8H Algebra 1: Unit 9 Exponential Functions

1 8H Algebra 1: Unit 9 Exponential Functions We will: • interpret key features of exponential function graphs and tables. • prove that exponential functions grow by equal factors over equal intervals. • create equations arising from exponential functions and use them to solve problems. • use the properties of exponents to transform expressions for exponential functions. • recognize situations in which a quantity grows or decays by a constant percent rate per unit interval. • interpret the parameters in an exponential function in terms of context. • observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly. Name__________________________ Ms. Cirillo 8H Algebra 1 2 Lesson 1: Graphs of Exponential Functions Aim: Students will be able to graph exponential functions. One of the major nonlinear functions we will study this year is the exponential function. Exponential functions contain the variable, x, in the exponent. A numerical exponent tells us how many times the base is used as a factor (multiplication!) Task- Evaluate each of the following expressions. a) 24 = d) 21 = g) 2-2 b) 23 = e) 20 = h) 2-3 Essential Question- Describe the type of number 2100 is. Essential Question- Describe the type of number 2-100 is. c) 22 = f) 2-1 = i) 2-4 = ! Exercise #1- On the set of axes below, graph the exponential function 𝑓(𝑥) = 2 . State the y-intercept of the function. Is there an x-intercept of the function? Why or why not? Is this function increasing or decreasing? Describe the pattern you see in the table and graph. 3 " ! Exercise #2- On the set of axes below, graph the exponential function 𝑓(𝑥) = '#( . State the y-intercept of the function. Is this function increasing or decreasing? Describe the pattern you see in the table and graph. 4 " Exercise #3- Complete a table of values for the exponential function 𝑓(𝑥) = $ the function. State the y-intercept of the function. Is this function increasing or decreasing? Describe the pattern you see in your table and graph. (2)! 5 . Then, graph 6 Key Idea- Exponential Functions Exponential functions show a pattern in their tables and graphs, but it is not a constant rate of change pattern like linear functions. Therefore, exponential functions are nonlinear curves. y = a(b)x a: y-intercept b: growth or decay factor Exponential functions grow by equal factors over equal intervals. o b > 1: the function strictly increases and shows exponential growth. o 0 < b < 1: the function strictly decreases and shows exponential decay. An exponential function continuously approaches but never touches the x-axis. The x-axis therefore is called an asymptote. J 7 Algebra Regents Question Graph the function If on the set of axes below. , determine if when . Justify your answer. 8 Lesson 2: Writing Equations of Exponential Functions Aim: Students will be able to write equations of exponential functions from tables, graphs, and real world applications. There are many things in the real world that grow faster as they grow larger or decrease slower as they get smaller. These types of phenomena represent exponential growth and decay. Task- The number of people who have heard a rumor often grows exponentially. Consider a rumor that starts with 3 people and where the number of people who have heard it doubles each day that it spreads. a) Why does it make sense that the number of people who have heard a rumor will grow exponentially? b) Fill in the table below for the number of people, N, who knew the rumor after it has spread a certain number of days, d. c) Write an equation giving the number of people who know the rumor, N, given the number of days it has been spreading, d. d) How many people would know the rumor after 20 days? How reasonable is this result? 9 Recall- Exponential growth can be VERY fast as exponential functions grow by equal factors (multiplication!) over equal intervals. The general form of an exponential function equation is y = a(b)x, where a is the y-intercept (initial/starting amount) and b is the growth or decay factor. ------------------------------------------------------------Exercise #1- Write the exponential function equation shown by each table or graph. a) c) x 0 1 2 3 4 f(x) 2 6 18 54 162 b) d) x 1 2 3 4 5 f(x) 80 20 5 1.25 0.3125 10 Exercise #2- Real World Applications a) The graph to the right represents a bacteria population after x days. Write an exponential function that represents the population. Using your equation, find the population after 12 hours. b) Rania owns 4 rabbits. She expects the number of rabbits to double each year. Write an equation to model this situation. Using your equation, determine how many rabbits Rania will own after 5 years. c) The NCAA Basketball Championship (also known as March Madness) is a tournament in which teams play against one another with only the winning teams progressing to the next round. The tournament starts with 64 teams. Write an equation to model this situation. Using your equation, determine how many teams will play at the start of round 5. 11 d) An application developer released a new app to be downloaded. The table below gives the number of downloads for the first four weeks after the launch of the app. Write an exponential equation that models these data. Use this model to predict how many downloads the developer would expect in the 26th week if this trend continues. Round your answer to the nearest download. Would it be reasonable to use this model to predict the number of downloads past one year? Explain your reasoning. e) A laboratory technician studied the population growth of a colony of bacteria. He recorded the number of bacteria every other day, as shown in the table below. Write an exponential equation that models these data. 12 Lesson 3: Exponential Growth and Decay Functions- Percent Rates Task #1- You are out to dinner at The Cheesecake Factory with your friends. The bill totals $80 and you would like to leave a 20% tip. How much money should you leave the waitress in total? a) Find the total amount of money you should leave by first finding 20% of $80 and then adding it to $80. b) Find the total amount of money you should leave with a single multiplication. Why does this work? Task #2- The number of bacteria in a colony has an initial amount of 100 bacteria. The bacteria grows by 50% each hour. Complete the table below to model this situation. Number of hours Number of bacteria 0 1 2 3 Is this an example of exponential growth? If so, what is the growth factor? 4 13 o Task #3- A cup of coffee has a temperature of 200 F. It is cooling down such that its temperature is decreasing at a constant rate of 10% per minute. Complete the table below to model this situation. Number of minutes Temperature, o F 0 1 2 3 4 Is this an example of exponential decay? If so, what is the decay factor? -------------------------------------------------------------------------------------------Key Idea- When a problem increases or decreases by a percent rate, this is an example of exponential growth or decay. o Increase by a percent rate, r: The growth factor is found by computing 1 + r. o Decrease by a percent rate, r: The decay factor is found by computing 1 – r (remember, this is what remains in the problem!) You will notice, the “b” in the equation is replaced by (1 + r) or (1 – r). -------------------------------------------------------------------------------------------Exercise #1- Real World Applications a) If the population of a small village in Maine grows by approximately 6% per year, write an equation to model this situation. 14 b) If there were 12 bacteria specimens and the rate of growth is 80% per hour, write an equation to model this situation. How many bacteria specimens will there be after 11 hours? Round to the nearest hundred bacteria. c) The value of a car is $22,500. The car depreciates (decreases) in value by 12% every year. Write a function that represents the value of the car, V(t), after t years. Find the value of the car after 6 years, to the nearest cent. d) Charlotte deposited $3,000 in an account at the Merrick National Bank, earning 4.5% interest compounded annually. She made no deposits or withdrawals. Write an equation that can be used to find B, the balance in her account, after t years. Find, to the nearest cent, the balance in her account after 3 years. 15 Exercise #2- Multiple Choice Practice a) The current population of a town is 10,000. If the population, P, increases by 20% each year, which equation could be used to find the population after t years? (1) P = 10,000(0.2)t (2) P = 10,000(0.8)t (3) P = 10,000(1.2)t (4) P = 10,000(1.8)t b) The country of Benin in West Africa has a population of 9.05 million people. The population is growing at a rate of 3.1% each year. Which function can be used to find the population 7 years from now? (1) f(t) = (9.05 x 106)(1 – 0.31)7 (2) f(t) = (9.05 x 106)(1 + 0.31)7 (3) f(t) = (9.05 x 106)(1 + 0.031)7 (4) f(t) = (9.05 x 106)(1 - 0.031)7 c) Platon was given $3000 when he turned 2 years old. His parents invested it at a 2% interest rate compounded annually. No deposits or withdrawals were made. Which expression can be used to determine how much money Platon had in the account when he turned 18? (1) 3000(1 + 0.02)16 (2) 3000(1 - 0.02)16 (3) 3000(1 + 0.02)18 (4) 3000(1 - 0.02)18 d) A student invests $500 for 3 years in a savings account that earns 4% interest per year. No further deposits or withdrawals were made during this time. Which statement does not yield the correct balance in the account at the end of 3 years? (1) 500(1.04)3 (2) 500(1 - .04)3 (3) 500(1.04)(1.04)(1.04) (4) 500 + 500(.04) + 520(.04) + 540.8(.04) 16 Lesson 4: Identifying and Interpreting Parts of Exponential Functions Task- For each of the following exponential functions, state its y-intercept and whether it is increasing or decreasing. equation y-intercept increasing/decreasing 𝑓(𝑥) = 125(1.5)! 2 ! 𝑦 = 8. 0 3 3 ! 𝑃(𝑡) = 56 . 0 2 𝑦 = 2! 3 ! 𝑦 = 22 . 0 4 5 ! 𝑦 = 256 . 0 2 -------------------------------------------------------------------------------------------Key Idea- Percent of Change To determine the percent of change for an exponential function: o b > 1 à subtract b - 1 o 0 < b < 1 à subtract 1 – b 17 Exercise #1- Given the exponential equations, answer each of the following. a) y = 50(1.15)% What is the y-intercept (initial value)? __________________ Does this equation represent exponential growth or decay? __________________ What is the growth or decay factor? ___________________ What is the percent of change? b) y = 800(1.018)% . What is the y-intercept (initial value)? __________________ Does this equation represent exponential growth or decay? __________________ What is the growth or decay factor? ___________________ What is the percent of change? c) y = 21,000(0.75)% . What is the y-intercept (initial value)? __________________ Does this equation represent exponential growth or decay? __________________ What is the growth or decay factor? ___________________ What is the percent of change? d) A = 100(1 − 0.12)& . What is the y-intercept (initial value)? __________________ Does this equation represent exponential growth or decay? __________________ What is the growth or decay factor? ___________________ What is the percent of change? 18 Exercise #2- Multiple Choice Practice a) Some banks charge a fee on savings accounts that are left inactive for an extended period of time. The equation y = 5000(0.98)x represents the value, y, of one account that was left inactive for a period of x years. What is the y-intercept of this equation and what does it represent? (1) 0.98, the percent of money in the account initially (2) 0.98, the percent of money in the account after x years (3) 5000, the amount of money in the account initially (4) 5000, the amount of money in the account after x years b) The function V(t) = 1350(1.017)t represents the value V(t), in dollars, of a comic book t years after its purchase. The yearly rate of appreciation of the comic book is (1) 17% (2) 1.7% (3) 1.017% (4) 0.017% c) Kaylee has her money invested in a stock portfolio. The value, v(x), of her portfolio can be modeled with the function v(x) = 30,000(0.78)x, where x is the number of years since she made her investment. Which statement describes the rate of change of the value of her portfolio? (1) It decreases 78% per year. (2) It increases 78% per year. (3) It decreases 22% per year. (4) It increases 22% per year. d) In the equation , A is the total amount, P is the principal amount, r is the annual interest rate, and t is the time in years. Which statement correctly relates information regarding the annual interest rate for each given equation? 1) For , the principal amount of money is increasing at a 25% interest rate. 2) For , the principal amount of money is increasing at a 52% interest rate. 3) For , the principal amount of money is decreasing at a 14% interest rate. 4) For , the principal amount of money is decreasing at a 68% interest rate. 19 Exercise #3- Free Response a) The number of carbon atoms in a fossil is given by the function 𝑦 = 5100(0.95)! , where x represents the number of years since being discovered. In the function, explain what 5100 and 0.95 represent. b) The breakdown of a sample of a chemical compound is represented by the function , where represents the number of milligrams of the substance and t represents the time, in years. In the function , explain what 0.5 and 300 represents. 20 Lesson 5: Rewrite Exponential Functions with Exponent Rules 7th Grade Recap – Rules of Exponents • Multiplying Exponents Rule: Keep the base, Add the Exponents. a m × a n = a m+ n am = a m- n n a • Dividing Exponents Rules: Keep the Base, Subtract the exponents. • Power to a Power Exponent Rule: Keep the Base, Multiple the Exponents. ( a m )n = a mn or ( ab )n = a n b n • 0 Zero Exponent Rule: For any nonzero number a, a = 1 • -n Negative Exponents Rule: a = 1 an -------------------------------------------------------------------------------------------Multiplying 𝑎) 𝑥 ' ∙ 𝑥 $ b) 𝑏 $ ∙ 𝑏 ( Dividing a) !" !# b) +$ c) +% Power to a Power a) (𝑥 - )$ b) (𝑥𝑦)' ! c) 6'( ∙ 6) c) (6#! )# d) 5#'* ∙ 5#'* ,&$ ,! d) (7)-! d) ! '( ! e) (5𝑥 # )' 21 Key Idea – Equivalent Exponential Expressions Exponential Functions can be rewritten using the rules of exponents. -------------------------------------------------------------------------------------------Model Example #1: The number of bacteria grown in a lab can be modeled by 𝑃(𝑡) = 300 ∙ 2$. , where 𝑡 is the number of hours. Which expression is equivalent to 𝑃(𝑡)? (1) 300 ∙ 8. (3) 300 ∙ 16. (2) 300. ∙ 2$ (4) 300#. ∙ 2#. Model Example #2: The growth of a certain organism can be modeled by 𝐶(𝑡) = 10(1.029)#$. , where 𝐶(𝑡) is the total number of cells after 𝑡 hours. Which function is approximately equivalent to 𝐶(𝑡)? (1) 𝐶(𝑡) = 240(. 083)#$. (2) 𝐶(𝑡) = 10(. 083). (3) 𝐶(𝑡) = 10(1.986). (4) 𝐶(𝑡) = 240(1.986)./#$ -------------------------------------------------------------------------------------------Algebra Regents Questions a) A computer application generates a sequence of musical notes using the function 𝑓(𝑛) = 6(16)0 , where 𝑛 is the number of the note in the sequence and 𝑓(𝑛) is the note frequency in hertz. Which function will generate the same note sequence as 𝑓(𝑛)? (1) 𝑔(𝑛) = 12(2)$0 (2) 𝑔(𝑛) = 6(2)$0 (3) 𝑔(𝑛) = 12(4)#0 (4) 𝑔(𝑛) = 6(8)#0 b) Mario's $15,000 car depreciates in value at a rate of 19% per year. The value, V, after t years can be modeled by the function . Which function is equivalent to the original function? 1) 3) 2) 4) 22 c) In an organism, the number of cells, , after d days can be represented by the function . This function can also be expressed as 1) 2) 3) 4) d) The population of a city can be modeled by after t years. Which function is approximately equivalent to 1) 2) 3) 4) , where ? is the population -------------------------------------------------------------------------------------------Model Example #3: Nora inherited a savings account that was started by her grandmother 25 years ago. This scenario is represented by the function A(t) = 5000(1.013)t + 25, where A(t) represents the value of the account, in dollars, t years after the inheritance. Which function below is equivalent to A(t)? (1) A(t) = 5000[(1.013)t]25 (2) A(t) = 5000[(1.013)t + (1.013)25] (3) A(t) = (5000)t(1.013)25 (4) A(t) = 5000(1.013)t(1.013)25 23 Algebra Regents Questions a) The expression is equivalent to 1) 2) 3) 4) b) A laboratory technician used the function following expressions: I. II. The function is equivalent to 1) I, only 2) II, only 3) I and III 4) II and III to model her research. Consider the III. c) Jacob and Jessica are studying the spread of dandelions. Jacob discovers that the growth over t weeks can be defined by the function . Jessica finds that the growth function over t weeks is . Calculate the number of dandelions that Jacob and Jessica will each have after 5 weeks. Based on the growth from both functions, explain the relationship between and . 24 Lesson 6: Comparing Linear and Exponential Functions Here is a classic riddle! Mr. Jones has an apple orchard. He hires his friend, Arthur, to pick apples and offers him two payment options: o Option 1- $1.50 per bushel of apples picked. o Option 2- 1 cent for picking one bushel, 3 cents for picking two bushels, 9 cents for picking 3 bushels and so on, with the amount of money tripling for each additional bushel picked. Task- Complete a table of values for each option. number of bushels 1 2 3 4 5 6 7 payment ($) Which option represents a linear function? How do you know? number of bushels 1 2 3 4 5 6 7 Which option represents an exponential function? How do you know? If Arthur picks 6 bushels of apples, which option should he choose? If Arthur picks 18 bushels of apples, which option should he choose? What does this tell us about exponential growth? payment ($) 25 Key Idea- Linear v Exponential Functions Linear functions have a constant rate of change. Linear functions grow by equal differences over equal intervals. Exponential functions do not have a constant rate of change. Exponential functions grow by equal factors over equal intervals. Key words for exponential functions à factor, percent rate of change, double/triple/half/etc. -------------------------------------------------------------------------------------------Exercise #1- State if each situation could be modeled by a linear or exponential function. a) a population of bacteria that doubles every hour b) a bank account balance that grows at a rate of 5% per year, compounded annually c) the cost of a cell phone service that charges a base amount plus 20 cents per minute d) the concentration of medicine in a person’s body that decays by a factor of one-third every hour e) the amount of money in Christina’s piggy bank which she adds $10 to each week f) the amount of money in a savings account where $150 is deducted every month g) the amount of money in a certificate of deposit that gets 4% interest each year *h) the amount of money in Jake’s wallet that increases and decreases by a different amount of each week 26 Exercise #2- Free Response Practice a) Two functions are shown in the table below. Which function is linear? How do you know? Which function is exponential? How do you know? b) Jonathan has invented a new app for smart phones that two companies are interested in purchasing for a two-year contract. • • Company A is offering him $10,000 for the first month and will increase the amount each month by $5,000. Company B is offering him $500 for the first month and will double their payment each month from the previous month. Monthly payments are made at the end of each month. For which monthly payment will company B’s payment first exceed company A’s payment? 27 Algebra Regents Question Mike knows that (3, 6.5) and (4, 17.55) are points on the graph of an exponential function, g(x), and he wants to find another point on the graph of this function. First, he subtracts 6.5 from 17.55 to get 11.05. Next, he adds 11.05 and 17.55 to get 28.6. He states that (5, 28.6) is a point on g(x). Is he correct? Explain your reasoning. 28 Unit 10 Review Sheet – Exponential Functions ! ! # 1. Graph 𝑦 = (4)# on the set of axes below. 2. Graph 𝑦 = & ' on the set of axes below. a) State the y-intercept of the function. a) State the y-intercept of the function. b) Is this function increasing or decreasing? b) Is this function increasing or decreasing? c) Describe the pattern you see in your table and graph. c) Describe the pattern you see in your table and graph. " $ 29 3. Write an equation to represent each exponential function. a) x 1 2 3 4 5 f(x) 6 12 24 48 96 b) c) The New York Volleyball Association invited 64 teams to compete in a tournament. After each round, half of the teams were eliminated. Write an equation to represent the number of teams, t, that remained in the tournament after r rounds. 4. Amber bought an antique dresser for $500. If the value of the dresser increases by 6% each year, write an equation to determine the value, V(t), of the dresser after t years. - 5. Peter opened a retirement account with $36,500. His account grew at a rate of 3 $ % per year compounded annually. He made no deposits or withdrawals on the account. Write an equation to model this situation. At the end of 20 years, what was the account worth, to the nearest dollar? 30 6. On January 1, 1999, the price of gasoline was $1.39 per gallon. If the price of gasoline increased by 0.5% per month, write an equation to model this situation. What was the cost of one gallon of gasoline, to the nearest cent, on January 1 one year later? 7. A car was purchased for $25,000. Research shows that the car has an average yearly depreciation rate of 17%. Create a function that will determine the value, V(t), of the car t years after purchase. Determine, to the nearest cent, how much the car will depreciate from year 3 to year 4. 8. Given the equation 𝑦 = 3(0.6)! . a) What is the y-intercept? b) Does this equation represent exponential growth or decay? How do you know? c) What is the growth or decay factor? d) What is the percent of change? 31 - ! 9. Given the equation 𝑦 = 4 '#( . a) What is the y-intercept? b) Does this equation represent exponential growth or decay? How do you know? c) What is the growth or decay factor? d) What is the percent of change? 10. The value, v(t), of a car depreciates according to the function v(t) = 22,500(0.85)t, where t is the time, in years, since the car was purchased. In the function v(t), explain what the 22,500 and the 0.85 represent. 11. Multiple Choice- A population, p(x), of turkeys in a certain area is represented by the function p(x) = 17(1.15)2x. Which function below is equivalent to p(x)? (1) p(x) = (19.55)2x (2) p(x) = 17(1.3225)x (3) p(x) = 17(1.35)x (4) p(x) = 1.15(17)x 32 12. Multiple Choice- Corinne inherited a savings account that was started by her grandmother 5 years ago. This scenario is represented by the function B(t) = 2000(1.02)t + 5, where B(t) represents the value of the account, in dollars, t years after the inheritance. Which function below is equivalent to B(t)? 1) B(t) = 2000[(1.02)t]5 2) B(t) = 2000[(1.02)t + (1.02)5] 3) B(t) = (2000)t(1.02)5 4) B(t) = 2000(1.02)t(1.02)5 13. State if each function is linear and exponential. Explain your reasoning. a) x 1 2 3 4 f(x) 120 180 270 405 b) x 1 2 3 4 g(x) 7.5 11 14.5 18 c) The population of a town increases by 2% every 5 years. d) A radioactive substance decays by a factor of one-eighth each day. e) A gym charges a one-time membership fee plus a monthly fee of $19.99. 33 Blasts from the Past! 1. A function is defined as {(2, 3), (4, 5), (6, 7)}. State an ordered pair that could be added to the set of ordered pairs so that it remains a function. 2. Write the inequality represented by the graph shown below. 3. State the domain and range of the function shown below. 4. If g(x) = f(x – 5) + 3, state the transformations on f(x) that created g(x). 5. Simplify: (x – 4)2 + 3(2x + 6) 34 6. Find the average rate of change between hour 1 and hour 5. 7. Determine the x-intercept and y-intercept of the graph of the equation 2x + 4y = 12. 8. Solve for a: b = 10(a – 5) 9. The sum of three consecutive odd integers is equal to nine more than twice the third integer. Determine the three integers algebraically. 35 10. Dana went shopping for plants to put in her garden. She bought three roses and two daisies for $31.88. Later that day, she went back and bought two roses and one daisy for $18.92. If r represents the cost of one rose and d represents the cost of one daisy, write a system of equations that models this situation. Use your system of equations to algebraically determine both the cost of one rose and the cost of one daisy. If Dana had waited until the plants were on sale, she would have paid $4.50 for each rose and $6.50 for each daisy. Determine the total amount of money she would have saved by buying all of her flowers during the sale.