacteria amount? 3. Consider that you are trying to find the bacteria amount after the next half hour. a. What percent represents each of the previous half hour s amount? For example: When looking for the bacteria at the 2 nd half hour, what percentage of the bacteria does the 1 st half hour s bacteria represent? b. Is the amount increasing or decreasing? What percent represents the change? c. Explain the relationship between the two percentages in parts a & b and the bacteria amount. 4. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. Round to the nearest 100 th if necessary. 5. Write a recursive rule for the given situation. 6. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 7. Write an explicit rule for the given situation. Support your findings with work or an explanation. 8. How many hours will it take for the bacteria amount to reach 200,000? Show your work. 9. What will the bacteria amount be after 13 hours? Show your work. 37

38 In 1859, a small number of rabbits were introduced to Australia by English Settlers. The rabbits had no natural predators in Australia, so they reproduced rapidly and quickly became a serious problem for sheep and cattle, eating the grasses intended for them. The initial population was 100 rabbits and they reproduce at a rate of 80% yearly. Let r represent the rabbit population and t represent the time in years. 1. Identify the independent & dependent variables. 2. What is the value of the initial population? What percent represents the initial population? 3. Consider that you are trying to find the population after the next year. a. What percent represents each of the previous year s population? For example: When looking for year 2 s population, what percentage of rabbit s population does year 1 s population represent? b. Is the population increasing or decreasing? What percent represents the change? c. Explain the relationship between the two percentages in parts a & b and the rabbit population. 4. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. Round to the nearest whole if necessary. 5. Write a recursive rule for the given situation. 6. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 7. Write an explicit rule for the given situation. Support your findings with work or an explanation. 8. How many years will it take for the population to reach 1,000,000 rabbits? Show your work. 9. What will the rabbit population be after 12 years? Show your work. 38

39 In parts of the United States, wolves are being reintroduced to wilderness areas where they had become extinct. Suppose 20 wolves are released in northern Michigan, and the yearly growth rate for this population is expected to be 20%. Let w represent the wolves population and t represent the time in years. 1. Identify the independent & dependent variables. 2. What is the value of the initial population? What percent represents the initial population? 3. Consider that you are trying to find the population after the next year. a. What percent represents each of the previous year s population? For example: When looking for year 2 s population, what percentage of wolves population does year 1 s population represent? b. Is the population increasing or decreasing? What percent represents the change? c. Explain the relationship between the two percentages in parts a & b and the wolf population. 4. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. Round to the nearest whole if necessary. 5. Write a recursive rule for the given situation. 6. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 7. Write an explicit rule for the given situation. Support your findings with work or an explanation. 8. How many years will it take for the population to reach 75 wolves? Show your work. 9. What will the wolf population be after 17 years? Show your work. 39

40 The price of a yearbook is $25. However, the cost of printing increases every year resulting in the price of the yearbook increasing. The price of a yearbook needs to be raised 1.2% every two years to offset the increase in the cost to make the yearbooks. Let c represent the cost of the yearbook and t represent the time in years. 1. Identify the independent & dependent variables. 2. What is the value of the initial yearbook cost? What percent represents the initial yearbook cost? 3. Consider that you are trying to find the next year s yearbook cost. a. What percent represents each of the previous year s yearbook cost? For example: When looking for year 2 s cost, what percentage of yearbook s cost does year 1 s cost represent? b. Is the cost increasing or decreasing? What percent represents the change? c. Explain the relationship between the two percentages in parts a & b and the yearbook cost. 4. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. 5. Write a recursive rule for the given situation. 6. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 7. Write an explicit rule for the given situation. Support your findings with work or an explanation. 8. How many years will it take for the yearbook cost to reach $30? Show your work. 9. What will the yearbook cost after 19 years? Show your work. 40

41 Extensions Edna and Saul invested $2,500. Their investment doubled every 12 years. Let y represent the amount in dollars and x the time in years. 1. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. 2. Write a recursive rule for the given situation. 3. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 4. Write an explicit rule for the given situation. Support your findings with work or an explanation. 5. How many years will it take for the balance to reach $6,000? Show your work. 6. What will the balance be after 22 years? Show your work. The initial population of Alesha s hometown was 499. The population tripled every 5 years. Let y represent the population and x the time in years. 1. Create a table showing at least five terms for the situation. Show all your work on how you found each dependent variable s value. 2. Write a recursive rule for the given situation. 3. Graph your data on graph paper. a. Be sure to label your graph and choose an appropriate scale. b. Is your graph continuous or discrete? Support your reasoning. 4. Write an explicit rule for the given situation. Support your findings with work or an explanation. 5. How many years will it take for the population to reach 11,000? Show your work. 6. What will the population be after 26 years? Show your work. 41