Unit 3A Modeling with Exponential Functions

Transcription

1 Common Core Math 2 Unit A Modeling with Exponential Functions Name: Period: Estimated Test Date: Unit A Modeling with Exponential Functions 1

2 2 Common Core Math 2 Unit A Modeling with Exponential Functions

3 Common Core Math 2 Unit A Modeling with Exponential Functions Main Concepts Page # Study Guide 4-6 Vocabulary 7 Equivalent Forms of exponential Expressions 7-10 Simplifying Radicals 11-1 Operations with Radicals Rational Exponents and Radicals Exponential Growth and Decay 22-1 Test Review 8-9 Homework Answers 40-

4 Common Core Math 2 Unit A Modeling with Exponential Functions Common Core Standards A-SSE. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. A-CED.1 A-CED. A-REI.2 F-BF.1 F-IF-7 N-RN.2 Create equations and inequalities in one variable and use them to solve problems. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Unit Description In this unit, students will continue their exploration of exponential functions begun in Math I. Students will extend the properties of exponents to rational exponents. Students will continue to explore transformations of exponential functions, adding vertical stretches and compressions. Students will also use exponential functions and equations to solve more complex real world problems (that may involve rational exponents), interpreting values in context. Essential Questions By the end of this unit, I will be able to answer the following questions: Exponential functions model real world problems, of growth and decay, such as monetary growth, population increases or decreases, car values, half-life, etc. One type of function does not fit all situations in life. Exponential functions can be written as explicit expressions or using a recursive process. 4

5 Common Core Math 2 Unit A Modeling with Exponential Functions Unit Skills I can: Solving exponential equations The method of graphing or making a table to solve an exponential equation is appropriate when an exact answer is not required. Graphs and tables of exponential functions The general form of an exponential function is f(x) = a b x, where a is the initial value and b is the base/common ratio. (F-BF.1) Every exponential function has a horizontal asymptote. (F-IF.4) The graph of f(x) + k is shifted k units vertically as compared to the parent graph of f(x). If k is positive, the graph shifts up. If k is negative, the graph shifts down. The graph of f(x + k) is shifted k units horizontally as compared to the parent graph of f(x). If k is positive, the graph shifts left. If k is negative, the graph shifts right. The graph of k f(x) is scaled vertically by a factor of k as compared to the parent graph of f(x) (i.e. the point (x, y) maps to the point (x, k y). If the absolute value of k is greater than 1, then the graph is stretched vertically. If the absolute value of k is less than 1, then the graph is compressed vertically. Applications of exponential functions Exponential functions model real world problems of growth and decay including, but not limited to, monetary growth, population increases or decreases, car values, and half-life. The equation for exponential growth is given by y = a(1 + r) x and the equation for exponential decay is given by y = a(1 r) x, where a is the initial amount, r is the growth or decay rate in decimal form, and x is the number of time intervals that have passed. 5

6 Common Core Math 2 Unit A Modeling with Exponential Functions Vocabulary: Define each word and give examples and notes that will help you remember the word/phrase. Common Difference Example and Notes to help YOU remember: Common Ratio Example and Notes to help YOU remember: Domain Example and Notes to help YOU remember: Explicit function Example and Notes to help YOU remember: Practical Domain Example and Notes to help YOU remember: Theoretical Domain Example and Notes to help YOU remember: Recursive function Example and Notes to help YOU remember: 6

7 Common Core Math 2 Unit A Modeling with Exponential Functions Equivalent Forms of Exponential Expressions Before we begin today s lesson, how much do you remember about exponents? Use expanded form to write the rules for the exponents. OBJECTIVE 1 Multiplying Exponential Expressions 2 4 y 4 y SUMMARY: a m a n = OBJECTIVE 2 Dividing Exponential Expressions (Remember: x x = 1) 6 2 y 10 y SUMMARY: am an = OBJECTIVE Negative Exponential Expressions: Simplify 2 WAYS using expanded form AND the rule from OBJECTIVE 2 2 y y SUMMARY: 1 a n = OBJECTIVE 4 Exponential Expressions Raised to a Power ( 6 ) 2 (y ) 4 (12m) 5 SUMMARY: (a m ) n = SUMMARY: (a b) n = We ve learned how to simplify exponential expressions in the past and reviewed those just now. Next we need to use those properties to find some missing values. 7

8 Common Core Math 2 Unit A Modeling with Exponential Functions Find the value of x in each of the following expressions. 5 x 5 2 = x = x = 4 (5 ) x = 5 6 ( 2 ) x = 1 (4 x ) 1 2 = = 54 5x x 12 = x = x = 1 ( x ) 2 = x = 1 Find the values of x and y in each of the following expressions. 5 x 2 y = (5 ) ( 2 2 x) = 6 2y (x2 5 ) = 26 5 y (5 6) 2 = 5 x 6 y (2 x 6 ) 4 = y ( x 4 2 ) = 4 y 8

9 Common Core Math 2 Unit A Modeling with Exponential Functions These problems will have more than one correct solution pair for x and y. Find at least solution pairs. (5 x )(5 y ) = 5 12 Possible Solutions Option 1 Option 2 Option ( x ) y = Possible Solutions Option 1 Option 2 Option 4 x 4 y = 1 Possible Solutions Option 1 Option 2 Option When there are so many rules to keep track of, it s very easy to make careless mistakes. To help you guard against that, it helps to become a critical thinker. Take a look at the expanded and simplified examples below. One of them has been simplified correctly and there s an error in the other two. Identify the correctly simplified example with a. For the incorrectly simplified examples, write the correct answer and provide suggestions so that the same mistake is not made again. x 2 x = x (4x)(x) = 4x 2 50c 2 d 2 5cd 5 = 45c2 d You ve seen some of the more common mistakes that can happen when simplifying exponential expressions, and you may have made similar mistakes in the past. For each of the next rows of problems, complete one of the problems correctly and two of the problems incorrectly. For the incorrect problems, try to use errors that you think might go unnoticed if someone wasn t paying close attention. When you finish, you ll switch papers with two different neighbors (one for each row) so that they can check your work, find, fix, and write suggestions for how those mistakes can be avoided. (2x 2 y ) 5 (x) 2 (x 2 ) x 2 y 6 2xy 2 8x 2 y x x ( 2xy) 4 9

10 Common Core Math 2 Unit A Modeling with Exponential Functions Properties of Exponents ( Kuta Software Infinite Algebra I) 10

11 Common Core Math 2 Unit A Modeling with Exponential Functions Simplifying Radicals & Basic Operations We are familiar with taking square roots ( ) or with taking cubed roots ( ), but you may not be as familiar with the elements of a radical. n x = r An index in a radical tells you how many times you have to multiply the root times itself to get the radicand. For example, in 81 = 9, 81 is the radicand, 9 is the root, and the index is 2 because you have to multiply the root by itself twice to get the radicand (9 9 = 9 2 = 81). When a radical is written without an index, there is an understood index of =? Radicand Index Root is because = = x 5 =? Radicand Index Root is because = 5 = 2x Yes you can use your calculator to do this, but for some of the more simple problems, you should be able to figure them out in your head. To use your calculator Step 1: Type in the index. Step 2: Press MATH Step : Choose 5: x Step 4: Type in the radicand. 5 24y m 4 n 8 144v 8 BE CAREFUL THAT YOUR VARIABLE ONLY STANDS FOR A POSITIVE NUMBER. For instance, a 2 = a, if a 0, but if a < 0, then a 2 = a (the opposite of a), since the square root sign always indicates the positive square root. Since there is no way for us to know if a is positive or negative, we use absolute value. So, a 2 = a Example: a b 2 = a a 2 b 2 = a b a a cannot be negative because we would not have been able to take the square root of a. However, b could be negative, so use absolute value signs. 11

12 Common Core Math 2 Unit A Modeling with Exponential Functions BUT not every problem will work out that nicely! Try using your calculator to find an exact answer for 24 = The calculator will give us an estimation, but we can t write down an irrational number like this exactly it can t be written as a fraction and the decimal never repeats or terminates. The best we can do for an exact answer is use simplest radical form. Here are some examples of how to write these in simplest radical form. See if you can come up with a method for doing this. Compare your method with your neighbor s and be prepared to share it with the class. (Hint: do you remember how to make a factor tree?) 12 = 2 24 = = 2 Simplifying Radicals: 1) 2) Examples: 2 16x ) 8x 15x 5 80n x 2x y x y z 192x y z x z 12

13 Common Core Math 2 Unit A Modeling with Exponential Functions 1

14 Common Core Math 2 Unit A Modeling with Exponential Functions Multiplying Radicals When written in radical form, it s only possible to write two multiplied radicals as one if the index is the same. As long as this requirement is met, 1) multiply the 2) multiply the ) Simplify! x 2 y 5xy 2 6 8x y 10xy x y 15x y x 2x 40x y 4 27x 9x y xy 5x 2 50y

15 Common Core Math 2 Unit A Modeling with Exponential Functions Adding & Subtracting Radicals You ve been combining like terms in algebraic expressions for a long time! Show your skills by simplifying the following expressions. 2x x + 4x = y 2x + y 6y = Usually we say that like terms are those that contain the same variable expression, but they can also contain the same radical expression. When you add or subtract radicals, you can only do so if they contain the same index and radicand. Just like we don t change the variable expression when we add or subtract, we re not going to change the radical expression either. All we are going to do is add or subtract the coefficients. Always simplify the radical before you decide that you can t add or subtract x - 20x a 5 8a a a y 7 2y

16 Common Core Math 2 Unit A Modeling with Exponential Functions Simplify each expression xy xy Simplify 2 7 A. -15 x B. -15x C. -5x D. -5 x Multiply. Simplify. A. 25 B. 5 5 C D. 5 A garden has width 1 and length 7 1. What is the perimeter of the garden in simplest radical form? 16

17 Common Core Math 2 Unit A Modeling with Exponential Functions HW Operations with Radicals 17

18 Common Core Math 2 Unit A Modeling with Exponential Functions Rational Exponents & Radicals Raising a number to the power of ½ is the same as performing a familiar operation. Let s take a look at the graph of y = x 1 2 to discover that operation. Step 1: Type x 1 2 into the y= screen on your graphing calculator. Step 2: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step : Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Where have you seen this relationship before? Summarize your findings in a sentence. Step 4: Type x 1 into the y= screen on your graphing calculator. Step 5: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step 6: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Have you seen this relationship before? Summarize your findings in a sentence. Step 7: Type 25 x into the y= screen on your graphing calculator. Step 8: Adjust your table so that the values go up by ½ and begin at 0. Verify that your table contains the same values as the rest of your class. Step 9: Discuss with your classmates the pattern you see. Use the table below to help you see the pattern. (One row has been completed for you). Summarize your findings in the space beside the table. X (exponent) X (exponent) as a fraction with a denominator of 2 Y 1 (25 x ) Rewrite Y 1 as a power of 25 with fraction exponents Rewrite Y 1 as a power of ( 25) How could you use this pattern to find the value of 6 /2? Check your answer in the calculator. 18

19 Common Core Math 2 Unit A Modeling with Exponential Functions How could you use this pattern to find the value of 27 2/? Check your answer in the calculator. How could you use this pattern to find the value of 81 5/4? Check your answer in the calculator. Step 10: Generally speaking, how can you find the value of an expression containing a rational exponent. Use the expression a m/n to help you in your explanation. You try: Rewrite each of the following expressions in radical form. 2 2 x 27) 4 ( ( 16x) 5 y -9/ a 4 ( 2 5) 5 x 1.2 Now, reverse the rule you developed to change radical expressions into rational expressions. 5 2 ( 6) x 19

20 Common Core Math 2 Unit A Modeling with Exponential Functions Earlier in this unit, you learned that when written in radical form, it s only possible to write two multiplied radicals as one if the index is the same. However, if you convert the radical expressions into expressions with rational exponents, you CAN multiply or divide them (as you saw in your warm-up)! Give it a try Write your final answer as a simplified radical. 12 y 4 y 6 ( a2 b ) 4 (2 a) a 4 x 12 y 2 64x 512x x 8 7 x x ( x y 2 27x 9 ) 6 How does the idea of simplifying radicals relate to the idea of rational exponents? There are several ways to approach this. Develop your own method for calculating simplest radical form of an expression without converting to radical form until the very last step! a 2 b 6 4 c 10 5 d 25 Describe your method for simplifying radicals from rational exponents. Share your method with the class. 20

21 Common Core Math 2 Unit A Modeling with Exponential Functions Radical and Rational Exponents (Kuta Software Infinite Algebra 2) Kuta Software - Infinite Algebra 2 Name Radicals and Rational Exponents Date Write each expression in radical form. 17) (6v) ) m 1 2 1) 7 2) Write each expression in exponential form. 5 ) 2 4) ) ( 4 m) 20) ( 6x ) 4 2 5) 6 6) ) 4 v 22) 6 p Write each expression in exponential form. 7) ( 10) 8) 6 2 2) ( a) 4 24) 1 ( k ) 5 9) ( 4 2) 5 10) ( 4 5) 5 Simplify ) 9 26) ) 2 12) ) ) 6 2 Write each expression in radical form. 1) (5x) ) (5x) ) (x 6 ) 0) (9n 4 ) ) (64n 12 ) - 2) (81m 6 ) ) a 6 5 2a pkhurta0 lsao jf2tjw6a2rkee rlxlzcg.w A 4Akl2 l 0rwiVgChPtlso hrsemsteurovzeqdp.7 o omia2dkek 7wLijtuhF AIUnNf4iBnyi0t2eU GAHlGgBe4blrGaj n2y.i KDuUtoaN rsqoq ft8wgarrqes LILsC6.E j dakldl0 XrCilgFhEt TsS Kr sekswejrhvfexds.u Y BMjajdYe7 RwZigtUhB RIUnJfSiNn5iatzeO mail2gvewb2rnam 22U.g 15) (10n) 21

22 Common Core Math 2 Unit A Modeling with Exponential Functions Investigation : Exponential Growth & Decay Materials Needed: Graphing Calculator (to serve as a random number generator) To use the calculator s random integer generator feature: 1. Type any number besides zero into your calculator, press STO, MATH,, ENTER, ENTER 2. Press MATH 5 1, 6 ) You can use numbers other than 1 and 6 here. The calculator will choose numbers between and including these numbers when you press enter. Continue pressing enter for more numbers. Investigation: 1. Choose a recorder to collect the class s data on the board. You ll copy the data down in your table later. 2. Everyone in the class should stand so that the recorder can count everyone and record the number of people standing in a table for Stage 0.. Use your calculator to find a random integer between 1 and 6. If you roll a 1, sit down. Before proceeding, allow time for the recorder to count the number of people still standing. When the recorder is finished counting, (s)he will let you know. 4. Repeat step until fewer than people are standing (or you run out of room on the table). 5. Record the data in your table. Stage People Standing Questions: 1. What is your initial value for this set of data? What does it represent in the investigation? 2. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain.. Based on your answer to Question 2, find the r OR d for the data you collected. Show the process you used to do so. 4. Could you estimate your answer to Question without conducting the exploration? If so, how? 5. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 6. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the stage of the investigation and let f(x) equal the number of people standing in that stage. 22

23 Common Core Math 2 Unit A Modeling with Exponential Functions Investigation 2: Half of a radioactive substance decays every 5 years. How much will remain of a 12 milligram sample after 50 years? Complete the table. Years Remaining radioactive substance Questions: 7. What is your initial value for this set of data? What does it represent in the investigation? 8. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 9. Based on your answer to Question 8, find the r OR d for the data you collected. Show the process you used to do so. 10. Could you estimate your answer to Question 9 without filling in the table? If so, how? 11. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 12. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of 5 year increments in the investigation and let f(x) equal the amount of radioactive substance remaining. 1. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of years in the investigation and let f(x) equal the amount of radioactive substance remaining. Use your equation to determine how much radioactive substance will remain after 500 years. 14. When will there be exactly 5 milligrams of the radioactive substance? Determine your answer to the nearest month. 15. Compare Investigation 1 and Investigation 2. What are the similarities and differences? 2

24 Common Core Math 2 Unit A Modeling with Exponential Functions Investigation : You invest money in a savings account that earns 2.5% interest annually. How much money will you have at the end of 10 years if you begin with $1000? Complete the table. Years Money in your account Questions: 16. What is your initial value for this set of data? What does it represent in the investigation? 17. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 18. Based on your answer to Question 17, find the r OR d for the data you collected. Show the process you used to do so. 19. Could you estimate your answer to Question 18 without filling in the table? If so, how? 20. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 21. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of years in the investigation and let f(x) equal the amount of money in the account. 22. You find a bank that will pay you % interest annually, so you consider moving your account. Your current bank decides you re a good customer and offers you a special opportunity to compound your interest semiannually!!! (They make it sound like it s a really good deal, so you re curious). You don t play around with your money, so you ask what exactly that means. They explain that you ll still get 2.5% interest, but they ll give you 1.25% interest at the end of June and 1% interest at the end of December. You want to see if you make more money than you would if you just switched banks, so you do the calculations. Which bank is giving you a better deal? Explain your answer. 24

25 Common Core Math 2 Unit A Modeling with Exponential Functions When you write an equation for a situation and use it to make predictions, you assume that other people who use it will understand the situation as well as you do. That s not always the case when you take away the context, so we sometimes need to provide some additional information to accompany the equation. 2. The domain of a function is the set of all the possible input values that can be used when evaluating it. If you remove your functions in Questions 6 and 1 and 21 from the context of this situation and simply look at the table and/or graph of the function, what numbers are part of the theoretical domain of the function? Will this be the case with all exponential functions? Why or why not? 24. When you consider the context, however, not all of the numbers in the theoretical domain really make sense. We call the numbers in the theoretical domain that make sense in our situation the practical domain. For instance, in the first investigation, our input values are Stages. If you look at the tables you created, what numbers would be a part of the practical domain for these investigations? Investigation 1 Investigation 2 Investigation Make note of any similarities and differences and explain why they exist. 25

26 Common Core Math 2 Unit A Modeling with Exponential Functions Summary of Findings The results of Investigation 1 & 2 simulate exponential decay because the values are being multiplied by the same value between 0 and 1 each time. The results of Investigation simulate exponential growth because the values are being multiplied by the same value greater than 1 each time. As you may have noticed in Questions 4, the common ratio is related to the probability that you will be randomly assigned a 1 out of 6 options. Since this probability is 1:6, or 1 6, or 0.16, or % that fraction of the class will sit down. These numbers do not represent our common ratio, however. That s because when 1 of 6 people sit down, that leaves 5:6, or 5 or 0.8 or 8% of the class still standing. To find the common ratio (b) using this probability 6 (p) or any percentage for that matter, you can use the following equations. The probability or percent MUST be written as a decimal to use this formula. Exponential Growth b = 1 + p This is because you begin with 100% (1 when written as a decimal) and add the same percentage each time. Exponential Decay b = 1 p This is because you begin with 100% (1 when written as a decimal) and subtract the same percentage each time. In problems 5, 11, and 20, you wrote recursive functions for the exponential data. Make sure you include the starting value of your data (a) this will always be where the independent variable is 0. NEXT = NOW b, Starting at a y = ab x For each of these Investigations, we came up with functions with theoretical domains of all real numbers. In fact, all exponential functions will have domains of all real numbers. However, when we considered the context, we realized that the practical domain for these the first investigation only involves the whole numbers because all of the other real numbers would not work as stages. We also saw that in most scenarios where the domain is related to time, the domain is only positive values. 26

27 Common Core Math 2 Unit A Modeling with Exponential Functions HW: ZOMBIES! SCENARIO 1 A rabid pack of zombies is growing exponentially! After an hour, the original zombie infected 5 people total, and those 5 zombies went on to infect 5 people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine how many newly infected zombies will be created after 4 hours. If possible, draw a diagram, create a table, a graph, and an equation. SCENARIO 2 During this attack, a pack of 4 zombies walked into town last night around midnight. Each zombie infected people total, and those zombies went on to infect people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine the number of newly infected zombies at 6 am this morning. If possible, draw a diagram, create a table, a graph, and an equation. SCENARIO At 9:00 am, the official count of the zombie infestation was Every hour the number of zombies quadruples. Around what time did the first zombie roll into town? 27

28 Common Core Math 2 Unit A Modeling with Exponential Functions Guided Practice : Exponential Growth & Decay 1. You decide to conduct an investigation in a room of 100 standing students who were to randomly choose a number between and including 1 through 20. If they chose a multiple of 4, they were to sit down. You record the number of people still standing after each turn. a. What is the probability of choosing a multiple of 4? b. What is the common ratio for this investigation? c. What is the initial value of this investigation? d. Write a recursive equation for the investigation. e. Write an explicit equation for the investigation. f. Using your equations, how many stages of the investigation will occur before there are fewer than people standing? 2. The amount of radioactive ore in a sample can be modeled by the equation y = 20(0.997) x, where x represents years and y is the amount of ore remaining in milligrams. a. What was the initial amount of radioactive ore? b. Is this an example of exponential growth or decay? c. What percentage of the ore is being lost or gained according to this model? d. When will there be half of the initial amount of the radioactive ore?. Complete the following table. Explicit Function Recursive Function Initial Value Common Ratio Growth or Decay? y = 2() x NEXT = NOW 0.5, Start at y = ab x growth 28

29 Common Core Math 2 Unit A Modeling with Exponential Functions 4. The height of a plant can be modeled by the table below. Day Height (in) a) What is the initial height of the plant? Explain how you found your answer? b) What is the common ratio? Explain how you found your answer. c) Is this an example of exponential growth or decay? How could you find the answer if you only had the common ratio? d) Write the recursive function for this situation. e) Write the explicit function for this situation. f) How tall will the plant be on the 12 th day of data collection? How tall is this in inches, feet, yards, and miles? Does this seem realistic to you? g) On what day will the plant first be over 100 yards tall? 5. When you opened a savings account on your 15 th birthday, you deposited most of the money from your summer job ($2000). The banker who helped you informed you that you would received 1.5% interest each year. a. How much money will you have in the account when you turn 21? b. Use the properties of exponents to determine the monthly percentage interest rate that you could have gotten from the bank that would have given you the same amount of money when you turned For each scenario, write an explicit equation, define your variables, and determine the practical and theoretical domain. a) The town of Braeford was first established in 1854 when it had a population of 24. Since then it has grown by a percentage of 1.25% each year. b) A species of bacteria reproduces exactly once each hour on the hour. At this exact time, each organism present divides into two organisms. One of these bacteria is placed into a petri-dish at 8:00 this morning. 29

30 Common Core Math 2 Unit A Modeling with Exponential Functions HW : Exponential Growth & Decay 1. In 1990, Florida s population was about 1 million. Since1990, the state s population has grown about 1.7% each year. This means that Florida s population is growing exponentially. Year Population a) Write an explicit function in the form y = ab x that models the values in the table. b) What does x represent in your function? c) What is the a value in the equation and what does it represent in this context? d) What is the b value in the equation and what does it represent in this context? 2. Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8.1% per year. In 1985, such hospital costs were an average of $460 per day. a) Write an equation to model the cost of hospital care. Let x = the number of years after b) Find the approximate cost per day in c) When was the cost per day $1000 d) When was the cost per day $2000?. To treat some forms of cancer, doctors use Iodine-11 which has a half-life of 8 days. If a patient received 12 millicuries of Iodine-11, how much of the substance will remain in the patient 2 weeks later? 0

31 Common Core Math 2 Unit A Modeling with Exponential Functions 4. Suppose your parents deposited $1500 in an account paying 6.5% interest compounded annually when you were born. a) Find the account balance after 18 years. b) What would be the difference in the balance after 18 years if the interest rate in the original problem was 8% instead of 6.5%? c) What would be the difference in the balance if the interest was 6.5% and was compounded monthly instead of annually? 5. Since 1980, the number of gallons of whole milk each person in the US drinks in a year has decreased 4.1% each year. In1980, each person drank an average of 16.5 gallons of whole milk per year. Year Population a) Write a recursive function for the data in the table. b) Write an explicit function in the form y = ab x that models the values in the table. Define your variables. c) According to this same trend, how many gallons of milk did a person drink in a year in 1970? 6. The model y = (0.982) x represents the population in Washington, D.C. x years after a) How many people were there in 1990? b) What percentage growth or decay does this model imply? c) Write a recursive function to represent the same model as the provided explicit function. d) Suppose the current trend continues, predict the number of people in DC in 201. Suppose the current trend continues, when will the population of DC be approximately half what it was in 1990? 1

32 Common Core Math 2 Unit A Modeling with Exponential Functions Unit A Test Review Simplify each expression, and write your final answer with rational exponents. 1. 6s 2 (s 6 ) 1/ 2. 2k 2/ 1 4 k5/6 4. x x Simplify each expression, and write your final answer in simplest radical form. 4. m 1/2 m 4/ 5. (12n 2 24n 1/4 ) x 8 8x 7. Explain why 16 1/4 = is a true statement. Fill in the blank to make each statement true. 8. 2x 6 = 8x 9. (10x - )/ = 5x 10. ( ) 2 = 2x 11. ( ) - = 8m 9 Write each expression in simplified radical form m 2 n m 4 n 1. k k 6/ y x y (b 5) 2 (b 5) Explain how to calculate the value of 81 /4 without using a calculator. 2

33 Common Core Math 2 Unit A Modeling with Exponential Functions 19. The function y = 290,000 (0.92)x represents the value of an old home that has been abandoned by its owners x years ago. Find the decay rate of the old home. 20. If the population in 1995 for a Town A is 1,500, and is increasing at a rate of 2.% every 5 years, what is the projected population of the Town A in 2025? If the population in 2001 for a Town B is 100,000, and is increasing at a rate of 1.5% every year, what is the projected population of the Town A in 2012? 22. If the population in 1999 for a Town C is 95,000, and is decreasing at a rate of 2.5% every 2 years, what is the projected population of the Town C in 200? 2. If the population in 2005 for a Town D is 5,500, and is decreasing at a rate of 1.9% every year, what is the projected population of the Town C in 2009?

34 Common Core Math 2 Unit A Modeling with Exponential Functions Homework Answers Properties of Exponents; Page 10 1) 4m 5 11) 1 2) 2m 1 ) 8 12) 16x 8 r 1) 256 4) 8n 14) 16a 6 5) 8k 5 15) 81k 16 6) 4x 2 16) 1 7) 6y 2 x 4xy 8) 4v 4 u 2 17) 1 9) 12 2b 4 ab 10) x5 y 2 18) x4 y 2 19) y 2x 4 20) 1 9m 2 21) 1 2r 22) 1 4x 5 2) n 24) ) m 7 26) 2x2 yz 7 27) z xy 2 28) 2h k j 4 29) 4m2 n p 0) x7 yz Simplifying Square Roots; pages 1 THEY DON T HAVE THE VEGAS IDEA. Operations with Radicals; pages 17 1) 5 2) 5x 2 2 ) 11 4) 4 6 I DON T KNOW AND I DON T CARE 5) ) 2xy 5y 7) 7 x + y 8) Radicals and Rational Exponents; pages 21 9) ) 6nt n 11) ) ) 0n 4 t 2 7t 14) 59 15) ) ) 7 2) ( 4) 4 ) ( 2) 5 4) ( 7) 4 5) ( 6) 6 6) 2 7) ) ) ) ) ) ) 4 ( 5X) 5 14) 1 5X 15) ( 10n) 5 16) ( a) 6 17) ( 6v) 18) 1 m 19) m 4 20) (6x) 4 21) v ) (6p) 1 2 2) (a) 4 24) (k) ) 26) ) 10 28) ) x 0) n 2 1) 1 2n 2 2) 9m 4

35 Common Core Math 2 Unit A Modeling with Exponential Functions Zombies; Page Hours New zombies y = 1(5) x 2. Hours New zombies y = 4() x. Hours New zombies 9 am am am am am 64 4 am 16 am 4 2 am 1 y = 1684(4) -7 = 1 Exponential Growth and Decay; pages a. y = 1(1.017) x b. years since 1990 c. 1 mil in FL in 1990 d growth of 100% + 1.7% 2. a. y = 460(1.081) x b. x = 27 y = $ c. x = 9.97 yrs y = $1000 at end of 1994 d. x = yrs y = $2000 at end of 200. y = 12(.5) 14/8 =.57 millicuries 4. a. $ b. $14.05 c. $ a. NEXT = NOW * start at 16.5 b. y = 16.5(1.041) x c gal 6. a b. 18% decay c. NEXT = NOW * start at d e Test Review; pages s k 2. 2x m m ,887,872n 6 4 n 6. 8x 2x x x x 2 or 2 5 2x x 2 or 2 5 2x m or 2 1 m mn 5m 6 1. k k y 2 4 y (b 5) Answers vary 19. 8% 20. 1, , , ,094 5

36 6 Common Core Math 2 Unit A Modeling with Exponential Functions