Common Core Math I Unit 5: Exponential Functions

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1 Common Core Math I Unit 5: Exponential Functions Concept Pages Study Guide 1 2 Sequences and NOW-NEXT Forms of Linear and Exponential Growth 3 10 Explicit Form of Exponential Functions Exponential Growth Linear and Exponential Growth Review 21 Translations of Exponential Functions 22 Exponential Decay Review & Practice Assessment Students will understand that exponential relationships can be modeled using tables, graphs, and equations the solutions to an exponential equation are represented by the points on a graph a sequence of numbers where the ratio between consecutive numbers is constant is called a geometric sequence growing functions have a common ratio, r, value greater than 1 exponential function can be evaluated for a given domain patterns can help determine if a model is arithmetic or geometric the NOW-NEXT form can be used to model geometric sequences the explicit form (y=) can be used to model exponential functions they can translate between the recursive (NOW-NEXT) and explicit forms in modeling situations compound interest can be calculated using the explicit form of an exponential function the difference between linear and exponential functions, and how they can be used to model arithmetic and geometric sequences parts of an expression may be interpreted to tell you about a given situation a parent exponential function may be translated vertically or horizontally decaying functions have a common ratio, r, value between 0 and 1 declining sequences have a common ratio and subsequent terms can be determined using this ratio exponential models may be created to represent real-world situations half-life is a form of exponential decay and how to determine half-life depreciation can be modeled using exponential functions Essential Questions: What does exponential growth and decay look like in a real-world situation? Why is rate of growth or decay so important? Why is it important to be able to recognize the difference between linear and exponential functions? How does the exponent impact the output? 1 P a g e

2 A-CED.2 A-REI.10 A-SSE.1 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x 2 ) 2 - (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 - y 2 )(x 2 + y 2 ). F-BF.1 F-BF.2 F-BF.3 F-IF.2 F-IF.4 F-IF.5 F-IF.6 F-IF.7e F-IF.8b F-IF.9 F-LE.1c F-LE.2 F-LE.3 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. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Note: At this level, focus on linear and exponential functions. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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, and trigonometric functions, showing period, midline, and amplitude. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between situations that can be modeled with linear functions and with exponential functions c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 2 P a g e

3 Problem Situation: The Brown Tree Snake The Brown Tree Snake is responsible for entirely wiping out over half of Guam s native bird and lizard species as well as two out of three of Guam s native bat species. The Brown Tree Snake was inadvertently introduced to Guam by the US military due to the fact that Guam is a hub for commercial and military shipments in the tropical western Pacific. It will eat frogs, lizards, small mammals, birds and birds' eggs, which is why Guam s bird, lizard, and bat population has been affected. Adapted from Global Invasive Species Database The number of snakes for the first five years is summarized by the following sequence: 1, 5, 25, 125, 625, 1. What are the next three terms of the sequence? 2. What is the initial term of the sequence? 3. How would you write the sequence as a NOW-NEXT statement? 4. What is the pattern of change? 5. Do you think the sequence above is an arithmetic sequence? Why or why not? Growing Sequences: Review of Arithmetic Sequences An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance, 2, 5, 8, 11, 14,... and 7, 3, 1, 5,... are arithmetic, since you add 3 in the first sequence and add 4 in the second sequence, respectively, at each step. Arithmetic sequences can be modeled with linear functions. The number added at each stage of an arithmetic sequence is called the common difference d, because if you subtract (find the difference of) successive terms, you'll always get this common value. For example, find the common difference and the next term of the following sequence: 3, 11, 19, 27, To find the common difference, subtract a pair of consecutive terms = 8 ; = 8 ; = 8 ; = 8 The difference is always 8, so d = 8. Then the next term is = 43. For arithmetic sequences, the common difference is d, and the first term is often referred to as the initial term of the sequence. 6. Write a NOW-NEXT statement for the arithmetic sequence 3, 11, 19, 27, Write an explicit function for the arithmetic sequence in slope-intercept form. Let x = the term number. e.g. (2,11). In the Brown Tree Snake sequence, the rate of change is not arithmetic as shown below. 1, 5, 25, 125, 625, = 4 ; 25 5 = 20 ; = 100 ; = 500 The difference is not a common number; therefore, the sequence is not arithmetic. So, what kind of sequence is this? 1, 5, 25, 125, 625, = 5 ; 5 5 = 25; 25 5 = 125; = P a g e

4 The initial term of the Brown Tree Snake is 1 and the rate of change is that of multiplication by 5 each time in order to generate the next terms of the sequence. This type of sequence is called a geometric sequence. A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16 and 81, 27, 9, 3, 1, 1/3... are geometric sequences, since you multiply by 2 (or divide by ½ ) in the first sequence and multiply by 1/3 (or divide by 3) in the second sequence, respectively, at each step. Geometric sequences can be modeled with exponential functions. 8. Write a NOW-NEXT sequence to model the geometric sequence 1, 5, 25, 125, 625 The number multiplied at each stage of a geometric sequence is called the common ratio r, because if you divide consecutive terms, you'll always get this common value. So, let s determine the common ratio r of the Brown Tree Snake Sequence. 1, 5, 25, 125, /1 = 5 ; 25/5 = 5 ; 125 /25 = 5 ; 625/125 = 5 The common ratio of the Brown Tree Snake is r = 5. Find the initial term and the common ratio of other geometric sequences. Use the initial term and the common ratio to write a NOW-NEXT statement for each sequence. 9. 1/2, 1, 2, 4, 8... Initial term: Common ratio: NEXT = 10. 2/9, 2/3, 2, 6, 18 Initial term: Common ratio: NEXT = Now it is time for you to determine if the following sequences are arithmetic or geometric. Practice with Sequences For a sequence or NOW-NEXT statement, write arithmetic and the common difference or geometric and the common ratio. If a sequence is neither arithmetic nor geometric, write neither , 6, 18, 54, ; common = , 34, 54, 74, ; common = 13. 4, 16, 36, 64, ; common = 14. 9, 109, 209, 309, ; common = 15. 1, 3, 9, 27, ; common = 16. START = 7, NEXT = NOW(2) ; common = 17. START = 7, NEXT = NOW 2 ; common = 18. START = 3, NEXT=NOW( 1 ) ; common = 5 4 P a g e

5 The following story has been told in different ways. Versions of the story date back over 1000 years. When the creator of the game of chess showed his invention to the ruler of the country, the ruler was so impressed that he gave the inventor the right to choose his own reward. The man asked the king for the following: that for the first square of the chess board, he would receive one grain of rice, two grains for the second square, four on the third square, and so forth, doubling the amount each time. The ruler quickly accepted the inventor's offer, and thought that the man hadn t asked for much. However, the king s treasurer explained that it would take more than all the rice in the kingdom to give the inventor the reward. The story usually ends with the inventor becoming the new king or being executed. 1. Write a NOW-NEXT statement to model this situation. 2. How much rice will he get for each square? Write your answer in each square. Start from the top left. Work from left to right and top to bottom. If you re having problems doing this, explain why you re having problems. 3. How much total rice will he get after 2 squares? After 3 squares? After 4 squares? Do you see any patterns? 4. How much rice will he get in total? If you can t write the number as a base 10 number, can you write it using exponents? 5. Let x = the number of the square and y = the number of grains of rice for the square. For example, (1, 1), (2, 2), (3, 4), (4, 8). Write an explicit equation to describe how much rice he will be given for a certain square. 6. Can you write an equation to describe the total number of grains of rice he will receive for a certain number of squares? One solution to your equation would be (2, 3). For the first two squares, he receives 3 grains of rice in total. 7. On average, a pound of rice contains about 22,700 grains of rice. If y = tons of rice, and x = number of grains of rice, write an equation to find how many tons of rice (2000 lbs = 1 ton) you would have if you had x grains of rice. Is the equation linear or exponential? How do you know? Give context to any numbers in your equation. Trivia: In 2009, the world produced about 850,000,000 tons of rice. 5 P a g e

6 Who Wants to Be Rich? Students at a local school want to have a quiz show called Who Wants to Be Rich? Contestants will be asked a series of questions. A contestant will play until he or she misses a question. The total prize money will grow with each question answered correctly. Lucy and Pedro are on the prize winnings committee and have different view of how prize winning should be awarded. Their plans are outlined below for your consideration. Review them by answering the questions following the plans. Remember that the committee has a fixed amount of money to use for this quiz show. 1. Lucy proposes that a contestant receives $5 for answering the first question correctly. For each additional correct answer, the total prize would increase by $10. a. For Lucy s proposal, complete the table below. Number of questions Total prize b. Sketch the graph of correctly answered questions Be sure to title your graph and label the axes. c. How much money would a contestant win if he or she correctly answered 6 questions? d. How much money would a contestant win if he or she correctly answered 9 questions? e. How many questions would a contestant need to answer correctly to win at least $50? f. How many questions would a contestant need to answer correctly to win at least $75? g. How is this table growing? Is this a linear or exponential growth pattern? 6 P a g e

7 2. Pedro also proposes that the first question should be worth $5. However, he thinks a contestant s winnings should double with each subsequent answer. a. For Pedro s proposal, complete the table below. Number of questions Total prize b. Sketch the graph of correctly answered questions c. How much money would a contestant win if he or she correctly answered 6 questions? d. How much money would a contestant win if he or she correctly answered 9 questions? e. How many questions would a contestant need to answer correctly to win at least $50? f. How many questions would a contestant need to answer correctly to win at least $75? g. How is this table growing? Is this a linear or exponential growth pattern? 3. Which plan is better for the contestants? Explain your reasoning. 4. Which plan is better for the school? Explain your reasoning. Adapted from Growing, Growing, Growing Exponential Relationships, Connected Mathematics 2, Pearson Publishing, P a g e

8 Independent Practice: Charity Donations Mari s wealthy Great-aunt Sue wants to donate money to Mari s school for new computers. She suggests three possible plans for her donations. Plan 1: Great-aunt Sue s first plan is give money in the following way: 1, 2, 4, 8... She will continue the pattern in this table until day 12. Complete the table to show how much money the school would receive each day. Day Donation $1 $2 $4 $8 Plan 2: Great-aunt Sue s second plan is to give funds in the following way: 1, 3, 9, She will continue the pattern in this table until day 8. Complete the table to show how much money the school would receive each day. Day Donation $1 $3 $9 $27 Plan 3: Great-aunt Sue s third plan is to give money in the following way: 1, 4, 16, She will continue the pattern in this table until day 6. Complete the table to show how much money the school would receive each day. Day Donation $1 $4 $16 $64 Graph each plan on the same graph to the right. 1. How much does each plan give the school on day 6? 2. What is the common ratio (growth rate) for each plan? a. Plan 1 b. Plan 2 c. Plan 3 3. Which plan should the school choose? Why? 4. Which plan will give the school the greatest total amount of money? 8 P a g e

9 Jason is planning to swim in a charity swim-a-thon. Several relatives have agreed to sponsor him in this charity event. Each of their donations is explained below. Grandfather: I will give you $1 for the first 1 lap, $3 for the 2 nd lap, $5 for the 3 rd lap, $7 for the 4 th lap, and so on. Father: I will give you $1 for the first lap, $1.50 for the 2 nd lap, $2.25 for the 3 rd lap, $3.38 for the 4 th lap, and so on. Aunt June: I will give you $2 for the first lap, $3.50 for the 2 nd lap, $5 for the 3 rd lap, $6.50 for the 4 th lap, and so on. Uncle Bob: I will give you $1 for the first lap, $1.25 for the 2 nd lap, $1.56 for the 3 rd lap, $1.95 for the 4 th lap, and so on. 5. Decide whether each donation sequence is exponential, linear, or neither. a. Grandfather s Plan b. Father s Plan c. Aunt June s Plan d. Uncle Bob s Plan 6. Complete the table for each sequence below. Grandfather s Plan # of Laps Donation $1 $3 $5 $7 Father s Plan # of Laps Donation $1 $1.50 $2.25 $3.38 Aunt June s Plan # of Laps Donation $2 $3.50 $5 $6.50 Uncle Bob s Plan # of Laps Donation $1 $1.25 $1.56 $ P a g e

10 7. Graph each table on the graph on the next page. Label each line or curve. Title the graph and label the axes. 8. Use either the table or graph to determine the total money Jason will raise for each plan if he swims 10 laps. a. Grandfather s Plan b. Father s Plan c. Aunt June s Plan d. Uncle Bob s Plan Adapted from Growing, Growing, Growing, Exponential Relationships, Connected Mathematics 2, Pearson, P a g e

11 Evaluate each exponential function for the domain { 1, 0, 1}. As the domain values increase, what do the range values do? 1. f(x) = 5 x 2. y = 5 x 3. g(t) = ( 2 3 ) t 4. y = ( 2 3 ) x 5. Mental math: Match each exponential function with its graph. Which input values did you check mentally? y = ( 1 3 ) x y = ( 1 3 ) x y = 3 x y = 3 x A B C D Evaluate the exponential functions for the domain { 2, 1, 0, 1, 2}. Sketch the function s graph. 6. y = 2 x 7. y = ( 1 2 ) x 8. Would the graph change if there were no parentheses in #7? Explain. 9. Tribbles reproduce very quickly and double in number every 30 minutes. Kirk has 3 tribbles. The function t(h) = 3 2 h where h is the number of half-hour periods and t(h) is the number of tribbles, models this situation. How many tribbles will Kirk have in 2 hours? Bonus 1: Each ordered pair represents a value from a geometric sequence. (3, 7) means that the 3 rd number in the sequence is 7. For each of the following pairs of points, calculate the common ratio and calculate the first 5 numbers of the sequence. A. {(1, 5), (2, 15)} B. {(1, 3), (2, 6)} C. {(1, 1), (3, 4)} D. {(3, 8), (5, 18)} Bonus 2: Write a function rule for each geometric sequence in the form of y = a(r) x. Let y = the value of the term. Let x = the term number. 11 P a g e

12 12 P a g e Evaluate each function rule for the domain { -3, -2, -1, 0, 1, 2, 3}. What characteristics do you notice about exponential functions?

13 Bacteria Growth If you don t brush your teeth regularly, it won t take long for large colonies of bacteria to grow in your mouth. Suppose a single bacterium lands on your tooth and starts multiplying by a factor of 4 every hour. 1. Complete the table below to model the bacteria growth over several hours. Hours Expanded Form Exponential Form Evaluated Form (fill in the blank) Graph the data in the table below. Be sure to label your graph and axes. 2. Is this graph linear or exponential? 3. What is the initial value in this situation? 3. What is the common ratio in this situation? 4. What do the first & third columns have in common? 5. What are the similarities and differences between the entries in the third column of the table? 6. Use the pattern you saw in the third column to predict how many bacteria there will be after 10 hours. Continue the pattern in the table or on the graph to verify your answer. 7. If y is the number of bacteria after x hours, write a rule that will allow you to calculate how many bacteria there are at any time. 8. Write the NOW-NEXT form to show the pattern of growth. NEXT = NOW 13 P a g e

14 y = the number of bacteria produced in that hour x = the number of hours r = the common ratio or rate of change a1 = the initial term of the sequence or the starting point Use the above information to write the explicit form of the exponential function y = a1 r x. Notice how similar it is to the NOW-NEXT recursive form. START = 1 ; NEXT = NOW r y = a1 r x y = 1 4 x 9. After how many hours will there be at least 1,000,000 bacteria in the colony? 10. Suppose that instead of 1 bacterium, 50 bacteria land in your mouth. Write an explicit equation which describes the number of bacteria y in this colony after x hours. 11. What is different in this equation from the equation in step 7? 12. Using your new equation, determine the number of bacteria in the colony after 8 hours and after 10 hours. Which method for determining the number of bacteria is easier for you? Using a table, graph, NOW- NEXT, or explicit equation? Explain. 14 P a g e

15 Guided Practice: More Bacteria The bacteria E. coli often causes illness among people who eat the infected food. Suppose a single E. coli bacterium in a batch of ground beef begins doubling every 10 minutes. 1. Complete the table below to determine how many bacteria there will be after 10, 20, 30, 40, and 50 minutes have elapsed (assuming no bacteria die). 10-min Period Number of 1 2 Bacteria 2. Graph the data on the table. Be sure to title your graph and label your axes. 3. Write two rules that can be used to calculate the number of bacteria in the food after any number of 10-minute periods. START = ; NEXT = NOW y = a1 r x 4. What is the initial value? y = x 5. What is the common ratio? 15 P a g e

16 6. How many times would the bacteria double in 2 hours? 7. Use your rule(s) to determine the number of bacteria after 2 hours. 8. When will the number of bacteria reach at least 100,000? Students at a high school conducted an experiment to examine the growth of mold. They set out a shallow pan containing a mixture of chicken broth, gelatin, and water. Each day, the students recorded the area of the mold in square millimeters. The students wrote the exponential equation m = 50(3 d ) to model the growth of the mold. In this equation, m is the area of the mold in square millimeters after d days. 9. What is the area of the mold at the start of the experiment? 10. What is the growth factor or common ratio? 11. What is the area of the mold after 5 days? 12. On which day will the area of the mold reach 6,400 mm 2? 13. An exponential equation can be written in the form y = a(b x ), where a and b are constant values. a. What value does b have in the mold equation? What does this value represent? b. What value does a have in the mold equation? What does this value represent? Lesson adapted from Growing, Growing, Growing Exponential Relationships, Connected Mathematics 2, Pearson, P a g e

17 Independent Practice: Killer Plants Ghost Lake is a popular site for fishermen, campers, and boaters. In recent years, a certain water plant has been growing on the lake at an alarming rate. The surface area of Ghost Lake is 25,000,000 square feet. At present, 1,000 square feet are covered by the plant. The Department of Natural Resources estimates that the area is doubling every month. 1. Complete the table below. Number of Months Area Covered in Square 1,000 Feet 2. Use the data to graph the situation. Be sure to label your axes and title your graph. 3. Write 2 equations (NOW-NEXT and y =) to represent the growth pattern of the plant on Ghost Lake. 4. Explain what information the variables and numbers in your equations represent. 5. How much of the lake s surface will be covered with the water plant by the end of a year? 6. How much of the lake s surface was covered by the water plant 6 months ago? 7. In how many months will the plant completely cover the surface of the lake? 17 P a g e

18 Loon Lake has a killer plant problem similar to Ghost Lake. Currently, 5,000 square feet of the lake is covered with the plant. The area covered is growing by a factor of 1.5 each year. 8. Complete the table to show the area covered by the plant for the next 5 years. Number of Years Area Covered in Square 5,000 Feet 9. Graph the data. Be sure to label your axes and title your graph. 10. Write 2 equations (NOW-NEXT and y =) to represent the growth pattern of the plant on Ghost Lake. 11. Explain what information the variables and numbers in your equations represent. 12. How much of the lake s surface will be covered with the plant by the end of 7 years? 13. The surface area of the lake is approximately 5 acres. How long will it take before the lake is completely covered if one acre is 43,560 square feet? FYI: One acre is about the size of a football field with the end zones cut off Adapted from Growing, Growing, Growing Exponential Relationships, Connected Mathematics 2, Pearson, P a g e

19 Exponential functions can be in the form of y=ar nx. As with linear functions, (x, y) represent all solutions where x is an input (domain) value and y is an output (range) value. The variable a represents the y-intercept (the value of y when x = 0). For many word problems, a is the initial amount. Notice that a is not part of the base. The variable r is the base of the power and it represents the common ratio. r > 0 and r 1. The common ratio can be used to calculate the percent of change. The values of a and r also allow us to quickly determine if the output values are increasing or decreasing. When a > 0, if 0<r<1, the exponential function is a decreasing (decay) function and the percent of change is 1 r. When a > 0, if r>1, the exponential function is an increasing (growth) function and the percent of change is r 1. The number of times change occurs (the number of times a is multiplied by r) is represented by nx. n, is the ratio of changes/x-unit. Unless translated, exponential functions have an asymptote at y = 0 (the x-axis). The asymptote is a line that the function approaches, but never intersects. One example of exponential growth is compound interest. Is each equation an exponential function? If so, is it growth or decay? Identify the initial amount and the percent of increase or the percent of decrease. 1. y = 5 2 x 2. y = 3 ( 2) x 3. y = x 4. y = 1.5 ( 1 5 ) x 5. y = x 6. y = x Compound Interest Once in a while, you hear about people winning the lottery. The winnings can be several millions of dollars. The big money winners are usually paid in annual installments for about 20 years. But some of the smaller prizes are awarded in a matter of weeks. What do you think you would do if you won the lottery? Sue s uncle gave her a lottery ticket on her 18 th birthday and she won!! In the lottery payoff scheme, she has two payoff choices: Option 1 is to receive a single $20,000 payment now. Option 2 is to receive a single $40,000 payment in ten years. Which option should she take? But wait, the word is out and several banks have called to tell you about their investment plans for Option 1. One bank has offered a special 10-year certificate of deposit paying 8% interest compounded annually. Should Sue take option 1 and invest with this bank? How do you represent and reason about functions involved in investments paying compound interest anyway? It s not really that difficult. Let s look at each option above. 19 P a g e

20 Option 2 is really easy; in 10 years from now, you ll receive 40,000. Option 1 is not as clear. If you take option 1, you ll receive 20,000 and that is it. However, if you invest this money in the special 10 year certificate of deposit, you ll receive more. Let s find out how much more. The basic NOW-NEXT equation would be NEXT = NOW (NOW) At the end of year 1, the balance is: 20,000 + (0.08 x 20,000) = 20,000 + = At the end of year 2, the balance is: + (0.08 x ) = + = At the end of year 3, the balance is: + (0.08 x ) = + = Use this NOW-NEXT equation to determine how much money Sue will have in 10 years with Option 1. But there has GOT to be an easier way to do this! We know the initial value is $20,000 and we know she s investing it for 10 years, but what is the common ratio? Calculate the value by taking a look at the final account balances Sue had for the first 3 years. That makes sense! Each year when Sue s interest is added, she keeps 100% of what she had the previous year and also adds 8% - so the final balance is 100% + 8% = 108% of what she had at the beginning of the year. Change 108% to a decimal and you have 1.08 for the common ratio or growth factor. Now that you know the math behind it, we can write this as an explicit function and add in the information that we have: y = a (1 + r ) t ; a = initial amount; 1 is there to represent 100% as a decimal ; r = the % increase as a decimal t = the time in years ; y = balance in $ Substitute in the values that we have y = 20,000( ) 10 = 20,000(1.08) 10 = 43, Which option should Sue take? Some would advise her to take option 1 and to Invest the funds in the special CD. However, if she doesn t want to invest her money, then she would be better off, taking option 2. The down side here is that she will have to wait 10 years for her money. Which option would you advise Sue to take? Why? YOUR TURN Write the NOW-NEXT and explicit formulas for the following compound interest problems. 1) You have an initial investment of $15,000 to be invested at a 6% interest rate compounded annually. What is the investment worth at the end of 5 years? What is the investment worth at the end of 15 years? 2) You have an initial investment of $7,000 to be invested at a 4.5% interest rate compounded annually. What is the investment worth at the end of 20 years? What is the investment worth at the end of 30 years? Not all banks compound annually. Use your knowledge of vocabulary and variables to write an exponential function for the following situations dealing with compound interest. The initial deposit is$5,000. Carefully think about how much change is occurring and how often it is occurring. Let x = the number of years. Let y = the account s balance in $. Use a calculator to evaluate each function rule to calculate the balance after 50 years and after 60 years. 3. 4% annual interest compounded annually 4. 4% annual interest compounded semiannually 5. 4% annual interest compounded quarterly 6. 4% annual interest compounded monthly 20 P a g e

21 Practice with Linear Functions versus Exponential Functions Exponential functions, like linear functions, can be expressed by rules relating x and y values and by rules relating NOW and NEXT y values when an x value increases in steps of 1. Compare the patterns of (x, y) values produced by these functions: y = 2(3 x ) and y = 3x + 2 by completing these tasks. 1. For each function write another rule using NOW and NEXT that could be used to produce the same pattern of (x, y) values. a. y = 2(3 x ) b. y = 3x How would you describe the similarities and differences in the relationships of x and y in terms of their function graphs, tables, and rules? a. Similarities and differences of function graphs b. Similarities and differences of function tables c. Similarities and differences of function rules 3. For each table, identify the function as linear or exponential. Write a function rule to describe the data. a. b. c. d. x y x y x y x y P a g e

22 Translation Investigation 1. Without evaluating the functions, look at the function rules for f(x), g(x), h(x), and j(x). Look at the function rules for # 2 5. Using what you ve learned about translating functions, how would you describe g(x), h(x) and j(x) as translations of f(x)? 2. y =f(x) = 2 x Where is the y-intercept? Where is the asymptote? x y Describe the domain and range of y = f(x). 3. y =g(x) = 2 x 1 What change occurred from the parent function y = 2 x? Where is the y-intercept now? Where is the asymptote now? x y Describe the domain and range of y = g(x). 4. y = h(x)= 2 x 1. What change occurred from the parent graph y = 2 x? Where is the y-intercept now? Where is the asymptote now? x y Describe the domain and range of y = h(x). 5. y = j(x) = 2 x 1 2. What change occurred from the parent graph y = 2 x? Where is the y-intercept now? Where is the asymptote now? x y Describe the domain and range of y = j(x). 6. Describe in your own words how n(x) will be a translation of f(x). n(x) = f(x h) + k. Write each of the previous function in this way. 7. What happens when h > 0? h < 0? h = 0? 8. What happens when k > 0? k < 0? k = 0? 22 P a g e

23 Exponential functions show decay when the initial value is positive and the common ratio is between 0 and 1. They are decreasing functions that decrease by a certain percentage. For example, radioactive elements decay over time. The time it takes for half of any radioactive substance to decay is called the half-life. Different elements have different half-lives. After one half-life, 50% of the substance is gone and 50% remains. The time it takes a person s body to process out half of a drug or other substance is also called a half-life. If variables are not defined, carefully define variables. Write a function rule to model each situation and answer the question. 1. A hospital prepared a 100-mg supply of technetium-99m, which has a half-life of 6 hours. Write an exponential function to find the amount of technetium-99m that remains after 75 hours. 2. Arsenic-74 is used to locate brain tumors. It has a half-life of 17.5 days. Write an exponential decay function for a 90- gram sample. Use the function to find the amount remaining after 6 days. 3. A town has a population of 50,000. The town has been decreasing at an average annual rate of 1.25%. Find the estimated population in 10 years. 4. Assume that kidneys can filter out 25% of a drug in the blood every 4 hours. Bob takes one 1000-mg dose of the drug. A blood test is able to detect the presence of this medicine if there is at least 0.1 mg in the body. How many days will it take before the test will come back negative? Will the kidneys ever completely remove the drug from Bob s blood? Explain your answer. 5. Technetium 99m is a radioactive isotope with a half life of about 6 hours. Bob is given 20 mci during a medical procedure. Let y = mci of technetium 99m. Let x = number of half lives. a. Write a function rule to model this situation. b. How many half-lives occur in one day? c. Evaluate the function rule to find out how many mci of technetium-99m are still in Bob s body after one day. d. About what percent of the technetium-99m decays in one day? Depreciation can also be modeled by exponential functions when objects lose value by a certain percentage over time. Vehicles and equipment depreciate over time. The cost of a new truck is $32,000. It depreciates at a rate of 15% per year. This means that it loses 15% of its value each year. Start by making a table of values. Start with the value you know start with 32,000 at time t = 0. Then we multiply the value of the car by 85% for each passing year. (Since the car loses 15% of its value, which means that it keeps 85% of its value). Remember that 85% means that we multiply by the decimal Number of Years Value of the Truck ($) 32, P a g e

24 Graph the data on the coordinate grid below. Remember to label your axes. 6. Now let us write the equation for the data. Initial value: Percentage rate of depreciation: Equation: 7. 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. 8. 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: 9. 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 10. Write an exponential function to model the situation and graph it. Remember to label the axes. 11. Estimate when the ATV will have a value of $ P a g e

25 Exponential Function Practice Declare variables, and create a function rule to model each of the following. 1. Jill deposited $200 into a savings account with a 3% annual percentage rate compounded quarterly. Let b(x) = balance in $. Let x = # of years. 2. Lucy deposited $200 into a savings account with a 3% annual percentage rate compounded monthly. Let b(x) = balance in $. Let x = # of years. 3. (Calculator inactive)if neither person deposits more money into the account, who will have a higher balance in 20 years? 4. (Calculator active) What is each person s balance in 10 years? Assume each of the following are increasing or decreasing in an exponential fashion. Let x = # of hours. Write a function rule to model each situation. State whether it is exponential decay or growth. 5. When the experiment started, there were 50 cells. After 2 hours, there were 55 cells. 6. When the experiment started, there were 10,000 cells. After 2.5 hours, there were 9,880 cells. 7. When the experiment started, there were 500 mci of a radioactive element with a half-life of 7.2 hours. 8. When the experiment began, there were 4815 cells. The number of cells will increase by 2.3% every 42 hours. 9. When the experiment began there were 500 cells. The number of cells will decrease at an average rate of 2.9% per hour. 10. (Bonus). If after 10 years, the balances do not match your answers for #4, how could knowledge of exponential functions explain any small discrepancies? Declare variables and write an equation to model each of the following situations. 11. The population of Mathland in 1997 was 48,000. Since 1997, Mathland s population has increased by 13.2% annually. 12. The population of Quadratictown in 1997 was 160,000. Since 1997, Quadratictown s population has decreased by 8.5% annually. 13. Use a calculator to find the populations of Mathland and Quadratictown in P a g e

26 Unit 5 Practice Assessment 1 5: There are 10 tribbles present now. The number of tribbles triples every 15 minutes. Assume that the tribbles don t die. 1. How many will be present in 30 minutes? Show or explain your work. 2. Write a Now-Next rule that shows how to use the number of tribbles present at any time to predict the number that will be present 30-minutes later. 3. Write a rule in exponential standard form that can be used to calculate the number of tribbles present after any number of 15-minute periods. y = the number of tribbles. x = the number of 15-minute periods. 4. Use your function rule to complete the table. Number of 15-minute time periods Number of tribbles present 5. How many tribbles will be present in 2 hours? Explain or show your work. 6. A radioactive isotope has a half-life of 1 hour. You start with 80 mg of the isotope. Let x = the number of hours. Let y = the number of mg of the isotope. Write a function rule to model the situation. How much of the isotope should remain after 2 hours? Show or explain your work. What would the value of x be if you wanted to find out how much of the isotope would remain after 2 days? 7. A radioactive isotope has a half-life of 6 hours. You start with 240 mg of the isotope. Let x = the number of hours. Let y = number of mg of the isotope. Write a function rule to model the situation. Explain the meaning of the exponent in your function rule. How much of the isotope should remain after 12 hours? Show or explain your work. 8. A scientist uses the equation y = x to model the half-life of a radioactive isotope. If y = the number of mg of the isotope, and x = the number of minutes, what is known about the isotope and the experiment? How many mg of the isotope is there when the experiment begins? How long should it take to only have 10 mg of the isotope left? How long should it take for half of the current amount of the isotope to decay? 26 P a g e

27 9. Complete the table and graph y = 3 x x y 10. Complete the table and graph y = ( 1 4 )x x y 9. y 10. y x x 11. Adin deposits $1000 into a savings account with an annual percentage rate of 8%. Let y = his balance in dollars. Let x = the number of years. Write an equation to calculate his balance if the interest is compounded annually. Write an equation to calculate his balance if the interest is compounded semiannually. Write an equation to calculate his balance if the interest is compounded quarterly. Write an equation to calculate his balance if the interest is compounded monthly. 12. On January 1, 2013, the population of Mathland was 50,000. Scientists predict that the population will increase by 9% each year. Write an equation to model the situation. Let y = the number of people in Mathland. Let x = the number of years since What is the value of x if you want to predict the number of people in Mathland in 2022? Show or explain your work. 13. The scientists in the neighboring town of Quadraticville have written the following equation to model the town s population. y = x y = the number of people in Quadraticville and x = the number of years since 2013 Is the town s population expected to increase or decrease? Show or explain your work. By what percent is the town s population increasing or decreasing? Show or explain your work. What was the town s population in 2013? Show or explain your work. Based on the function rule, what will the town s population be on January 1, 2014? Show or explain your work. 27 P a g e

28 14. A scientist uses the equation y = x to model the half-life of a radioactive isotope. If y = the number of mg of the isotope, and x = the number of hours, what is known about the isotope and the experiment? How many mg of the isotope is there when the experiment begins? How long should it take to only have 50 mg of the isotope left? How long should it take for half of the current amount of the isotope to decay? 15. Write a NOW-NEXT statement for the following geometric sequence. 125, 25, 5, What are the next 2 numbers in the sequence? 16. Write a NOW-NEXT statement for the following geometric sequence. 6, 24, 96, What are the next 2 numbers in the sequence? 17. Write an exponential function rule for the data. {(0, 6), (1, 24), (2, 96)} 18. Write an exponential function rule for the data {(0, 125), (1, 25), (2, 5)} 19. Tamika and Ron dropped a ball and measured rebound heights (in feet) after each bounce. They found that the rule y = 15(0.8) x could be used to predict the rebound height of the ball, where y is the bounce height and x is the bounce number. a. From what initial height did they drop the ball? b. What does 0.8 represent in the problem? c. How high will the ball bounce on the third bounce? Explain or show your work. 20. Which is the best equation for the graph labeled I? Explain. Which is the best equation for the graph labeled II? Explain. a) y = 8(0.25) x b) y = 8(2) x c) y = 2(0.25) x d) y = 2(2) x e) y = 20(0.25) x f) y = 20(2) x 8 - For each situation below, write the explicit equation in function notation and then solve. Let f(x) = the population (# of individuals). Let x = time in units (hours, days, or years). For each situation, also identify the percent of change, identify the amount of time it takes for the change to occur, and state what the coefficient of x represents. 21. An alien amoeba colony is growing exponentially and had a population of 10,000 when it was first observed. Three hours later, the population was 80,000. What was the population six hours after it was first observed? What will be the population in 12 hours? In 24 hours? 22. The population of P floyd, an alien city found on the dark side of the moon, has grown at a rate of 3.2% each year for the last 10 years. If the population 10 years ago was 25,000, what is the population today? 23. A population of alien bacteria grows by 35% every hour. If the population begins with 100 alien specimens, how many are there after 6 hours? How many will there be in 18 hours? 24. The population in the town of Alien Acres is presently 42,500. The town has been growing at a steady annual rate of 2.7%. Find the number of years ago that the population was 30, P a g e