Locker LESSON 6.5 Comparing Properties of Linear Functions Common Core Math Standards The student is epected to: F-IF.9 Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Mathematical Practices MP. Modeling Language Objective Students will eplain how to compare linear functions that are represented in different was. ENGAGE Essential Question: How can ou compare linear functions that are represented in different was? Possible answer: You can find and compare their slopes, intercepts, domains, and ranges. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and different tpes of jobs that might pa a commission to salespeople. Then preview the Lesson Performance Task. PAGE 9 Houghton Mifflin Harcourt Publishing Compan Image Credits: auremar/ Shutterstock Name Class Date 6.5 Comparing Properties of Linear Functions Essential Question: How can ou compare linear functions that are represented in different was? Eplore Comparing Properties of Linear Functions Given Algebra and a Description Comparing linear relationships can involve comparing relationships that are epressed in different was. Dan s Plumbing and Kim s Plumbing have different was of charging their customers. The function D (t) = 35t represents the total amount in dollars that Dan s Plumbing charges for t hours of work. Kim s Plumbing charges $35 per hour plus a $0 flat-rate fee. Define a function K (t) that represents the total amount Kim s Plumbing charges for t hours of work and then complete the tables. The function K (t) = 35t + 0 represents the total amount in dollars that Kim s Plumbing charges for t hours of work. Cost for Dan s Plumbing Cost for Kim s Plumbing Resource Locker t D (t) = 35t (t, D (t) ) t K (t) = (t, K (t) ) 0 0 (0, 0) 0 1 35 (1, 35) 1 70 (, 70) 3 105 (3, 105) 3 What domain and range values for the functions D (t) and for K (t) are reasonable in this contet? Eplain. 35t + 0 0 (0, 0) 75 (1, 75) 110 (, 110) 15 (3, 15) The domain for both functions is nonnegative real numbers. A plumber can work for an fraction of an hour, but not for negative hours. The range for D (t) is nonnegative real numbers because a plumber can earn an fraction of a dollar, but not negative dollars. The range for K (t) is all real numbers greater than or equal to 0. Kim s Plumbing charges a minimum of $0 for work. Module 6 1 Lesson 5 Name Class Date 6.5 Comparing Properties of Linear Functions Essential Question: How can ou compare linear functions that are represented in different was? F-IF.9 Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Houghton Mifflin Harcourt Publishing Compan Image Credits: auremar/ Shutterstock Eplore Comparing Properties of Linear Functions Given Algebra and a Description Comparing linear relationships can involve comparing relationships that are epressed in different was. Dan s Plumbing and Kim s Plumbing have different was of charging their customers. The function D (t) = 35t represents the total amount in dollars that Dan s Plumbing charges for t hours of work. Kim s Plumbing charges $35 per hour plus a $0 flat-rate fee. Define a function K (t) that represents the total amount Kim s Plumbing charges for t hours of work and then complete the tables. The function K (t) = 35t + 0 represents the total amount in dollars that Kim s Plumbing charges for t hours of work. Resource Cost for Dan s Plumbing Cost for Kim s Plumbing 35t + 0 0 (0, 0) 0 (0, 0) 35 (1, 35) 75 (1, 75) 70 (, 70) 110 (, 110) 105 (3, 105) 15 (3, 15) t D (t) = 35t (t, D (t) ) t K (t) = (t, K (t) ) 0 0 1 1 3 3 What domain and range values for the functions D (t) and for K (t) are reasonable in this contet? Eplain. The domain for both functions is nonnegative real numbers. A plumber can work for an fraction of an hour, but not for negative hours. The range for D (t) is nonnegative real numbers because a plumber can earn an fraction of a dollar, but not negative dollars. The range for K (t) is all real numbers greater than or equal to 0. Kim s Plumbing charges a minimum of $0 for work. Module 6 1 Lesson 5 HARDCOVER PAGES 9 3 Turn to these pages to find this lesson in the hardcover student edition. 1 Lesson 6.5
Graph the two cost functions for the appropriate domain values. Compare the graphs. How are the alike? How are the different? Reflect 1. Discussion What information could be found about the two functions without changing their representation? the slope, or rate of change Eplain 1 Comparing Properties of Linear Functions Given Algebra and a Table A table and a rule are two was that a linear relationship ma be epressed. Sometimes it ma be helpful to convert one representation to the other when comparing two relationships. There are other times when comparisons are possible without converting either representation. Eample 1 Cost ($) 10 160 10 10 100 0 60 0 0 0 1 3 Time (h) Compare the initial value and the range for each of the linear functions ƒ () and g (). The domain of each function is the set of all real numbers such that 5. The table shows some ordered pairs for ƒ (). The function g () is defined b the rule g () = 3 + 7. The initial value is the output that is paired with the least input. The least input for ƒ () and g () is 5. The initial value of ƒ () is ƒ (5) = 0. The initial value of g () is g (5) = 3 (5) + 7 =. Since ƒ () is a linear function and its domain is the set of all real numbers from 5 to, its range will be the set of all real numbers from ƒ (5) to ƒ (). Since ƒ (5) = 0 and ƒ () = 3, the range of ƒ () is the set of all real numbers such that 0 ƒ () 3. Cost ($) 10 160 10 10 100 0 60 0 0 0 1 3 Time (h) Sample answer: The graphs have the same slope, but the have different -intercepts. Since g () is a linear function and its domain is the set of all real numbers from 5 to, its range will be the set of all real numbers from g (5) to g (). Since g (5) = and g () = 3 () + 7 = 31, the range of g () is the set of all real numbers such that g () 31. f () 5 0 6 7 3 Module 6 Lesson 5 Houghton Mifflin Harcourt Publishing Compan PAGE 30 EXPLORE Comparing Properties of Linear Functions Given Algebra and a Description INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP. Point out to students that when the complete tables, the will use the same values of the independent variable to create their graphs. Then the will compare and contrast the graphs. AVOID COMMON ERRORS Some students ma forget the meaning of the variables in a function. Remind students that, for a linear function of the forms f () = m + b, the slope is m and the -intercept is b. These values need to be considered in the contet and units of the problem s situation. EXPLAIN 1 Comparing Properties of Linear Functions Given Algebra and a Table PROFESSIONAL DEVELOPMENT Integrate Mathematical Process This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to use modeling. Students learn how to use various forms of linear functions to solve real-world problems, and to identif which smbolic forms are most useful for which kinds of problems. Students also work with real-world domains that are a subset of the real numbers. QUESTIONING STRATEGIES What does the term reasonable domain mean? The set of numbers that make sense for a given situation; a reasonable domain ma not include fractions, negative numbers, zero, or large numbers. Comparing Properties of Linear Functions
AVOID COMMON ERRORS If students have problems converting linear relationships, remind them that the ma or ma not be able to compare relationships in the given formats. Sometimes the ma need to convert one or both relationships into the same format, such as converting one or both into equations, tables, or graphs. B The domain of each function is the set of all real numbers such that 6 10. The table shows some ordered pairs for ƒ (). The function g () is defined b the rule g () = 5 +11. The initial value is the output that is paired with the least input. The least input for ƒ () and g () is 6. The initial value of ƒ () is f (6) = 36. The initial value of g () is g (6) = 5 (6) + 11 = 1. Since ƒ () is a linear function, and its domain is the set of all real numbers from to 10, its range will be the set of all real numbers from ƒ ( ) to ƒ (10). Since ƒ ( ) = and ƒ (10) =, the 6 6 36 6 f () 6 36 7 9 5 10 60 60 PAGE 31 range of ƒ () is the set of all real numbers such that 36 ƒ () 60. Since g () is a linear function and its domain is the set of all real numbers from 6 to 10, its range will be the set of all real numbers from g ( ) to g (10). Since g ( ) = and 6 6 1 10 61 g (10) = 5 ( ) + 11 =, the range of g () is the set of all real numbers such that 1 g () 61. Reflect. Discussion How can ou use a table of values to find the rate of change for a linear function? First, find the difference of the -values and the difference of the corresponding function values. Then divide the difference of the function values b the difference of the -values. Your Turn Houghton Mifflin Harcourt Publishing Compan 3. Find the rate of change for the linear function ƒ () that is shown in the table. f () 3 9 5 36 6 3 7 50 Use the ordered pairs (3, ) and (5, 36). Difference of -values: 5-3 = Difference of f ()-values: 36 - = 1 Divide the difference of the function values b the difference of the -values: 1 = 7 The rate of change is 7.. The rule for ƒ () in Eample 1B is ƒ () = 6. If the domains were etended to all real numbers, how would the slopes and -intercepts of ƒ () and g () = 5 + 11 in Eample 1B compare? The slope of f () is greater than the slope of g (). The -intercepts are different because the -intercept of f () is 0 and the -intercept of g () is 11. Module 6 3 Lesson 5 COLLABORATIVE LEARNING Small Group Activit Have students work in groups of three. Give students the following prompt: An emploee earns $.00 an hour. The function = gives the total pa the emploee will earn for working hours. Have the first student make a table of ordered pairs. The second student should make a graph for this function. The third student should eplain the relationships among the equation, the table, and the graph, and tell how each one describes the situation. Have students change roles so that each student has the opportunit to do each task. 3 Lesson 6.5
Eplain Comparing Properties of Linear Functions Given a Graph and a Description Information about a linear relationship ma have to be inferred from the contet given in the problem. Eample Write a rule for each function, and then compare their domain, range, slope, and -intercept. A rainstorm in Austin lasted for 3.5 hours, during which time it rained a stead rate of.5 mm per hour. The function A(t) represents the amount of rain that fell in t hours. The graph shows the amount of rain that fell during the same rainstorm in Dallas, D(t) (in millimeters), as a function of time t (in hours). Write a rule for each function. A (t) =.5t for 0 t 3.5 The line representing D(t) has endpoints at (0, 0) and (, 0). The slope of 0-0 D(t) is = 5. The -intercept is 0, so substituting 5 for m and 0 for b in - 0 = m + b produces the equation = 5. This can be represented b the function D (t) = 5t, for 0 t. The domains of each function both begin at 0 but end for different values of t, because the lengths of time that it rained in Austin and Dallas were not the same. The range for A (t) is 0 A (t) 15.75. The range for D (t) is 0 A (t) 0. The slope for D (t) is 5, which is greater than the slope for A (t), which is.5. The -intercepts of both functions are 0. Rainfall (mm) 1 (, 0) 16 1 1 10 6 0 1 3 Time (h) Houghton Mifflin Harcourt Publishing Compan Image Credits: Gett Images PAGE 3 EXPLAIN Comparing Properties of Linear Functions Given a Graph and a Description QUESTIONING STRATEGIES Can the graph of a linear function be a segment rather than a line or a ra? Eplain. Yes, if the domain is an interval, the graph will be a segment to reflect the situation. For a problem involving distance from a location where the -ais represents time, wh is the -intercept equal to 0? The initial value for both time and distance is 0, so the initial point is the origin, and both the - and -intercepts are equal to 0. Module 6 Lesson 5 DIFFERENTIATE INSTRUCTION Cognitive Strategies Several characteristics of two linear functions can be compared. Besides the slope and intercepts, there are specific domain and range values such as initial value and maimum and minimum values. Students must recognize that the ma need to represent the function in a different wa to facilitate comparisons. Guide students with questions such as: What are the independent and dependent variables? Will the points go upward or downward from left to right? How can ou find the equation for the relationship? Comparing Properties of Linear Functions
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Encourage students to think about what characteristics the can determine directl from each different tpe of representation of a function and what characteristics must be determined through calculation or reasoning. For eample, the maimum value of a function over an interval can be read from its graph but must be calculated if the function equation is given. B One group of hikers hiked at a stead rate of 6.5 kilometers per hour for hours. The function ƒ (t) represents the distance this group of hikers hiked in t hours. The graph shows the distance a second group of hikers hiked, g (t) (in kilometers), as a function of t (in hours). Write a rule for each function. ƒ (t) = 6.5t for 0 t The line representing g (t) has endpoints at (0, 0) and (, ). The slope of g (t) is.5 36 36-0 _ =..5-0 0 1 3 Time (h) The -intercept is 0, so substituting for m and 0 for b in = m + b produces the equation =. This can be represented b the function g (t) = t for 0 t.5. Distance (km) 36 (.5, 36) 3 0 16 1 The domains of each function both begin at 0 and end at different values of t. The range for ƒ (t) is 0 ƒ (t) 6 and the range for g (t) is 0 g (t) 36. The slope for g (t) is greater than the slope for f (t). The -intercepts are both 0. Houghton Mifflin Harcourt Publishing Compan Module 6 5 Lesson 5 LANGUAGE SUPPORT Connect Vocabular Students acquire academic language through modeling and practice. One particular construction the ma not hear in the same contet outside of mathematics is the term given. Students ma have heard epressions such as he was given a helmet, but the word given b itself ma be unfamiliar. Eplain to students that, in mathematics, given refers to the information that is provided to ou; the information that ou can use to answer the question. 5 Lesson 6.5
Reflect 5. What is the meaning of the -intercepts for the functions A (t) and D (t) in Eample A? The rainfall started at the same time that the rainstorm started. At that time, the total rainfall was 0 mm in both locations. Your Turn 6. An eperiment compares the heights of two plants over time. A plant was 5 cm tall at the beginning of the eperiment and grew 0.3 centimeters each da. The function ƒ (t) represents the height of the plant (in centimeters) after t das. The graph shows the height of the second plant, g (t) (in centimeters), as a function of time t (in das). Find the rate of change g (t) and compare it to the rate of change for ƒ (t). Rate of change g (t): 9-5 16-0 = 0.5 The rate of change for g (t) is 0.5 centimeters per da, which is less than the rate of change for f (t), which is 0.3 centimeters each da. Elaborate Time (das) 7. When would representing a linear function b a graph be more helpful than b a table? Sample answer: A graph would be more helpful than a table when approimating the value of a function at input values that are not given in a table. Height (cm) 9 (16, 9) 7 6 5 (0, 5) 3 1 0 6 10 1 1 16 1 PAGE 33 ELABORATE QUESTIONING STRATEGIES Do ou have to use the standard form of the equation to find the - and -intercepts? Eplain. No; however, using standard form will take fewer steps than using point-slope form directl to find the - and -intercepts. SUMMARIZE THE LESSON How do ou compare linear functions? It helps to represent each situation in the same wa. Sometimes the data can be represented best b function rules. At other times, a graph or a table ma be the best wa to represent the data.. When would representing a linear function b a table be more helpful than b a graph? Sample answer: When the rate of change is not an integer, such as when a function, f (), represents the cost of buing packages of printer paper at a rate of $3.59 per package. 9. Essential Question-Check-In How can ou compare a linear function represented in a table to one represented as a graph? Plot the ordered pairs in the table to convert the representation to a graph. Then compare the graphs slopes and intercepts. Houghton Mifflin Harcourt Publishing Compan Module 6 6 Lesson 5 Comparing Properties of Linear Functions 6
EVALUATE Evaluate: Homework and Practice Compare the initial value and the range for each of the linear functions ƒ () and g (). Online Homework Hints and Help Etra Practice 1. The domain of each function is the set of all real numbers such that 5. The table shows some ordered pairs for ƒ (). The function g () is defined b the rule g () = + 6. f () 5 ASSIGNMENT GUIDE The initial value of f () is 5. The initial value of g () is. 3 7 9 Concepts and Skills Eplore Comparing Properties of Linear Functions Given Algebra and a Description Eample 1 Comparing Properties of Linear Functions Given Algebra and a Table Eample Comparing Properties of Linear Functions Given a Graph and a Description Practice Eercises 13 15, 17, 0, Eercises 1 6, 16 17, 19 Eercises 7 1, 1, 1, 3 5 The range of f () is all real numbers such that 5 f () 11. The range of g () is all real numbers such that g () 11.. The domain of each function is the set of all real numbers such that 1. The table shows some ordered pairs for ƒ (). The function g () is defined b the rule g () = 7-3. The initial value of f () is 3. The initial value of g () is 53. The range of f () is all real numbers such that 3 f () 50. The range of g () is all real numbers such that 53 g () 1. 5 11 f () 3 9 3 10 11 6 1 50 CONNECT VOCABULARY Some students ma confuse the meaning of discrete with the meaning of discreet. Both words have the same pronunciation. Point out that discreet means careful, while discrete means separate or individual. Houghton Mifflin Harcourt Publishing Compan 3. The domain of each function is the set of all real numbers such that - -1. The function ƒ () is defined b the rule ƒ () = + 9. The table shows some ordered pairs for g (). The initial value of f () is 1. The initial value of g () is 10. The range of f () is all real numbers such that 1 f () 7. The range of g () is all real numbers such that 7 g () 10.. The domain of each function is the set of all real numbers such that 0. The function ƒ () is defined b the rule ƒ () = -3 + 15. The table shows some ordered pairs for g (). The initial value of f () is 15. The initial value of g () is 3. g () - 10-3 9 - -1 7 g () 0 3 1 19 15 The range of f () is all real numbers such that 3 f () 15. 3 11 The range of g () is all real numbers such that 7 g () 3. 7 Module 6 7 Lesson 5 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1 6 1 Recall of Information MP. Reasoning 7 Skills/Concepts MP. Reasoning 9 1 Skills/Concepts MP. Modeling 13 15 Skills/Concepts MP.6 Precision 16 17 Skills/Concepts MP. Reasoning 1 1 Recall of Information MP.6 Precision 7 Lesson 6.5
5. The domain of each function is the set of all real numbers such that 10 13. The table shows some ordered pairs for ƒ (). The function g () is defined b the rule g () = 1 + 1. The initial value of f () is. The initial value of g () is 17. The range of f () is all real numbers such that f () 53 _. The range of g () is of all real numbers such that 17 g () 37 _. 6. The domain of each function is the set of all real numbers such that 6. The function ƒ () is defined b the rule ƒ () = - 3 + 10. The table shows some ordered pairs for g (). The initial value of f () is 17. The initial value of g () is 1. The range of f () is all real numbers such that 11 f () 17. The range of g () is all real numbers such that 9 g () 1. f () 10 11 7_ 1 5 13 53_ g () 1 3 5 _ 51 _ 3 _ 1 6 9 PAGE 3 AVOID COMMON ERRORS If students have problems converting linear relationships, remind them that the ma be able to compare relationships in the given formats. However, sometimes the ma need to convert relationships to the same format, such as converting them both into equations, tables, or graphs. Write a rule for each function f and g, and then compare their domains, ranges, slopes, and -intercepts. 7. The function ƒ () has a slope of 6 and has a -intercept of 0. The graph shows the function g (). 16 f () = 6 + 0 1 - (-) g () slope: = 5; -intercept: - - 0 g () = 5 - The domain and range of the functions are the same. The slope of f () is greater than g (). The -intercept of f () is greater than the -intercept of g ().. The function ƒ () has a slope of -3 and has a -intercept of 5. The graph shows the function g (). f () = -3 + 5 - - (6) g () slope: _ - 0 = - 5_ ; -intercept: 6 g () = - 5_ + 6 The domain and range of the functions are the same. The slope of f () is steeper than the slope of g (). The -intercept of g () is greater than the -intercept of f (). -16 - - 0 16 - -16 - g() g() - 0 - Houghton Mifflin Harcourt Publishing Compan Module 6 Lesson 5 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 19 1 Skills/Concepts MP. Reasoning 3 Skills/Concepts MP. Reasoning 5 3 Strategic Thinking MP. Reasoning Comparing Properties of Linear Functions
AUDITORY CUES Have students work in pairs to quiz each other aloud about the various forms of a linear equation. Have one student state a linear equation, such as - 1 = 3 ( - ) or = +, and have the other tell which form the equation is in (slope-intercept, standard, or point-slope). Then have students reverse roles so that the first student states a form and the second student gives an equation in that form. PAGE 35 Write a rule for each function, and then compare their domains, ranges, slopes, and -intercepts. 9. Jeff, an electrician, had a job that lasted 5.5 hours, during which time he earned $3 per hour and charged a $5 service fee. The function J (t) represents the amount Jeff earns in t hours. Brendan also works as an electrician. The graph of B (t) shows the amount in dollars that Brendan earns as a function of time t in hours. J (t) = 3t + 5, 0 t 5.5 B (t) 160-0 Slope: = 30-0 -intercept: 0 B (t) = 30t + 0, 0 t Earnings ($) 10 160 10 10 100 0 60 0 0 The domain for both functions is real numbers that start at 0, but B (t) ends at and J (t) ends at 5.5. The range for B (t) is the set of all real numbers B (t), where 0 B (t) 160. The range for J (t) is the set of all real numbers J (t), where 5 J (t) 01. The slope of J (t) is greater than the slope of B (t). The -intercept of B (t) is greater than the slope of J (t). B(t) 0 1 3 Time (h) Houghton Mifflin Harcourt Publishing Compan 10. Apples can be bought at a farmer s market up to 10 pounds at a time, where each pound costs $1.10. The function a (w) represents the cost of buing w pounds of apples. 1 16 The graph of p (w) shows the cost in dollars of buing 1 1 w pounds of pears. 10 a (w) = 1.1w, 0 w 10 6 p(w) p (w) Slope: 1-0 10-0 = 1. -intercept: 0 p (w) = 1.w, 0 t 10 Weight (lb) The domain for both functions is real numbers from 0 to 10. The range for a (w) is real numbers a (w), where 0 a (w) 11. The range for p (w) is real numbers p (w), where 0 p (w) 1. The slope of p (w) is greater than the slope of a (w). The -intercept of both functions is 0. Cost ($) 0 1 3 5 6 7 9 Module 6 9 Lesson 5 9 Lesson 6.5
11. Biolog A gecko travels for 6 minutes at a constant rate of 19 meters per minute. The function g (t) represents the distance the gecko travels after t minutes. The graph of m (t) shows the distance in meters that a mouse travels after t minutes. g (t) = 19t, 0 t 6 m (t) = 37.5t, 0 t The domain for both functions is real numbers that begin at 0, but g (t) ends at 6 and m (t) ends at. The range for g (t) is real numbers g (t), where 0 g (t) 11. The range for m (t) is real numbers m (t), where 0 m (t) 300. The slope of m (t) is greater than the slope of g (t). The -intercept of both functions is 0. 1. Cind is buing a water pump. The bo for Pump A claims that it can move gallons per minute. The function A (t) represents the amount of water (in gallons) Pump A can move after t minutes. The graph of B (t) shows the amount of water in gallons that Pump B can move after t minutes. A (t) = t, t 0 B (t) = 5t, t 0 The domain and range for both functions are nonnegative real numbers. The slope of A (t) is greater than the slope of B (t). The -intercept of both functions is 0. Distance (m) Water pumped (gal) 50 00 350 300 50 00 150 100 50 m(t) 0 1 3 5 6 7 9 Time (min) 900 00 700 600 500 00 B(t) 300 00 100 0 6 10 1 1 16 1 Time (min) Houghton Mifflin Harcourt Publishing Compan Image Credits: Pete Orelup/Gett Images PAGE 36 QUESTIONING STRATEGIES What is a discrete linear function? A discrete linear function is a linear function whose graph consists of isolated points. How are discrete and continuous linear functions alike and how are the different? Discrete and continuous functions are alike in that the points on their graphs lie on a line. The differ in that the graph of a discrete function consists of isolated points, while the graph of a continuous function is an unbroken line or part of a line. Module 6 90 Lesson 5 Comparing Properties of Linear Functions 90
13. Erin is comparing two rental car companies for an upcoming trip. The function A (d) = 0.0d represents the total amount in dollars of driving a car d miles from compan A. Compan B charges $0.10 per mile and a $10 fee. a. Define a function B (d) that represents the total amount compan B charges for driving d miles and then complete the tables. Cost for Compan A Cost for Compan B d A (d) = 0.0d (d, A (d)) d B (d) = 0.10d + 10 (d, B (d)) 0 0 (0, 0) 0 10 (0, 10) 0 (0, ) 0 1 (0, 1) 0 (0, ) 0 1 (0, 1) b. Graph and label the two cost functions for all appropriate domain values. c. Compare the graphs. How are the alike? How are the different? Sample answer: the graphs have different slopes and different -intercepts. 1. Snow is falling in two cities. The function C (t) = t + represents the amount of snow on the ground, in centimeters, in Carlisle t hours after the snowstorm begins. There was cm of snow on the ground in York when the storm began and the snow accumulates at 1.5 cm per hour. 0 10 0 30 0 50 60 70 0 90 Distance (mi) a. Define a function Y (t) that represents the amount of snow on the ground after t hours in York and then complete the tables. Cost ($) 1 16 1 1 10 6 B(d) A(d) Houghton Mifflin Harcourt Publishing Compan Carlisle York t C (t) = t + (t, C (t)) t Y (t) = 1.5t + (t, Y (t)) 0 (0, ) 0 1 10 (1, 10) 1 1 (, 1) b. Graph and label the two cost functions for all appropriate domain values. c. Compare the graphs. How are the alike? How are the different? Sample answer: The graphs have the same -intercepts, but the have different slopes. Snow depth (cm) 1 16 1 1 10 6 (0, ) 9.5 (1, 9.5) 11 (, 11) C(t) Y(t) 0 1 3 Time (h) t Module 6 91 Lesson 5 91 Lesson 6.5
15. Gillian works from 0 to 30 hours per week during the summer. She earns $1.50 per hour. Her friend Emil also has a job. Her pa for t hours each week is given b the function e (t) = 13t, where 15 t 5. a. Find the domain and range of each function. The domain of g (t) is the set of real numbers t, where 0 t 30. The domain of e (t) is the set of real numbers t, where 15 t 5. The range of g (t) is the set of real numbers g (t), where 50 g (t) 375. The range of e (t) is the set of real numbers e (t), where 195 e (t) 35. b. Compare their hourl wages and the amount the earn per week. Gillian earns less per hour than Emil. Gillian earns from $50 to $375 per week and Emil earns from $195 to $35 per week. 16. The function A (p) defined b the rule A (p) = 0.13p + 15 represents the cost in dollars of producing a custom tetbook that has p pages for college A, where p B (p) 0 < p 500. The table shows some ordered pairs for B (p), where B (p) represents 0 the cost in dollars of producing a custom tetbook that has p pages for college B, 50 30 where 0 < p 500. For both colleges, onl full pages ma be printed. 100 36 Compare the domain, range, slope, and -intercept of the functions. Interpret the comparisons in contet. 150 The slope for A (p) is 0.13, while the slope for B (p) is 30 - = 0.1. The cost per page is 50-0 higher for college A than for college B. The -intercept for A (p) is 15, while the -intercept for B (p) is. The initial cost is greater for college B than for college A. The domains for each function are the same. Both can produce up to 500 pages. Ranges: A tetbook from college A costs between $15 and $0, and a tetbook from college B costs between $ and $. 17. Complete the table so that ƒ () is a linear function with a slope of and a -intercept of 7. Assume the domain includes all real numbers between the least f () and greatest values shown in the table. Compare ƒ () to g () = + 7 if the range - -1 of g () is -1 g () 11. -1 3 The domains, ranges, slopes and -intercepts are the same for 0 7 f () and g (). The represent the same function. 1 11 1. Which functions have a rate of change that is greater than the one shown in the graph? Select all that appl. a. ƒ () = 1_ 1_ - 5 The rate of change in 3 _ the graph is b. g () = + 6 1 > 3. 3 c. h () = 3_ 3_ - 9 - - 0 > 3 d. j () = 1_ - + 1_ 3 - e. k () = 1 > 3 _ All functions ecept the ones in choices a and d have a rate of change greater than 3, the one shown in the graph. Houghton Mifflin Harcourt Publishing Compan PAGE 37 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Provide students with a verbal description of a real-world situation that can be modeled b a function. Have them represent the function using a table, a graph, and a rule, and show how each form provides information about the function. Check students work. Module 6 9 Lesson 5 Comparing Properties of Linear Functions 9
JOURNAL Have students create a graphic organizer or idea map for the different forms used to represent functions, including tables, graphs, algebra rules, and verbal descriptions. Students should include the advantages of each form in their idea maps. PAGE 3 Houghton Mifflin Harcourt Publishing Compan 19. Does the function ƒ () = 5 + 5 with the domain 6 have the same domain as function g (), whose onl function values are shown in the table? Eplain. The functions do not have the same domain. The domain of f () includes all real numbers between 6 and, while the domain of g () is {6, 7, }. 0. The linear function ƒ () is defined b the table, and the linear function g () is shown in the graph. Assume that the domain of ƒ () includes all real numbers between the least and greatest values shown in the table. a. Find the domain and range of each function, and compare them. Domain of f (): -1 3-1 f () -7 0 - Domain of g (): -1 3 1-1 Range of f (): -7 f () 5 Range of g (): - g () 3 5 The domains are the same. The ranges are different. b. What is the slope of the line represented b each function? What is the -intercept of each function? The slope is 3 and the -intercept is - for f (), and the slope is and the -intercept is - for g (). 1. The linear function ƒ () is defined b ƒ () = - 1 + 6 for all real numbers, and the linear function g () is shown in the graph. a. Find the domain and range of each function, and compare them. The domain and range of both f () and g () is the set of all real numbers. b. What is the slope of the line represented b each function? What is the -intercept of each function? The slope is - 1_ and the -intercept is 6 for f (), and the slope is - 1_ and the -intercept is -6 for g (). H.O.T. Focus on Higher Order Thinking. Communicate Mathematical Ideas Describe a linear function for which the least value in the range does not occur at the least value of the domain (a function for which the least value in the range is not the initial value.) A line segment that has a negative slope has an initial value that is the greatest value in the range. The least value in the range will be paired with the greatest value in the domain. 3. Draw Conclusions Two linear functions have the same slope, same -intercept, and same -intercept. Must these functions be identical? Eplain our reasoning. The functions do not have to be identical. Each could have a different domain. - - g () 6 35 7 0 5 g() - 0 - - 0 - g() - Module 6 93 Lesson 5 93 Lesson 6.5
. Draw Conclusions Let ƒ () be a line with slope -3 and -intercept 0 with domain {0, 1,, 3}, and let g () = {(0, 0), (1, -1), (, -), (3, -9)}. Compare the two functions. The domains are the same, but the range of f () {0, - - -6}, while the range of g () is {0, -1 - -9}. Both have -intercepts at 0. The slope of f () is -3 and the slope of g () changes. The function f () linear, and the function g () is not a line. 5. Draw Conclusions Let ƒ () be a line with slope 7 and -intercept -17 with domain 0 5, and let g () = (0, -17), (1, -10), (, -3), (3, ), (, 11), (5, 1). Compare the two functions. The domains are not the same because the domain of f () is all real numbers between 0 and 5, while the domain of g () is onl the integers between 0 and 5. The range of f () is -17 f () 1 while the range of g (), {-17, -10, -3,, 11, 1}. Both have a -intercept of -17. The function f () is linear, but the function g () is not linear because it is not a line or a line segment. LANGUAGE SUPPORT Some students ma not be familiar with the terms commission or incentivize. Have volunteers use pla mone to demonstrate earning a commission and then eplain how it is calculated. Finall, have another volunteer eplain how earning a commission might incentivize someone to work harder to sell more. Lesson Performance Task Lindsa found a new job as an insurance salesperson. She has her choice of two different compensation plans. Plan F was described to her as a $50 base weekl salar plus a 10% commission on the amount of sales she made that week. The function ƒ () represents the amount Lindsa earns in a week when making sales of dollars with compensation plan F. Plan G was described to her with the 1000 900 00 700 600 graph shown. The function g () represents the amount Lindsa earns 500 in a week when making sales of dollars with compensation plan G. 00 300 Write a rule for the functions ƒ () and g (); then identif and 00 compare their domain, range, slope, and -intercept. Compare 100 the benefits and drawbacks of each compensation plan. Which compensation plan should Lindsa take? Justif our answer. 0 1000 000 3000 000 5000 Rule for f () : rate of change: 10% = 0.1 f () = 0.1 + 50 Domain: f () : All multiples of 0.01, where 0 < g () : All multiples of 0.01, where 0 < Their domains are the same. Slope: f () : 0.1 g () : 0.15 The slope of g () is greater than the slope of f (). Earnings ($) Rule for g () : rate of change: = g () = 0.15 + 300 Range: Sales ($) 600-300 000-0 = 0.15 f () : All multiples of 0.01, where f () 50 g () : All multiples of 0.01, where g () 300 The initial value for f ( ) is higher than the initial value for g ( ), but both can increase without limit. -intercept: f ( ) : 50 g ( ) : 300 The -intercept for f ( ) is greater than the -intercept for g ( ). Plan F has a greater base salar than plan G. If Lindsa doesn t make sales for one week, she would earn more mone with plan F than with plan G. Since the slope of g () is greater than the slope of f (), at some point Lindsa would earn more mone with plan G. This would actuall happen for sales in ecess of $3000. Lindsa should take compensation plan G because there is a better potential to earn more mone than with compensation plan F. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Have students graph both equations on the same coordinate grid and find their intersection, (3000, 750). Have students eplain what happens to the values to the right of the intersection point and what that means for Lindsa s salar. Module 6 9 Lesson 5 EXTENSION ACTIVITY The Internal Revenue Service considers a commission to be a supplemental wage. Have students research other eamples of supplemental wages. Then have them determine the current supplemental wages ta rate and then eplain whether the think the ta rate could affect Lindsa s choice in compensation plans. Students will find that commissions, bonuses, sick leave paments, severance paments, and taable prizes are a few of the tpes of supplemental wages. Supplemental wages that total more than $1 million were taed at a rate of 35% in 013, while totals less than $1 million could have been taed at a rate of 5% in 013. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Comparing Properties of Linear Functions 9