Algebra 2

NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S88 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 14 Linear and Exponential ModelsComparing Growth Rates Classwork Example 1 Linear Functions a Sketch points 𝑃1 04 and 𝑃2 412 Are there values of 𝑚 and 𝑏 such that the graph of the linear function described by 𝑓𝑥 𝑚𝑥 𝑏 contains 𝑃1 and 𝑃2 If so find those values If not explain why they do not exist b Sketch 𝑃1 04 and 𝑃2 0 2 Are there values of 𝑚 and 𝑏 so that the graph of a linear function described by 𝑓𝑥 𝑚𝑥 𝑏 contains 𝑃1 and 𝑃2 If so find those values If not explain why they do not exist NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S89 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exponential Functions Graphs c and d are both graphs of an exponential function of the form 𝑔𝑥 𝑎𝑏𝑥 Rewrite the function 𝑔𝑥 using the values for 𝑎 and 𝑏 that are required for the graph shown to be a graph of 𝑔 c 𝑔𝑥 d 𝑔𝑥 Example 2 A lab researcher records the growth of the population of a yeast colony and finds that the population doubles every hour a Complete the researchers table of data Hours into study 0 1 2 3 4 Yeast colony population thousands 5 02 205 2 27 4 12 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S90 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License b What is the exponential function that models the growth of the colonys population c Several hours into the study the researcher looks at the data and wishes there were more frequent measurements Knowing that the colony doubles every hour how can the researcher determine the population in halfhour increments Explain d Complete the new table that includes halfhour increments e How would the calculation for the data change for time increments of 20 minutes Explain f Complete the new table that includes 20minute increments Hours into study 0 1 3 2 3 1 4 3 5 3 2 Yeast colony population thousands 5 Hours into study 0 1 2 1 3 2 2 5 2 3 Yeast colony population thousands 5 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S91 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License g The researchers lab assistant studies the data recorded and makes the following claim Since the population doubles in 1 hour then half of that growth happens in the first half hour and the other half of that growth happens in the second half hour We should be able to find the population at 𝑡 1 2 by taking the average of the populations at 𝑡 0 and 𝑡 1 Is the assistants reasoning correct Compare this strategy to your work in parts c and e Example 3 A California Population Projection Engineer in 1920 was tasked with finding a model that predicts the states population growth He modeled the population growth as a function of time 𝑡 years since 1900 Census data shows that the population in 1900 in thousands was 1490 In 1920 the population of the state of California was 3554 thousand He decided to explore both a linear and an exponential model a Use the data provided to determine the equation of the linear function that models the population growth from 19001920 b Use the data provided and your calculator to determine the equation of the exponential function that models the population growth NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S92 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5000 10000 15000 20000 25000 30000 35000 40000 1880 1900 1920 1940 1960 1980 2000 2020 Population in Thousands Year California Population Growth 19002010 c Use the two functions to predict the population for the following years Projected Population Based on Linear Function 𝒇𝒕 thousands Projected Population Based on Exponential Function 𝒈𝒕 thousands Census Population Data and Intercensal Estimates for California thousands 1935 6175 1960 15717 2010 37253 Courtesy US Census Bureau d Which function is a better model for the population growth of California in 1935 and in 1960 e Does either model closely predict the population for 2010 What phenomenon explains the real population value NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S93 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 When a ball bounces up and down the maximum height it reaches decreases with each bounce in a predictable way Suppose for a particular type of squash ball dropped on a squash court the maximum height ℎ𝑥 after 𝑥 number of bounces can be represented by ℎ𝑥 65 1 3 𝑥 a How many times higher is the height after the first bounce compared to the height after the third bounce b Graph the points 𝑥 ℎ𝑥 for 𝑥values of 0 1 2 3 4 and 5 2 Australia experienced a major pest problem in the early 20th century The pest Rabbits In 1859 24 rabbits were released by Thomas Austin at Barwon Park In 1926 there were an estimated 10 billion rabbits in Australia Needless to say the Australian government spent a tremendous amount of time and money to get the rabbit problem under control To find more on this topic visit Australias Department of Environment and Primary Industries website under Agriculture a Based only on the information above write an exponential function that would model Australias rabbit population growth b The model you created from the data in the problem is obviously a huge simplification from the actual function of the number of rabbits in any given year from 1859 to 1926 Name at least one complicating factor about rabbits that might make the graph of your function look quite different than the graph of the actual function Lesson Summary Given a linear function of the form 𝐿𝑥 𝑚𝑥 𝑘 and an exponential function of the form 𝐸𝑥 𝑎𝑏𝑥 for 𝑥 a real number and constants 𝑚 𝑘 𝑎 and 𝑏 consider the sequence given by 𝐿𝑛 and the sequence given by 𝐸𝑛 where 𝑛 1234 Both of these sequences can be written recursively 𝐿𝑛 1 𝐿𝑛 𝑚 and 𝐿0 𝑘 and 𝐸𝑛 1 𝐸𝑛 𝑏 and 𝐸0 𝑎 The first sequence shows that a linear function grows additively by the same summand 𝑚 over equal length intervals ie the intervals between consecutive integers The second sequence shows that an exponential function grows multiplicatively by the same factor 𝑏 over equallength intervals ie the intervals between consecutive integers An increasing exponential function eventually exceeds any linear function That is if 𝑓𝑥 𝑎𝑏𝑥 is an exponential function with 𝑎 0 and 𝑏 1 and 𝑔𝑥 𝑚𝑥 𝑘 is a linear function then there is a real number 𝑀 such that for all 𝑥 𝑀 then 𝑓𝑥 𝑔𝑥 Sometimes this is not apparent in a graph displayed on a graphing calculator that is because the graphing window does not show enough of the graphs for us to see the sharp rise of the exponential function in contrast with the linear function NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 14 ALGEBRA I Lesson 14 Linear and Exponential ModelsComparing Growth Rates S94 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 After graduating from college Jane has two job offers to consider Job A is compensated at 100000 a year but with no hope of ever having an increase in pay Jane knows a few of her peers are getting that kind of an offer right out of college Job B is for a social media startup which guarantees a mere 10000 a year The founder is sure the concept of the company will be the next big thing in social networking and promises a pay increase of 25 at the beginning of each new year a Which job will have a greater annual salary at the beginning of the fifth year By approximately how much b Which job will have a greater annual salary at the beginning of the tenth year By approximately how much c Which job will have a greater annual salary at the beginning of the twentieth year By approximately how much d If you were in Janes shoes which job would you take 4 The population of a town in 2007 was 15000 people The town has gotten its fresh water supply from a nearby lake and river system with the capacity to provide water for up to 30000 people Due to its proximity to a big city and a freeway the towns population has begun to grow more quickly than in the past The table below shows the population counts for each year from 20072012 a Write a function of 𝑥 that closely matches these data points for 𝑥values of 0 1 2 3 4 and 5 b Assume the function is a good model for the population growth from 20122032 At what year during the time frame 20122032 will the water supply be inadequate for the population Year Years Past 2007 Population of the town 2007 0 15000 2008 1 15600 2009 2 16224 2010 3 16873 2011 4 17548 2012 5 18250 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 9 ALGEBRA I Lesson 9 Representing Naming and Evaluating Functions S49 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 9 Representing Naming and Evaluating Functions Classwork Opening Exercise Match each picture to the correct word by drawing an arrow from the word to the picture FUNCTION A function is a correspondence between two sets 𝑋 and 𝑌 in which each element of 𝑋 is matched to one and only one element of 𝑌 The set 𝑋 is called the domain of the function The notation 𝑓 𝑋 𝑌 is used to name the function and describes both 𝑋 and 𝑌 If 𝑥 is an element in the domain 𝑋 of a function 𝑓 𝑋 𝑌 then 𝑥 is matched to an element of 𝑌 called 𝑓𝑥 We say 𝑓𝑥 is the value in 𝑌 that denotes the output or image of 𝑓 corresponding to the input 𝑥 The range or image of a function 𝑓 𝑋 𝑌 is the subset of 𝑌 denoted 𝑓𝑋 defined by the following property 𝑦 is an element of 𝑓𝑋 if and only if there is an 𝑥 in 𝑋 such that 𝑓𝑥 𝑦 Example 1 Define the Opening Exercise using function notation State the domain and the range Elephant Camel Polar Bear Zebra NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 9 ALGEBRA I Lesson 9 Representing Naming and Evaluating Functions S50 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 2 Is the assignment of students to English teachers an example of a function If yes define it using function notation and state the domain and the range Example 3 Let 𝑋 1 2 3 4 and 𝑌 5 6 7 8 9 𝑓 and 𝑔 are defined below 𝑓 𝑋 𝑌 𝑔 𝑋 𝑌 𝑓 17 25 36 47 𝑔 1 5 2 6 1 8 29 37 Is 𝑓 a function If yes what is the domain and what is the range If no explain why 𝑓 is not a function Is 𝑔 a function If yes what is the domain and range If no explain why 𝑔 is not a function What is 𝑓2 If 𝑓𝑥 7 then what might 𝑥 be NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 9 ALGEBRA I Lesson 9 Representing Naming and Evaluating Functions S51 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 1 Define 𝑓 to assign each student at your school a unique ID number 𝑓 students in your school whole numbers Assign each student a unique ID number a Is this an example of a function Use the definition to explain why or why not b Suppose 𝑓Hilda 350123 What does that mean c Write your name and student ID number using function notation 2 Let 𝑔 assign each student at your school to a grade level a Is this an example of a function Explain your reasoning b Express this relationship using function notation and state the domain and the range 𝑔 students in the school grade level Assign each student to a grade level NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 9 ALGEBRA I Lesson 9 Representing Naming and Evaluating Functions S52 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Let ℎ be the function that assigns each student ID number to a grade level ℎ student ID number grade level Assign each student ID number to the students current grade level a Describe the domain and range of this function b Record several ordered pairs 𝑥 𝑓𝑥 that represent yourself and students in your group or class c Jonny says This is not a function because every ninth grader is assigned the same range value of 9 The range only has 4 numbers 9 10 11 12 but the domain has a number for every student in our school Explain to Jonny why he is incorrect NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 9 ALGEBRA I Lesson 9 Representing Naming and Evaluating Functions S53 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Which of the following are examples of a function Justify your answers a The assignment of the members of a football team to jersey numbers b The assignment of US citizens to Social Security numbers c The assignment of students to locker numbers d The assignment of the residents of a house to the street addresses e The assignment of zip codes to residences f The assignment of residences to zip codes g The assignment of teachers to students enrolled in each of their classes h The assignment of all real numbers to the next integer equal to or greater than the number i The assignment of each rational number to the product of its numerator and denominator 2 Sequences are functions The domain is the set of all term numbers which is usually the positive integers and the range is the set of terms of the sequence For example the sequence 1 4 9 16 25 36 of perfect squares is the function 𝐿𝑒𝑡 𝑓 positive integers perfect squares Assign each term number to the square of that number a What is 𝑓3 What does it mean b What is the solution to the equation 𝑓𝑥 49 What is the meaning of this solution c According to this definition is 3 in the domain of 𝑓 Explain why or why not d According to this definition is 50 in the range of 𝑓 Explain why or why not 3 Write each sequence as a function a 1 3 6 10 15 21 28 b 1 3 5 7 9 c 𝑎𝑛1 3𝑎𝑛 𝑎1 1 where 𝑛 is a positive integer greater than or equal to 1 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S95 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 15 Piecewise Functions Classwork Opening Exercise For each real number 𝑎 the absolute value of 𝑎 is the distance between 0 and 𝑎 on the number line and is denoted 𝑎 1 Solve each one variable equation a 𝑥 6 b 𝑥 5 4 c 2𝑥 3 10 2 Determine at least five solutions for each twovariable equation Make sure some of the solutions include negative values for either 𝑥 or 𝑦 a 𝑦 𝑥 b 𝑦 𝑥 5 c 𝑥 𝑦 Exploratory Challenge 1 For parts ac create graphs of the solution set of each twovariable equation from Opening Exercise 2 a 𝑦 𝑥 b 𝑦 𝑥 5 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S96 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c 𝑥 𝑦 d Write a brief summary comparing and contrasting the three solution sets and their graphs For parts ej consider the function 𝑓𝑥 𝑥 where 𝑥 can be any real number e Explain the meaning of the function 𝑓 in your own words f State the domain and range of this function g Create a graph of the function 𝑓 You might start by listing several ordered pairs that represent the corresponding domain and range elements NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S97 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License h How does the graph of the absolute value function compare to the graph of 𝑦 𝑥 i Define a function whose graph would be identical to the graph of 𝑦 𝑥 5 j Could you define a function whose graph would be identical to the graph of 𝑥 𝑦 Explain your reasoning k Let 𝑓1𝑥 𝑥 for 𝑥 0 and let 𝑓2𝑥 𝑥 for 0 Graph the functions 𝑓1 and 𝑓2 on the same Cartesian plane How does the graph of these two functions compare to the graph in part g Definition The absolute value function 𝑓 is defined by setting 𝑓𝑥 𝑥 for all real numbers Another way to write 𝑓 is as a piecewise linear function 𝑓𝑥 𝑥 𝑥 0 𝑥 𝑥 0 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S98 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 1 Let 𝑔𝑥 𝑥 5 The graph of 𝑔 is the same as the graph of the equation 𝑦 𝑥 5 you drew in Exploratory Challenge 1 part b Use the redrawn graph below to rewrite the function 𝑔 as a piecewise function Label the graph of the linear function with negative slope by 𝑔1 and the graph of the linear function with positive slope by 𝑔2 as in the picture above Function 𝑔1 The slope of 𝑔1 is 1 why and the 𝑦intercept is 5 therefore 𝑔1𝑥 𝑥 5 Function 𝑔2 The slope of 𝑔2 is 1 why and the 𝑦intercept is 5 why therefore 𝑔2𝑥 𝑥 5 Writing 𝑔 as a piecewise function is just a matter of collecting all of the different pieces and the intervals upon which they are defined 𝑔𝑥 𝑥 5 𝑥 5 𝑥 5 𝑥 5 Exploratory Challenge 2 The floor of a real number 𝑥 denoted by 𝑥 is the largest integer not greater than 𝑥 The ceiling of a real number 𝑥 denoted by 𝑥 is the smallest integer not less than 𝑥 The sawtooth number of a positive number is the fractional part of the number that is to the right of its floor on the number line In general for a real number 𝑥 the sawtooth number of 𝑥 is the value of the expression 𝑥 𝑥 Each of these expressions can be thought of as functions with the domain being the set of real numbers NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S99 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License a Complete the following table to help you understand how these functions assign elements of the domain to elements of the range The first and second rows have been done for you 𝒙 𝒇𝒍𝒐𝒐𝒓𝒙 𝒙 𝒄𝒆𝒊𝒍𝒊𝒏𝒈𝒙 𝒙 𝒔𝒂𝒘𝒕𝒐𝒐𝒕𝒉𝒙 𝒙 𝒙 48 4 5 08 13 2 1 07 22 6 3 2 3 𝜋 b Create a graph of each function 𝒇𝒍𝒐𝒐𝒓𝒙 𝒙 𝒄𝒆𝒊𝒍𝒊𝒏𝒈𝒙 𝒙 𝒔𝒂𝒘𝒕𝒐𝒐𝒕𝒉𝒙 𝒙 𝒙 c For the floor ceiling and sawtooth functions what would be the range values for all real numbers 𝑥 on the interval 01 The interval 12 The interval 2 1 The interval 15 25 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S100 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Relevant Vocabulary PIECEWISE LINEAR FUNCTION Given a number of nonoverlapping intervals on the real number line a real piecewise linear function is a function from the union of the intervals to the set of real numbers such that the function is defined by possibly different linear functions on each interval ABSOLUTE VALUE FUNCTION The absolute value of a number 𝑥 denoted by 𝑥 is the distance between 0 and 𝑥 on the number line The absolute value function is the piecewise linear function such that for each real number 𝑥 the value of the function is 𝑥 We often name the absolute value function by saying Let 𝑓𝑥 𝑥 for all real numbers 𝑥 FLOOR FUNCTION The floor of a real number 𝑥 denoted by 𝑥 is the largest integer not greater than 𝑥 The floor function is the piecewise linear function such that for each real number 𝑥 the value of the function is 𝑥 We often name the floor function by saying Let 𝑓𝑥 𝑥 for all real numbers 𝑥 CEILING FUNCTION The ceiling of a real number 𝑥 denoted by 𝑥 is the smallest integer not less than 𝑥 The ceiling function is the piecewise linear function such that for each real number 𝑥 the value of the function is 𝑥 We often name the ceiling function by saying Let 𝑓𝑥 𝑥 for all real numbers 𝑥 SAWTOOTH FUNCTION The sawtooth function is the piecewise linear function such that for each real number 𝑥 the value of the function is given by the expression 𝑥 𝑥 The sawtooth function assigns to each positive number the part of the number the noninteger part that is to the right of the floor of the number on the number line That is if we let 𝑓𝑥 𝑥 𝑥 for all real numbers 𝑥 then 𝑓 1 3 1 3 𝑓 1 1 3 1 3 𝑓100002 002 𝑓03 07 etc NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S101 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Explain why the sawtooth function 𝑠𝑎𝑤𝑡𝑜𝑜𝑡ℎ𝑥 𝑥 𝑥 for all real numbers 𝑥 takes only the fractional part of a number when the number is positive 2 Let 𝑔𝑥 𝑥 𝑥 where 𝑥 can be any real number In otherwords 𝑔 is the difference between the ceiling and floor functions Express 𝑔 as a piecewise function 3 The Heaviside function is defined using the formula below 𝐻𝑥 1 𝑥 0 0 𝑥 0 1 𝑥 0 Graph this function and state its domain and range 4 The following piecewise function is an example of a step function 𝑆𝑥 3 5 𝑥 2 1 2 𝑥 3 2 3 𝑥 5 a Graph this function and state the domain and range b Why is this type of function called a step function 5 Let 𝑓𝑥 𝑥 𝑥 where 𝑥 can be any real number except 0 a Why is the number 0 excluded from the domain of 𝑓 b What is the range of f c Create a graph of 𝑓 d Express 𝑓 as a piecewise function e What is the difference between this function and the Heaviside function 6 Graph the following piecewise functions for the specified domain a 𝑓𝑥 𝑥 3 for 5 𝑥 3 b 𝑓𝑥 2𝑥 for 3 𝑥 3 c 𝑓𝑥 2𝑥 5 for 0 𝑥 5 d 𝑓𝑥 3𝑥 1 for 2 𝑥 2 e 𝑓𝑥 𝑥 𝑥 for 5 𝑥 3 f 𝑓𝑥 𝑥 if 𝑥 0 𝑥 1 if 𝑥 0 g 𝑓𝑥 2𝑥 3 if 𝑥 1 3 𝑥 if 𝑥 1 NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 15 ALGEBRA I Lesson 15 Piecewise Functions S102 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM3TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Write a piecewise function for each graph below a b Graph of 𝒃 c d 1 1 Mathematical Functions Exponential and Linear 1 Exponential Functions Graphs c and d are both graphs of an exponential function of the form gx abx We need to rewrite the function gx using the values for a and b that are required for the graph shown to be a graph of g 11 Part c From the given graph we have the points P1 2 05 P2 0 2 At x 0 g0 ab0 a 2 a 2 At x 2 using g2 05 g2 2b2 05 2 b2 05 b2 4 b 2 Thus the function for this graph is gx 2 2x 12 Part d We are given two points P1 1 2 P2 2 27 4 At x 1 g1 ab1 2 a b 2 At x 2 g2 ab2 27 4 Solving the system of equations a 2b ab2 27 4 1 Substituting a2b into the second equation 2bb2 274 2b3 274 b 32 Thus a3 and the function for this graph is gx 332x 2 Linear Functions We now move on to analyzing linear functions 21 Part a We are given two points P1 04 and P2 412 To find the slope m of the line passing through these points m 12440 2 Now using the point P1 04 we can find b fx 2x b 420b b4 Thus the equation of the line is fx 2x 4 22 Part b We are given two points P1 04 and P2 02 both with the same xvalue but different yvalues Since a linear function can only have one yintercept no such linear function exists that passes through both points Part a Completing the Table The population doubles every hour starting at 5 thousand at hour 0 The completed table is as follows Hours into Study 0 1 2 3 4 Population thousands 5 10 20 40 80 Part b Exponential Function The population grows exponentially doubling every hour The exponential function that models this growth is Pt 5 2t Where Pt is the population in thousands after t hours 5 is the initial population in thousands 2t represents the doubling every hour Part c Population in HalfHour Increments To find the population at halfhour increments we use the same exponential growth function Pt 5 2t but substitute t with the respective halfhour time values At t 12 P12 5 212 5 2 707 thousand At t 32 P32 5 232 5 2 2 1414 thousand Part d Completing the Table for HalfHour Increments The completed table for halfhour increments is Hours into Study 0 12 1 32 2 52 3 Population thousands 5 707 10 1414 20 2828 Part e Time Increments of 20 Minutes For increments of 20 minutes which is 13 of an hour we use the same exponential growth formula The exponent t is now a fraction representing time in hours For example at t 13 the population is P13 5 213 634 thousand Part f Completing the Table for 20Minute Increments The completed table for 20minute increments is Hours into Study 0 13 23 1 43 53 2 Population thousands 5 634 803 10 1266 1595 20 Part g Analyzing the Assistants Claim The assistant claims that since the population doubles in 1 hour half of that growth happens in the first halfhour and the other half happens in the second halfhour Therefore the population at t 12 can be estimated by averaging the populations at t 0 and t 1 This suggests the following calculation P12 P0 P12 5 102 75 thousand However as calculated in part c the actual population at t 12 is approximately 707 thousand Why the Assistants Reasoning is Incorrect The assistants reasoning is incorrect because it assumes linear growth where half of the total growth occurs in the first half of the time interval However population growth in this case is exponential meaning the rate of growth increases over time The correct approach as shown in part c involves using the exponential growth formula which accounts for the fact that more growth happens towards the end of the interval In conclusion the population at t 12 is not the average of the populations at t 0 and t 1 rather it follows the exponential model Pt 5 2t article amsmath graphicx Population Growth and Exponential Problems California Population Problem In 1920 an engineer was tasked with modeling Californias population growth based on census data from 1900 and 1920 The available data is Population in 1900 1490 thousand people Population in 1920 3554 thousand people The engineer decided to test two models a linear and an exponential model a Linear Function A linear function has the general form ft mt b where t is the number of years after 1900 We know that f0 1490 and f20 3554 We can set up the system of equations 1490 m 0 b b 1490 3554 m 20 1490 Solving for m 3554 1490 20m m 2064 20 1032 Therefore the linear function modeling the population growth is ft 1032t 1490 b Exponential Function An exponential function has the general form gt P0 ekt where P0 1490 and we know that g20 3554 Thus we have 3554 1490 e20k Dividing both sides by 1490 3554 1490 e20k 2385 e20k Now applying the natural logarithm ln2385 20k k ln2385 20 00439 Therefore the exponential function is gt 1490 e00439t 5 c Population Projections We will use the two functions to predict the population for the years 1935 1960 and 2010 For the linear function ft 1032t 1490 For 1935 t 35 f35 1032 35 1490 5102 For 1960 t 60 f60 1032 60 1490 7682 For 2010 t 110 f110 1032 110 1490 12802 For the exponential function gt 1490 e00439t For 1935 t 35 g35 1490 e0043935 1490 4682 6976 For 1960 t 60 g60 1490 e0043960 1490 8538 12724 For 2010 t 110 g110 1490 e00439110 1490 3267 48677 The complete projection table is Year Linear Projection ft thousands Exponential Projection gt thousands Actual Population thousands 1935 5102 6976 6175 1960 7682 12724 15717 2010 12802 48677 37253 d Better Model for 1935 and 1960 The exponential function is a better model for 1935 as the prediction of 6976 is closer to the actual population of 6175 compared to the linear prediction of 5102 However for 1960 both models deviate with the exponential model over estimating and the linear model underestimating but the exponential prediction is closer 6 e Prediction for 2010 The exponential model overestimates the population in 2010 while the linear model underestimates it Neither model fits the actual value well A likely explanation for the real population is that the growth rate did not continue exponentially due to factors like migration public policies and resource limita tions which constrained the population growth 7 Squash Ball Bouncing Problem The function for the height after x bounces is given by hx 65 13x a Height Comparison between First and Third Bounces To find how many times higher the height after the first bounce is compared to the height after the third bounce we calculate h1 65 13 2167 h3 65 133 241 Ratio h1h3 2167241 9 Thus the height after the first bounce is approximately 9 times greater than the height after the third bounce b Graphing the Points We calculate the heights for the values x 0 1 2 3 4 5 h0 65 h1 2167 h2 722 h3 241 h4 080 h5 027 The points are 0 65 1 2167 2 722 3 241 4 080 5 027 Australian Rabbit Problem a Exponential Function for Rabbit Growth We know that in 1859 there were 24 rabbits and by 1926 there were approxi mately 10 billion rabbits We assume the exponential model Pt P0ekt where P0 24 P67 10 109 and t 67 Thus we have 10 109 24 e67k Dividing both sides by 24 41666666667 e67k Taking the natural logarithm ln41666666667 67k k ln41666666667 67 0311 Therefore the exponential function modeling the rabbit population growth is Pt 24e0311t b Complicating Factors A major complicating factor would be resource limitations food space etc As the population grows available resources become scarce which slows down population growth Additionally human interventions such as the introduction of predators or population control measures could drastically affect the growth rate article amsmath Solutions Problems 3 and 4 Nayder Almeida Problem 3 Job Offer Comparison a Which job will have a greater annual salary at the beginning of the fifth year By approximately how much Job A Offers 100000 per year with no increase SA5 100 000 Job B Starts at 10000 with a 25 yearly increase The formula for the salary is SBn 10 000 125n1 9 In the fifth year SB5 10 000 1254 24 41406 Conclusion Job A has a higher salary in the 5th year with a difference of approximately 100 000 24 41406 75 58594 b Which job will have a greater annual salary at the beginning of the tenth year By approximately how much Job A remains at 100000 For Job B SB10 10 000 1259 74 70536 Conclusion Job A still has a higher salary with a difference of 100 000 74 70536 25 29464 c Which job will have a greater annual salary at the beginning of the twentieth year By approximately how much Job A remains at 100000 For Job B SB20 10 000 12519 457 62489 Conclusion Job B has a higher salary in the 20th year with a difference of 457 62489 100 000 357 62489 d Which job would you take This depends on personal preference Job A offers immediate stability while Job B could be more rewarding in the long term Problem 4 Population Growth a Write a function for x that matches the data points The data suggests exponential population growth The general formula for this growth is Px P0 1 rx Where P0 15 000 population in 2007 r 004 growth rate of 4 per year Thus the population function is Px 15 000 104x b In what year during 20122032 will the water supply be inade quate 10 The water supply can support up to 30000 people To find when this will occur we solve 30 000 15 000 104x Dividing both sides by 15000 2 104x Taking the natural logarithm of both sides ln2 x ln104 x ln2 ln104 06931 00392 1768 Therefore the population will exceed 30000 around the year 2007 18 2025 The water supply will become inadequate around the year 2025 11 Exercise Solutions Your Name September 30 2024 Exercise 1 Define f to assign each student at your school a unique ID number a Is this an example of a function Use the definition to explain why or why not Solution Yes this is an example of a function According to the defi nition of a function each element in the domain students in the school is associated with exactly one element in the range unique ID number No two students share the same ID and each student has only one ID making this a function b Suppose fHilda 350123 What does that mean Solution This means that the function f assigns the unique ID number 350123 to the student named Hilda In other words Hildas ID number is 350123 c Write your name and student ID number using function notation Solution Lets assume your name is John Doe and your student ID number is 123456 Then the function notation would be fJohn Doe 123456 1 Exercise 2 Let g assign each student at your school to a grade level a Is this an example of a function Explain your reasoning Solution Yes this is an example of a function Each student in the school is assigned to exactly one grade level This means that every element in the domain students corresponds to exactly one element in the range grade levels b Express this relationship using function notation and state the domain and the range Solution g students in the school grade level Domain All students in the school Range All possible grade levels eg 9th grade 10th grade etc Exercise 3 Let h be the function that assigns each student ID number to a grade level a Describe the domain and range of this function Solution Domain All student ID numbers in the school Range All possible grade levels eg 9th grade 10th grade etc b Record several ordered pairs x hx that represent yourself and students in your group or class Solution Suppose the ID numbers and grade levels are as follows 123456 A 123457 B 123458 C 123459 D 2 c Jonny says This is not a function because every ninth grader is assigned the same range value of 9 The range only has 4 numbers 9 10 11 12 but the domain has a number for every student in our school Explain to Jonny why he is incorrect Solution Jonny is incorrect because a function can have the same range value for multiple different domain values Each unique student ID is mapped to a single grade level even though many students can share the same grade level 3 Problem Set 1 Which of the following are examples of a function Justify your answers a The assignment of the members of a football team to jersey numbers Solution Yes this is a function because each player domain is assigned to exactly one jersey number range b The assignment of US citizens to Social Security numbers Solution Yes this is a function because each citizen domain is assigned to exactly one Social Security number range c The assignment of students to locker numbers Solution Yes this is a function because each student domain is as signed to exactly one locker number range d The assignment of the residents of a house to the street addresses Solution No this is not a function because multiple residents domain can share the same street address range e The assignment of zip codes to residences Solution No this is not a function because multiple residences domain can share the same zip code range f The assignment of residences to zip codes Solution Yes this is a function because each residence domain is assigned to exactly one zip code range 4 g The assignment of teachers to students enrolled in each of their classes Solution No this is not a function because one teacher can have many students and a student can be in more than one class h The assignment of all real numbers to the next integer equal to or greater than the number Solution Yes this is a function because each real number domain is assigned to exactly one integer range i The assignment of each rational number to the product of its numerator and denominator Solution Yes this is a function because each rational number domain has a unique product range 5 Problem Set 2 Sequences are functions The domain is the set of all term numbers usually positive integers and the range is the set of terms of the sequence a What is f3 What does it mean Solution If the sequence is 1 4 9 16 25 36 perfect squares then f3 9 This means that the 3rd term of the sequence is 9 b What is the solution to the equation fx 49 What is the meaning of this solution Solution The solution is x 7 because 7 squared is 49 This means that the 7th term of the sequence is 49 c According to this definition is 3 in the domain of f Explain why or why not Solution No 3 is not in the domain of f because the domain consists of positive integers only d According to this definition is 50 in the range of f Explain why or why not Solution No 50 is not in the range of f because 50 is not a perfect square Problem Set 3 Write each sequence as a function 6 a 1 3 6 10 15 21 28 Solution The sequence can be expressed as fn nn 1 2 This represents the sum of the first n positive integers b 1 3 5 7 9 Solution The sequence can be expressed as fn 2n 1 This represents the sequence of odd numbers c an1 3an a1 1 where n is a positive integer greater than or equal to 1 Solution The sequence can be expressed as fn 3n1 This represents a geometric sequence with a common ratio of 3 and initial term of 1 7 Exercise Solutions Your Name September 30 2024 1 1 a x 6 Solution x 6 or x 6 b x 5 4 Solution 1 x 5 4 x 9 2 x 5 4 x 1 c 2x 3 10 Solution No solution exists as the absolute value cannot yield a negative result No solution 2 2 a y x Solutions x 3 y 3 x 2 y 2 x 0 y 0 x 2 y 2 x 5 y 5 1 b y x 5 Solutions x 2 y 3 x 5 y 0 x 7 y 2 x 1 y 6 x 10 y 5 c x y Solutions y 4 x 4 y 3 x 3 y 0 x 0 y 2 x 2 y 6 x 6 3 Exploratory Challenge 1 Figure 1 Letter A Green Letter B Blue Letter C Purple 2 d Comparison of Solution Sets The graphs of y x y x5 and x y have similar shapes as they all represent absolute value relationships However y x 5 shifts the graph horizontally to the right The graph of x y is the reflection of y x across the line y x e Explanation of the Function fx The function fx x takes any real number x and returns its absolute value converting negative numbers into positive ones while leaving non negative numbers unchanged f Domain and Range Domain All real numbers Range Only non negative numbers 0 g Graph of the Function fx x x y The graph of fx x is in the shape of a V with the vertex at the point 0 0 3 h Comparison of the Absolute Value Function with the Graph of y x The graph of the absolute value function fx x is identical to the graph of y x Both show that the value of y is the absolute value of x The graph is symmetric about the yaxis forming a V with the vertex at the point 0 0 Therefore there is no difference between the two graphs i Function Identical to the Graph of y x 5 The function that would have a graph identical to that of y x 5 is fx x 5 This graph is the same as the graph of y x but shifted 5 units to the right with the vertex at the point 5 0 j Function Identical to the Graph of x y A function whose graph would be identical to the graph of x y is fy y The graph of x y shows x as the absolute value of y which is identical to the graph of y x except that the variables x and y are swapped The graph would have the same V shape but stretched along the yaxis instead of the xaxis 4 x y Graph of y x x y Graph of y x 5 5 x y Graph of x y k Graph of the Functions f1x and f2x The functions f1x x for x 0 and f2x x for x 0 are two pieces of a piecewise function The graph of these functions together is the graph of the absolute value function fx x since the absolute value of x results in x for negative x and in x for positive x 6 Graph of the Functions f1x and f2x x y f1x x f2x x 4 Exploratory Challenge 2 41 a Function Table x x x sawtoothx x x 48 4 5 08 13 2 1 07 22 2 3 02 3 3 3 0 2 2 2 0 3 3 3 0 π 3 4 01416 Table 1 Values of the floor ceiling and sawtooth functions 7 42 b Function Graphs Floor and ceiling functions Floor function Ceiling function 2 1 1 2 3 4 5 05 1 x fx x x 43 c Interval Analysis For the given intervals the values of the functions floorx ceilingx and sawtoothx are Interval 0 1 x 0 x 1 sawtoothx x Interval 1 2 x 1 x 2 sawtoothx x 1 Interval 2 1 x 2 x 1 sawtoothx x 2 Interval 15 25 x 1 for x 15 2 and x 2 for x 2 25 x 2 for x 15 2 and x 3 for x 2 25 9 sawtoothx x 1 for x 15 2 and sawtoothx x 2 for x 2 25 10