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NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S118 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 18 Graphs of Exponential Functions and Logarithmic Functions Classwork Opening Exercise Complete the following table of values of the function 𝑓𝑥 2𝑥 We want to sketch the graph of 𝑦 𝑓𝑥 and then reflect that graph across the diagonal line with equation 𝑦 𝑥 𝒙 𝒚 𝟐𝒙 Point 𝒙𝒚 on the graph of 𝒚 𝟐𝒙 3 2 1 0 1 2 3 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S119 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License On the set of axes below plot the points from the table and sketch the graph of 𝑦 2𝑥 Next sketch the diagonal line with equation 𝑦 𝑥 and then reflect the graph of 𝑦 2𝑥 across the line NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S120 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 1 Complete the following table of values of the function 𝑔𝑥 log2𝑥 We want to sketch the graph of 𝑦 𝑔𝑥 and then reflect that graph across the diagonal line with equation 𝑦 𝑥 𝒙 𝒚 𝐥𝐨𝐠𝟐𝒙 Point 𝒙𝒚 on the graph of 𝒚 𝐥𝐨𝐠𝟐𝒙 1 8 1 4 1 2 1 2 4 8 On the set of axes below plot the points from the table and sketch the graph of 𝑦 log2𝑥 Next sketch the diagonal line with equation 𝑦 𝑥 and then reflect the graph of 𝑦 log2𝑥 across the line NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S121 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 2 Working independently predict the relation between the graphs of the functions 𝑓𝑥 3𝑥 and 𝑔𝑥 log3𝑥 Test your predictions by sketching the graphs of these two functions Write your prediction in your notebook provide justification for your prediction and compare your prediction with that of your neighbor NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S122 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Now lets compare the graphs of the functions 𝑓2𝑥 2𝑥 and 𝑓3𝑥 3𝑥 Sketch the graphs of the two exponential functions on the same set of axes then answer the questions below a Where do the two graphs intersect b For which values of 𝑥 is 2𝑥 3𝑥 c For which values of 𝑥 is 2𝑥 3𝑥 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S123 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License d What happens to the values of the functions 𝑓2 and 𝑓3 as 𝑥 e What happens to the values of the functions 𝑓2 and 𝑓3 as 𝑥 f Does either graph ever intersect the 𝑥axis Explain how you know 4 Add sketches of the two logarithmic functions 𝑔2𝑥 log2𝑥 and 𝑔3𝑥 log3𝑥 to the axes with the graphs of the exponential functions from Exercise 3 then answer the questions below a Where do the two logarithmic graphs intersect b For which values of 𝑥 is log2𝑥 log3𝑥 c For which values of 𝑥 is log2𝑥 log3𝑥 d What happens to the values of the functions 𝑔2 and 𝑔3 as 𝑥 e What happens to the values of the functions 𝑔2 and 𝑔3 as 𝑥 0 f Does either graph ever intersect the 𝑦axis Explain how you know g Describe the similarities and differences in the behavior of 𝑓2𝑥 and 𝑔2𝑥 as 𝑥 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S124 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Sketch the graphs of the functions 𝑓𝑥 5𝑥 and 𝑔𝑥 log5𝑥 2 Sketch the graphs of the functions 𝑓𝑥 1 2 𝑥 and 𝑔𝑥 log1 2𝑥 3 Sketch the graphs of the functions 𝑓1𝑥 1 2 𝑥 and 𝑓2𝑥 3 4 𝑥 on the same sheet of graph paper and answer the following questions a Where do the two exponential graphs intersect b For which values of 𝑥 is 1 2 𝑥 3 4 𝑥 c For which values of 𝑥 is 1 2 𝑥 3 4 𝑥 d What happens to the values of the functions 𝑓1 and 𝑓2 as 𝑥 e What are the domains of the two functions 𝑓1 and 𝑓2 4 Use the information from Problem 3 together with the relationship between graphs of exponential and logarithmic functions to sketch the graphs of the functions 𝑔1𝑥 log1 2𝑥 and 𝑔2𝑥 log3 4𝑥 on the same sheet of graph paper Then answer the following questions a Where do the two logarithmic graphs intersect b For which values of 𝑥 is log1 2𝑥 log3 4𝑥 c For which values of 𝑥 is log1 2𝑥 log3 4𝑥 d What happens to the values of the functions 𝑔1 and 𝑔2 as 𝑥 e What are the domains of the two functions 𝑔1 and 𝑔2 5 For each function 𝑓 find a formula for the function ℎ in terms of 𝑥 a If 𝑓𝑥 𝑥3 find ℎ𝑥 128𝑓 1 4 𝑥 𝑓2𝑥 b If 𝑓𝑥 𝑥2 1 find ℎ𝑥 𝑓𝑥 2 𝑓2 c If 𝑓𝑥 𝑥3 2𝑥2 5𝑥 1 find ℎ𝑥 𝑓𝑥 𝑓𝑥 2 d If 𝑓𝑥 𝑥3 2𝑥2 5𝑥 1 find ℎ𝑥 𝑓𝑥 𝑓𝑥 2 6 In Problem 5 parts c and d list at least two aspects about the formulas you found as they relate to the function 𝑓𝑥 𝑥3 2𝑥2 5𝑥 1 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 18 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions S125 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 For each of the functions 𝑓 and 𝑔 below write an expression for i 𝑓𝑔𝑥 ii 𝑔𝑓𝑥 and iii 𝑓𝑓𝑥 in terms of 𝑥 a 𝑓𝑥 𝑥 2 3 𝑔𝑥 𝑥12 b 𝑓𝑥 𝑏 𝑥𝑎 𝑔𝑥 𝑏 𝑥 𝑎 for two numbers 𝑎 and 𝑏 when 𝑥 is not equal to 0 or 𝑎 c 𝑓𝑥 𝑥1 𝑥1 𝑔𝑥 𝑥1 𝑥1 when 𝑥 is not equal to 1 or 1 d 𝑓𝑥 2𝑥 𝑔𝑥 log2𝑥 e 𝑓𝑥 ln𝑥 𝑔𝑥 𝑒𝑥 f 𝑓𝑥 2 100𝑥 𝑔𝑥 1 2 log 1 2 𝑥 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Rational Functions S62 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 11 Rational Functions Classwork Opening Exercise Factor each expression completely a 9𝑥4 16𝑥2 b 2𝑥3 5𝑥2 8𝑥 20 c 𝑥3 3𝑥2 3𝑥 1 d 8𝑥3 1 Example 1 Simplify the expression 𝑥2 5𝑥 6 𝑥 3 to lowest terms and identify the values of 𝑥 that must be excluded to avoid division by zero NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Rational Functions S63 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 1 Simplifying Rational Expressions to Lowest Terms 1 Simplify each rational expression to lowest terms specifying the values of 𝑥 that must be excluded to avoid division by zero a 𝑥2 6𝑥 5 𝑥2 3𝑥 10 b 𝑥3 3𝑥23𝑥 1 𝑥3 2𝑥2 𝑥 c 𝑥2 16 𝑥2 2𝑥 8 d 𝑥2 3𝑥 10 𝑥3 6𝑥2 12𝑥 8 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Rational Functions S64 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License e 𝑥3 1 𝑥2 1 Example 2 Let 𝑓𝑥 2𝑥4 6𝑥3 6𝑥2 2𝑥 3𝑥2 3𝑥 Simplify the rational expression 2𝑥4 6𝑥3 6𝑥2 2𝑥 3𝑥2 3𝑥 to lowest terms and use the simplified form to express the rule of 𝑓 Be sure to indicate any restrictions on the domain Exercise 2 2 Determine the domain of each rational function and express the rule for each function in an equivalent form in lowest terms a 𝑓𝑥 𝑥 22𝑥 3𝑥 1 𝑥 2𝑥 1 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Rational Functions S65 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License b 𝑓𝑥 𝑥2 6𝑥 9 𝑥 3 c 𝑓𝑥 3𝑥3 75𝑥 𝑥3 15𝑥2 75𝑥 125 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Rational Functions S66 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 For each pair of functions 𝑓 and 𝑔 find the domain of 𝑓 and the domain of 𝑔 Indicate whether 𝑓 and 𝑔 are the same function a 𝑓𝑥 𝑥2 𝑥 𝑔𝑥 𝑥 b 𝑓𝑥 𝑥 𝑥 𝑔𝑥 1 c 𝑓𝑥 2𝑥2 6𝑥 8 2 𝑔𝑥 𝑥2 6𝑥 8 d 𝑓𝑥 𝑥2 3𝑥 2 𝑥 2 𝑔𝑥 𝑥 1 e 𝑓𝑥 𝑥 2 𝑥2 3𝑥 2 𝑔𝑥 1 𝑥 1 f 𝑓𝑥 𝑥4 1 𝑥2 1 𝑔𝑥 𝑥2 1 g 𝑓𝑥 𝑥4 1 𝑥2 1 𝑔𝑥 𝑥2 1 h 𝑓𝑥 𝑥4 𝑥 𝑥2 𝑥 𝑔𝑥 𝑥3 1 𝑥 1 i 𝑓𝑥 𝑥4 𝑥3 𝑥2 𝑥2 𝑥 1 𝑔𝑥 𝑥2 2 Determine the domain of each rational function and express the rule for each function in an equivalent form in lowest terms a 𝑓𝑥 𝑥4 𝑥2 b 𝑓𝑥 3𝑥 3 15𝑥 6 c 𝑓𝑥 𝑥2 𝑥 2 𝑥2 𝑥 d 𝑓𝑥 8𝑥2 2𝑥 15 4𝑥2 4𝑥 15 e 𝑓𝑥 2𝑥3 3𝑥2 2𝑥 3 𝑥3 𝑥 f 𝑓𝑥 3𝑥3 𝑥2 3𝑥 1 𝑥3 𝑥 3 For each pair of functions below calculate 𝑓𝑥 𝑔𝑥 𝑓𝑥 𝑔𝑥 𝑓𝑥 𝑔𝑥 and 𝑓𝑥 𝑔𝑥 Indicate restrictions on the domain of the resulting functions a 𝑓𝑥 2 𝑥 𝑔𝑥 𝑥 𝑥 2 b 𝑓𝑥 3 𝑥 1 𝑔𝑥 𝑥 𝑥3 1 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S148 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 22 Choosing a Model Classwork Opening Exercise a You are working on a team analyzing the following data gathered by your colleagues 11 5 0 105 15 178 43 120 Your coworker Alexandra says that the model you should use to fit the data is 𝑘𝑡 100 sin15𝑡 105 Sketch Alexandras model on the axes at left on the next page b How does the graph of Alexandras model 𝑘𝑡 100 sin15𝑡 105 relate to the four points Is her model a good fit to this data c Another teammate Randall says that the model you should use to fit the data is 𝑔𝑡 16𝑡2 72𝑡 105 Sketch Randalls model on the axes at right on the next page NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S149 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Alexandras Model Randalls Model d How does the graph of Randalls model 𝑔𝑡 16𝑡2 72𝑡 105 relate to the four points Is his model a good fit to the data e Suppose the four points represent positions of a projectile fired into the air Which of the two models is more appropriate in that situation and why f In general how do we know which model to choose NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S150 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 1 The table below contains the number of daylight hours in Oslo Norway on the specified dates Date Hours and Minutes Hours August 1 16 56 1693 September 1 14 15 1425 October 1 11 33 1155 November 1 8 50 883 a Plot the data on the grid provided and decide how to best represent it b Looking at the data what type of function appears to be the best fit NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S151 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c Looking at the context in which the data was gathered what type of function should be used to model the data d Do you have enough information to find a model that is appropriate for this data Either find a model or explain what other information you would need to do so 2 The goal of the US Centers for Disease Control and Prevention CDC is to protect public health and safety through the control and prevention of disease injury and disability Suppose that 45 people have been diagnosed with a new strain of the flu virus and that scientists estimate that each person with the virus will infect 5 people every day with the flu a What type of function should the scientists at the CDC use to model the initial spread of this strain of flu to try to prevent an epidemic Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S152 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 An artist is designing posters for a new advertising campaign The first poster takes 10 hours to design but each subsequent poster takes roughly 15 minutes less time than the previous one as he gets more practice a What type of function models the amount of time needed to create 𝑛 posters for 𝑛 20 Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so 4 A homeowner notices that her heating bill is the lowest in the month of August and increases until it reaches its highest amount in the month of February After February the amount of the heating bill slowly drops back to the level it was in August when it begins to increase again The amount of the bill in February is roughly four times the amount of the bill in August a What type of function models the amount of the heating bill in a particular month Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S153 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 An online merchant sells used books for 500 each and the sales tax rate is 6 of the cost of the books Shipping charges are a flat rate of 400 plus an additional 100 per book a What type of function models the total cost including the shipping costs of a purchase of 𝑥 books Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so 6 A stunt woman falls from a tall building in an actionpacked movie scene Her speed increases by 32 fts for every second that she is falling a What type of function models her distance from the ground at time 𝑡 seconds Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S154 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 A new car depreciates at a rate of about 20 per year meaning that its resale value decreases by roughly 20 each year After hearing this Brett said that if you buy a new car this year then after 5 years the car has a resale value of 000 Is his reasoning correct Explain how you know 2 Alexei just moved to Seattle and he keeps track of the average rainfall for a few months to see if the city deserves its reputation as the rainiest city in the United States Month Average rainfall July 093 in September 161 in October 324 in December 606 in What type of function should Alexei use to model the average rainfall in month 𝑡 3 Sunny who wears her hair long and straight cuts her hair once per year on January 1 always to the same length Her hair grows at a constant rate of 2 cm per month Is it appropriate to model the length of her hair with a sinusoidal function Explain how you know Lesson Summary If we expect from the context that each new term in the sequence of data is a constant added to the previous term then we try a linear model If we expect from the context that the second differences of the sequence are constant meaning that the rate of change between terms either grows or shrinks linearly then we try a quadratic model If we expect from the context that each new term in the sequence of data is a constant multiple of the previous term then we try an exponential model If we expect from the context that the sequence of terms is periodic then we try a sinusoidal model Model Equation of Function Rate of Change Linear 𝑓𝑡 𝑎𝑡 𝑏 for 𝑎 0 Constant Quadratic 𝑔𝑡 𝑎𝑡2 𝑏𝑡 𝑐 for 𝑎 0 Changing linearly Exponential ℎ𝑡 𝑎𝑏𝑐𝑡 for 0 𝑏 1 or 𝑏 1 A multiple of the current value Sinusoidal 𝑘𝑡 𝐴 sin𝑤𝑡 ℎ 𝑘 for 𝐴 𝑤 0 Periodic NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 22 ALGEBRA II Lesson 22 Choosing a Model S155 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 4 On average it takes 2 minutes for a customer to order and pay for a cup of coffee a What type of function models the amount of time you wait in line as a function of how many people are in front of you Explain how you know b Find a model that is appropriate for this situation 5 An online ticketselling service charges 5000 for each ticket to an upcoming concert In addition the buyer must pay 8 sales tax and a convenience fee of 600 for the purchase a What type of function models the total cost of the purchase of 𝑛 tickets in a single transaction b Find a model that is appropriate for this situation 6 In a video game the player must earn enough points to pass one level and progress to the next as shown in the table below To pass this level You need this many total points 1 5000 2 15000 3 35000 4 65000 That is the increase in the required number of points increases by 10000 points at each level a What type of function models the total number of points you need to pass to level 𝑛 Explain how you know b Find a model that is appropriate for this situation 7 The southern white rhinoceros reproduces roughly once every three years giving birth to one calf each time Suppose that a nature preserve houses 100 white rhinoceroses 50 of which are female Assume that half of the calves born are female and that females can reproduce as soon as they are 1 year old a What type of function should be used to model the population of female white rhinoceroses in the preserve b Assuming that there is no death in the rhinoceros population find a function to model the population of female white rhinoceroses in the preserve c Realistically not all of the rhinoceroses survive each year so we assume a 5 death rate of all rhinoceroses Now what type of function should be used to model the population of female white rhinoceroses in the preserve d Find a function to model the population of female white rhinoceroses in the preserve taking into account the births of new calves and the 5 death rate NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S80 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 15 Structure in Graphs of Polynomial Functions Classwork Opening Exercise Sketch the graph of 𝑓𝑥 𝑥2 What will the graph of 𝑔𝑥 𝑥4 look like Sketch it on the same coordinate plane What will the graph of ℎ𝑥 𝑥6 look like Example 1 Sketch the graph of 𝑓𝑥 𝑥3 What will the graph of 𝑔𝑥 𝑥5 look like Sketch this on the same coordinate plane What will the graph of ℎ𝑥 𝑥7 look like Sketch this on the same coordinate plane NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S81 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 1 a Consider the following function 𝑓𝑥 2𝑥4 𝑥3 𝑥2 5𝑥 3 with a mixture of odd and even degree terms Predict whether its end behavior will be like the functions in the Opening Exercise or more like the functions from Example 1 Graph the function 𝑓 using a graphing utility to check your prediction b Consider the following function 𝑓𝑥 2𝑥5 𝑥4 2𝑥3 4𝑥2 𝑥 3 with a mixture of odd and even degree terms Predict whether its end behavior will be like the functions in the Opening Exercise or more like the functions from Example 1 Graph the function 𝑓 using a graphing utility to check your prediction c Thinking back to our discussion of 𝑥intercepts of graphs of polynomial functions from the previous lesson sketch a graph of an evendegree polynomial function that has no 𝑥intercepts d Similarly can you sketch a graph of an odddegree polynomial function with no 𝑥intercepts NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S82 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 2 The Center for Transportation Analysis CTA studies all aspects of transportation in the United States from energy and environmental concerns to safety and security challenges A 1997 study compiled the following data of the fuel economy in miles per gallon mpg of a car or light truck at various speeds measured in miles per hour mph The data are compiled in the table below Fuel Economy by Speed Speed mph Fuel Economy mpg 15 244 20 279 25 305 30 317 35 312 40 310 45 316 50 324 55 324 60 314 65 292 70 268 75 248 Source Transportation Energy Data Book Table 428 httpctaornlgovdatachapter4shtml a Plot the data using a graphing utility Which variable is the independent variable b This data can be modeled by a polynomial function Determine if the function that models the data would have an even or odd degree c Is the leading coefficient of the polynomial that can be used to model this data positive or negative d List two possible reasons the data might have the shape that it does NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S83 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Relevant Vocabulary EVEN FUNCTION Let 𝑓 be a function whose domain and range is a subset of the real numbers The function 𝑓 is called even if the equation 𝑓𝑥 𝑓𝑥 is true for every number 𝑥 in the domain Evendegree polynomial functions are sometimes even functions like 𝑓𝑥 𝑥10 and sometimes not like 𝑔𝑥 𝑥2 𝑥 ODD FUNCTION Let 𝑓 be a function whose domain and range is a subset of the real numbers The function 𝑓 is called odd if the equation 𝑓𝑥 𝑓𝑥 is true for every number 𝑥 in the domain Odddegree polynomial functions are sometimes odd functions like 𝑓𝑥 𝑥11 and sometimes not like ℎ𝑥 𝑥3 𝑥2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S84 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Graph the functions from the Opening Exercise simultaneously using a graphing utility and zoom in at the origin a At 𝑥 05 order the values of the functions from least to greatest b At 𝑥 25 order the values of the functions from least to greatest c Identify the 𝑥values where the order reverses Write a brief sentence on why you think this switch occurs 2 The National Agricultural Statistics Service NASS is an agency within the USDA that collects and analyzes data covering virtually every aspect of agriculture in the United States The following table contains information on the amount in tons of the following vegetables produced in the US from 19881994 for processing into canned frozen and packaged foods lima beans snap beans beets cabbage sweet corn cucumbers green peas spinach and tomatoes Vegetable Production by Year Year Vegetable Production tons 1988 11393320 1989 14450860 1990 15444970 1991 16151030 1992 14236320 1993 14904750 1994 18313150 Source NASS Statistics of Vegetables and Melons 1995 Table 191 httpwwwnassusdagovPublicationsAgStatistics19951996agr954pdf a Plot the data using a graphing utility b Determine if the data display the characteristics of an odd or evendegree polynomial function c List two possible reasons the data might have such a shape NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S85 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 The US Energy Information Administration EIA is responsible for collecting and analyzing information about energy production and use in the United States and for informing policy makers and the public about issues of energy the economy and the environment The following table contains data from the EIA about natural gas consumption from 19502010 measured in millions of cubic feet US Natural Gas Consumption by Year Year US natural gas total consumption millions of cubic feet 1950 577 1955 869 1960 1197 1965 1528 1970 2114 1975 1954 1980 1988 1985 1728 1990 1917 1995 2221 2000 2333 2005 2201 2010 2409 Source US Energy Information Administration httpwwweiagovdnavnghistn9140us2ahtm a Plot the data using a graphing utility b Determine if the data display the characteristics of an odd or evendegree polynomial function c List two possible reasons the data might have such a shape 4 We use the term even function when a function 𝑓 satisfies the equation 𝑓𝑥 𝑓𝑥 for every number 𝑥 in its domain Consider the function 𝑓𝑥 3𝑥2 7 Note that the degree of the function is even and each term is of an even degree the constant term is degree 0 a Graph the function using a graphing utility b Does this graph display any symmetry c Evaluate 𝑓𝑥 d Is 𝑓 an even function Explain how you know 5 We use the term odd function when a function 𝑓 satisfies the equation 𝑓𝑥 𝑓𝑥 for every number 𝑥 in its domain Consider the function 𝑓𝑥 3𝑥3 4𝑥 The degree of the function is odd and each term is of an odd degree a Graph the function using a graphing utility b Does this graph display any symmetry c Evaluate 𝑓𝑥 d Is 𝑓 an odd function Explain how you know NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 15 ALGEBRA II Lesson 15 Structure in Graphs of Polynomial Functions S86 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 6 We have talked about 𝑥intercepts of the graph of a function in both this lesson and the previous one The 𝑥intercepts correspond to the zeros of the function Consider the following examples of polynomial functions and their graphs to determine an easy way to find the 𝑦intercept of the graph of a polynomial function 𝑓𝑥 2𝑥2 4𝑥 3 𝑓𝑥 𝑥3 3𝑥2 𝑥 5 𝑓𝑥 𝑥4 2𝑥3 𝑥2 3𝑥 6 Lesson 11 Rational Functions Classwork Opening Exercise Factor each expression completely a 9x4 16x2 b 2x3 5x2 8x 20 c x3 3x2 3x 1 d 8x3 1 Example 1 Simplify the expression x2 5x 6x 3 to lowest terms and identify the values of x that must be excluded to avoid division by zero EUREKA MATH Lesson 11 Rational Functions engage ny S62 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE 130082015 Exercise 1 Simplifying Rational Expressions to Lowest Terms 1 Simplify each rational expression to lowest terms specifying the values of x that must be excluded to avoid division by zero a x2 6x 5 x2 3x 10 b x3 3x2 3x 1 x3 2x2 x c x2 16 x2 2x 8 d x2 3x 10 x3 6x2 12x 8 EUREKA MATH Lesson 11 Rational Functions engage ny S63 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE 130082015 e x3 1 x2 1 Example 2 Let fx 2x4 6x3 6x2 2x 3x2 3x Simplify the rational expression 2x4 6x3 6x2 2x 3x2 3x to lowest terms and use the simplified form to express the rule of f Be sure to indicate any restrictions on the domain Exercise 2 2 Determine the domain of each rational function and express the rule for each function in an equivalent form in lowest terms a fx x 22 x 3x 1 x 2x 1 EUREKA MATH Lesson 11 Rational Functions engage ny S64 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM3TE 130082015 b fx x2 6x 9 x 3 x 3x 3 x 3 x 3 c fx 3x3 75x x3 15x2 75x 125 3xx2 25 x 5x 5x 5 3xx 5x 5 x 5x 5x 5 3xx 5 x 52 1 For each pair of functions f and g find the domain of f and the domain of g Indicate whether f and g are the same function a fx x2x gx x b fx xx gx 1 c fx 2x2 6x 82 gx x2 6x 8 d fx x2 3x 2x 2 gx x 1 e fx x 2x2 3x 2 gx 1x 1 f fx x4 1x2 1 gx x2 1 g fx x4 1x2 1 gx x2 1 h fx x4 x x2 x gx x3 1 x 1 i fx x4 x3 x2 x2 x 1 gx x2 2 Determine the domain of each rational function and express the rule for each function in an equivalent form in lowest terms a fx x4 x2 x2 b fx 3x 3 15x 6 3x 1 35x 2 x 1 5x 2 c fx x2 x 2 x2 x x 2x 1 xx 1 x 2 x d fx 8x2 2x 15 4x2 4x 15 2x 34x 5 2x 32x 5 4x 5 2x 5 e fx 2x3 3x2 2x 3 x3 x 2x 3x2 9 xx2 9 2x 3 x f fx 3x3 x2 3x 1 x3 x 3x 1x2 3 xx2 3 3x 1 x 3x 9 x 3 For each pair of functions below calculate fx gx fx gx fx gx and fxgx Indicate restrictions on the domain of the resulting functions a fx 2x gx xx 2 b fx 3x 1 gx xx3 1 a Domain f IR x 0 Domain g IR fx gx b Domain f IR x 0 Domain g IR fx gx c Domain f IR Domain g IR fx gx d Domain f IR x 2 Domain g IR fx gx e Domain f IR x 2 and x 3 Domain g IR x 3 fx gx f Domain f IR x 3 and x 3 Domain g IR fx gx g Domain f IR Domain g IR fx gx h Domain f IR x 0 and x 3 Domain g IR fx gx i Domain f IR Domain g IR fx gx NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M3 ALGEBRA II Lesson 18 Graphs of Exponential Functions and Logarithmic Functions Classwork Opening Exercise Complete the following table of values of the function fx 2x We want to sketch the graph of y fx and then reflect that graph across the diagonal line with equation y x x y 2x Point x y on the graph of y 2x 3 18 3 18 2 14 2 14 1 12 1 12 0 1 0 1 1 2 1 2 2 4 2 4 3 8 3 8 EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M3 ALGEBRA II On the set of axes below plot the points from the table and sketch the graph of y 2x Next sketch the diagonal line with equation y x and then reflect the graph of y 2x across the line EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny S1 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M3 ALGEBRA II Exercises 1 Complete the following table of values of the function gx log2x We want to sketch the graph of y gx and then reflect that graph across the diagonal line with equation y x x y log2x Point x y on the graph of y log2x 18 3 18 3 14 2 14 2 12 1 12 1 1 0 1 0 2 1 2 1 4 2 4 2 8 3 8 3 On the set of axes below plot the points from the table and sketch the graph of y log2x Next sketch the diagonal line with equation y x and then reflect the graph of y log2x across the line EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny S12 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 2 Working independently predict the relation between the graphs of the functions fx 3x and gx log3x Test your predictions by sketching the graphs of these two functions Write your prediction in your notebook provide justification for your prediction and compare your prediction with that of your neighbor log3 9 x 3x 9 3x 32 x2 log3 0 x 3x 0 3x 0 30 1 log3 y x 3x y x 0 EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny S121 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka mathorg 3 Now lets compare the graphs of the functions f2 x 2x and f3 x 3x Sketch the graphs of the two exponential functions on the same set of axes then answer the questions below a Where do the two graphs intersect 0 1 b For which values of x is 2x 3x x 0 2x 3x c For which values of x is 2x 3x x 0 2x 3x EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny S122 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka mathorg d What happens to the values of the functions f2 and f3 as x Both functions are going up to infinity However one are faster than other So As x both f2 x and f3 x e What happens to the values of the functions f2 and f3 as x Both functions are close of the zero when x So As x both f2 x 0 and f3 x 0 f Does either graph ever intersect the xaxis Explain how you know No because both functions are exponential and their values never reach zero So 2x 0 and 3x 0 4 Add sketches of the two logarithmic functions g2 x log2 x and g3 x log3 x to the axes with the graphs of the exponential functions from Exercise 3 then answer the questions below a Where do the two logarithmic graphs intersect 10 b For which values of x is log2 x log3 x if x 3 then log2 x log3 x c For which values of x is log2 x log3 x if x 3 the log2 x log3 x d What happens to the values of the functions g2 and g3 as x When x is approaching infinity the values increase So g2 x and g3 x e What happens to the values of the functions g2 and g3 as x 0 As x is approaching zero the values nearch closen to zero by the way when x 0 g2 x and g3 x f Does either graph ever intersect the yaxis Explain how you know No Logarithms negative doesnt solution g Describe the similarities and differences in the behavior of f2 x and g2 x as x x f2 x g2 x EUREKA MATH Lesson 18 Graphs of Exponential Functions and Logarithmic Functions engage ny S123 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka mathorg 5x y 5x 5 y log5x log5 y sy s y 0 123 y 320 5 123 y y 8 log128 x log12 y 12y 8 2y 23 y 3 log34 2 y 12y 2 2y 23 y 3 y 3 when the graphs are together in 01 If x 0 then 12x 34x If x 0 then 12x 34x when x f3x 0 and f2x 0 f1 and f2 The functions f1 and f2 have domain NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M3 ALGEBRA II Problem Set 1 Sketch the graphs of the functions fx 5x and gx log5x Other Another page last 2 Sketch the graphs of the functions fx 12x and gx log12 x Another page last 3 Sketch the graphs of the functions f1x 12x and f2x 34x on the same sheet of graph paper and answer the following questions a Where do the two exponential graphs intersect When the graphs are together in 01 b For which values of x is 12x 34x If x 0 then 12x 34x c For which values of x is 12x 34x If x 0 then 12x 34x d What happens to the values of the functions f1 and f2 as x When x f3x 0 and f2x 0 e What are the domains of the two functions f1 and f2 The functions f1 and f2 have domain 4 Use the information from Problem 3 together with the relationship between graphs of exponential and logarithmic functions to sketch the graphs of the functions g1x log12x and g2x log34x on the same sheet of graph paper Then answer the following questions a Where do the two logarithmic graphs intersect 00 b For which values of x is log12x log34x When x 0 log12x log34x c For which values of x is log12x log34x When x 0 log12x log34x d What happens to the values of the functions g1 and g2 as x When x the functions g1 and g2 are e What are the domains of the two functions g1 and g2 Both functions have domain 0 5 For each function f find a formula for the function h in terms of x a If fx x3 find hx 128f14 x f2x 12814 x3 2x3 128164 x3 8x3 2x3 8x3 10x3 b If fx x2 1 find hx fx 2 f2 x 22 1 22 1 x2 4x 4 1 4 1 x2 4x 5 5 x2 4x c If fx x3 2x2 5x 1 find hx fx fx 2 d If fx x3 2x2 5x 1 find hx fx fx 2 Another page last Dont have space 6 In Problem 5 parts c and d list at least two aspects about the formulas you found as they relate to the function fx x3 2x2 5x 1 In the 5 part c it contains even power terms from fx In the 5 part d contains odd power terms from fx 5 a 128f14 x f2x 12814 x3 2x3 128164 x3 8x3 2x3 8x3 10x3 b fx 2 f2 x 22 1 22 1 x2 4x 4 1 4 1 x2 4x 5 5 x2 4x c fx fx2 x3 2x2 5x 9 x3 2x2 5x 92 x3 2x2 5x 9 x3 2x2 5x 92 4x2 22 2x2 9 d fx fx2 x3 2x2 5x 9 x3 2x2 5x 92 x3 2x2 5x 9 x3 2x2 5x 92 2x3 10x2 x3 5x For each of the functions f and g below write an expression for i fgx ii gfx and iii ffx in terms of x a fx x23 gx x12 b fx bxa gx bx a for two numbers a and b when x is not equal to 0 or a c fx x1x1 gx x1x1 when x is not equal to 1 or 1 d fx 2x gx log₂x e fx lnx gx ex f fx 2100x gx 12 log12 x 7 a i fgx fx12 x1223 x243 x8 i x8 ii gfx gx23 x2312 x243 x8 ii x8 iii ffx fx23 x2323 x49 iii x49 b i fgx fbx a bbx a a bbx bxb x ii gfx gbxa bxa a bxab a x iii ffx fbxa bbxa a bxab axa bxab ax a2 c i fgx fx1x1 x1x1 1 x1x1 1 x1 x1x1 x1 2x2 x ii gfx gx1x1 x1x1 1 x1x1 1 x1 x1x1 x1 2x2 x iii ffx fx1x1 x1x1 1 x1x1 1 x d fx 2x i fgx flog₂x 2log₂x x ii gfx g2x log₂2x xlog₂2 x iii ffx f2x 22x e i fgx fex lnex x ii gfx glnx elnx loge x ln x x iii ffx flnx lnlnx 8 i fgx 230012 log12 x 230012 log12 x 212 x x ii gfx 12 log 12 2300x 12 log 300x 12 x x iii ffx 23002300x 23002300x2 x 9 x 9 1 i fgx fx1x9 x 9 x 9 x 9 2x2 x ii gfx gx 1x 9 x 9 x 9 x 9 2x2 x iii ffx fx 9x 9 x 9x 9 x Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S148 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 Lesson 22 Choosing a Model Classwork Opening Exercise a You are working on a team analyzing the following data gathered by your colleagues 11 5 0 105 15 178 43 120 Your coworker Alexandra says that the model you should use to fit the data is 𝑘𝑡 100 sin15𝑡 105 Sketch Alexandras model on the axes at left on the next page b How does the graph of Alexandras model 𝑘𝑡 100 sin15𝑡 105 relate to the four points Is her model a good fit to this data c Another teammate Randall says that the model you should use to fit the data is 𝑔𝑡 16𝑡2 72𝑡 105 Sketch Randalls model on the axes at right on the next page Yes because it passes through in the four points 115 0 105 15 178 43 120 Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S149 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 d How does the graph of Randalls model 𝑔𝑡 16𝑡2 72𝑡 105 relate to the four points Is his model a good fit to the data Alexandras Model Randalls Model e Suppose the four points represent positions of a projectile fired into the air Which of the two models is more appropriate in that situation and why The more appropriate is the graph of Randall because the parabola passes through or is close in the points f In general how do we know which model to choose There is moment will be necessary the graph of Alexandra However in the other moment is Randall Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S150 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 Exercises 1 The table below contains the number of daylight hours in Oslo Norway on the specified dates Date Hours and Minutes Hours August 1 16 56 1693 September 1 14 15 1425 October 1 11 33 1155 November 1 8 50 883 a Plot the data on the grid provided and decide how to best represent it b Looking at the data what type of function appears to be the best fit Linear function Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S151 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 c Looking at the context in which the data was gathered what type of function should be used to model the data Periodic function d Do you have enough information to find a model that is appropriate for this data Either find a model or explain what other information you would need to do so It isnt possible because we dont know the maximum and minimum number of daylight hours in Oslo However it could be sinusoidal function 2 The goal of the US Centers for Disease Control and Prevention CDC is to protect public health and safety through the control and prevention of disease injury and disability Suppose that 45 people have been diagnosed with a new strain of the flu virus and that scientists estimate that each person with the virus will infect 5 people every day with the flu a What type of function should the scientists at the CDC use to model the initial spread of this strain of flu to try to prevent an epidemic Explain how you know b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so On the first day there are 45 people with the virus On the second day each person infects 5 people On the third day more 5 people will be infected The function that represents it is EXPONENTIAL This may be demonstrated like that 1ª day 45 people 45 x 50 2ª day 45 x 5 people 45 x 51 fn 45 x 5n1 3ª day 45 x 5 x 5 people 45 x 52 Yes this was explained in the a iten We know that each day 5 people more are infected in relation to last day and then we use fn 45 x 5n1 to calculate it because n represents the days reducing 1 We could to think like that f1 45 x 511 45 x 50 f2 45 x 521 45 x 51 f3 45 x 531 45 x 52 Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S152 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 3 An artist is designing posters for a new advertising campaign The first poster takes 10 hours to design but each subsequent poster takes roughly 15 minutes less time than the previous one as he gets more practice a What type of function models the amount of time needed to create 𝑛 posters for 𝑛 20 Explain how you know As the time gap between posters is narrowing in a linear fashion we ought to represent this situation with a linear function b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so Yes The number of hours needed to create posters can be made through a linear function fx10 n1 025 10 025n 025 025n 1225 So fx 025n 1225 4 A homeowner notices that her heating bill is the lowest in the month of August and increases until it reaches its highest amount in the month of February After February the amount of the heating bill slowly drops back to the level it was in August when it begins to increase again The amount of the bill in February is roughly four times the amount of the bill in August a What type of function models the amount of the heating bill in a particular month Explain how you know Sinusoidal function because exterior temperatures repeat periodically b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so We dont know the highest or lowest amount of the heating bill Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S153 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 5 An online merchant sells used books for 500 each and the sales tax rate is 6 of the cost of the books Shipping charges are a flat rate of 400 plus an additional 100 per book a What type of function models the total cost including the shipping costs of a purchase of 𝑥 books Explain how you know We use a linear function because the total cost increases when the number of books increases b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so This means when we buy an amount x of the books we may calculate using fx 63L 4 5 6100 X 5 5 30100 53L As the shipping cost 4 1L to find the function just sum 53L 4 1L 63L 4 6 A stunt woman falls from a tall building in an actionpacked movie scene Her speed increases by 32 fts for every second that she is falling a What type of function models her distance from the ground at time 𝑡 seconds Explain how you know As speed is increasing by every second her rate at which she gets closer to the ground is increasing linearly with quadratic function b Do you have enough information to find a model that is appropriate for this situation Either find a model or explain what other information you would need to do so We dont know the height of the building and how far she will fall So it cant be made Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S154 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 Lesson Summary If we expect from the context that each new term in the sequence of data is a constant added to the previous term then we try a linear model If we expect from the context that the second differences of the sequence are constant meaning that the rate of change between terms either grows or shrinks linearly then we try a quadratic model If we expect from the context that each new term in the sequence of data is a constant multiple of the previous term then we try an exponential model If we expect from the context that the sequence of terms is periodic then we try a sinusoidal model Model Equation of Function Rate of Change Linear 𝑓𝑡 𝑎𝑡 𝑏 for 𝑎 0 Constant Quadratic 𝑔𝑡 𝑎𝑡2 𝑏𝑡 𝑐 for 𝑎 0 Changing linearly Exponential ℎ𝑡 𝑎𝑏𝑐𝑡 for 0 𝑏 1 or 𝑏 1 A multiple of the current value Sinusoidal 𝑘𝑡 𝐴 sin𝑤𝑡 ℎ 𝑘 for 𝐴 𝑤 0 Periodic Problem Set 1 A new car depreciates at a rate of about 20 per year meaning that its resale value decreases by roughly 20 each year After hearing this Brett said that if you buy a new car this year then after 5 years the car has a resale value of 000 Is his reasoning correct Explain how you know Brett is wrong If the car loses 20 of its value each year it keeps 80 of its resale value each year So the right form to use is an exponential function V t P08t where P is the original value of the car when it was new t represents number of years the car has been owned and V t represents the value car in year t When t 5 the value of the car is P085 33P So after 5 years the car will cost 33 of your original price 2 Alexei just moved to Seattle and he keeps track of the average rainfall for a few months to see if the city deserves its reputation as the rainiest city in the United States Month Average rainfall July 093 in September 161 in October 324 in December 606 in What type of function should Alexei use to model the average rainfall in month 𝑡 We can use a sinusoidal function Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S155 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 3 Sunny who wears her hair long and straight cuts her hair once per year on January 1 always to the same length Her hair grows at a constant rate of 2 cm per month Is it appropriate to model the length of her hair with a sinusoidal function Explain how you know No The sinusoidal function will show that her hair gets longer and then slowly shrinks until original length Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S156 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 4 On average it takes 2 minutes for a customer to order and pay for a cup of coffee a What type of function models the amount of time you wait in line as a function of how many people are in front of you Explain how you know As the wait time increases by a 2 minutes for each person in line So to prove it we can use a linear function b Find a model that is appropriate for this situation If there is no one anymore before you your wait time is zero So the wait time T in minutes may be represented in Tx2x where x is the number of people ahead of you in line 5 An online ticketselling service charges 5000 for each ticket to an upcoming concert In addition the buyer must pay 8 sales tax and a convenience fee of 600 for the purchase a What type of function models the total cost of the purchase of 𝑛 tickets in a single transaction Its linear function because the total price is 10850 5400 b Find a model that is appropriate for this situation P Ticket price 50 Sales 8 the value of the ticket Convenience fee 6 Fx n P 8P Cn Fx n 50 008 x 50 6 54n 6 6 In a video game the player must earn enough points to pass one level and progress to the next as shown in the table below To pass this level You need this many total points 1 5000 2 15000 3 35000 4 65000 That is the increase in the required number of points increases by 10000 points at each level a What type of function models the total number of points you need to pass to level 𝑛 Explain how you know Its a quadratic function because the number of points increases by 10000 points at each level Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S157 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 b Find a model that is appropriate for this situation 15000 5000 10000 35000 15000 20000 65000 35000 30000 a12 b1 c 5000 a22 b2 c 15000 a32 b3 c 35000 a b c 5000 4a 2b c 15000 9a 3b c 35000 We already know the values in the three first lines and in the last lines then just change the numbers to letters Observe it below 4a 2b c a b c 10000 4a 2b c a b c 10000 3a b 10000 9a 3b c 4a 2b c 10000 9a 3b c 4a 2b c 10000 5a b 20000 We have a systems of equations that it will can solved with addiction or substitution We may use the second option b 10000 3a 5a 10000 3a 20000 5a 3a 20000 10000 2a 10000 a 100002 5000 b 10000 3 x 5000 b 10000 15000 5000 a b c 5000 5000 5000 c 5000 c 5000 fg 5000g2 5000g 5000 7 The southern white rhinoceros reproduces roughly once every three years giving birth to one calf each time Suppose that a nature preserve houses 100 white rhinoceroses 50 of which are female Assume that half of the calves born are female and that females can reproduce as soon as they are 1 year old a What type of function should be used to model the population of female white rhinoceroses in the preserve Because all female rhinoceroses reproduces each 3 years and half of those calves are assumed to be female the Lesson 22 M3 NYS COMMON CORE MATHEMATICS CURRICULUM ALGEBRA II Lesson 22 Choosing a Model S158 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM3TE130082015 population of female rhinoceroses increases by 16 every year So we should use an exponential function to calculate the population of female white rhinoceroses b Assuming that there is no death in the rhinoceros population find a function to model the population of female white rhinoceroses in the preserve The function is ft 50117t since 1 16 117 and the initial population is 50 female southern white rhinoceroses c Realistically not all of the rhinoceroses survive each year so we assume a 5 death rate of all rhinoceroses Now what type of function should be used to model the population of female white rhinoceroses in the preserve We should still use an exponential function d Find a function to model the population of female white rhinoceroses in the preserve taking into account the births of new calves and the 5 death rate Since 5 of the rhinoceroses die each year that means that 95 of them survive The new growth rate is 095117 111 and as function ft 50111t