Domain, Range, and End Behavior

Transcription

1 Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Also A-CED., A-CED.3 Mathematical Practices MP. Precision Language Objective With a partner, fill in a graphic organizer showing the domain, range, and end behavior of a function. ENGAGE Essential Question: How can ou determine the domain, range, and end behavior of a function? Possible answer: The domain consists of values for which the function is defined or on which the real-world situation is based. The range consists of the corresponding f () values. The end behavior describes what happens to the f () values as the values increase without bound or decrease without bound. Houghton Mifflin Harcourt Publishing Compan Name Class Date 1.1 Domain, Range, and End Behavior Essential Question: How can ou determine the domain, range, and end behavior of a function? Eplore Representing an Interval on a Number Line An interval is a part of a number line without an breaks. A finite interval has two endpoints, which ma or ma not be included in the interval. An infinite interval is unbounded at one or both ends. Suppose an interval consists of all real numbers greater than or equal to 1. You can use the inequalit 1 to represent the interval. You can also use set notation and interval notation, as shown in the table. Description of Interval Tpe of Interval Inequalit Set Notation Interval notation All real numbers from a to b, including a and b Finite a b { a b} [a, b] All real numbers greater than a Infinite > a { > a} (a, + ) All real numbers less than or equal to a Infinite a { a} (, a] For set notation, the vertical bar means such that, so ou read { 1} as the set of real numbers such that is greater than or equal to 1. For interval notation, do the following: Use a square bracket to indicate that an interval includes an endpoint and a parenthesis to indicate that an interval doesn t include an endpoint. For an interval that is unbounded at its positive end, use the smbol for positive infinit, +. For an interval that unbounded at its negative end, use the smbol for negative infinit, -. Alwas use a parenthesis with positive or negative infinit. So, ou can write the interval 1 as [1, + ). A Complete the table b writing the finite interval shown on each number line as an inequalit, using set notation, and using interval notation. Finite Interval Inequalit Resource Locker < PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss how the distance a car can travel is a function of the amount of gas in the car s gas tank. Have students identif the independent and dependent variables of the functional relationship. Then preview the Lesson Performance Task. Set Notation Interval Notation { 3 } [ 3, ] { 3 < } ( 3, ] Module 1 5 Lesson 1 Name Class Date 1.1 Domain, Range, and End Behavior Essential Question: How can ou determine the domain, range, and end behavior of a function? F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Also A-CED., A-CED.3 Houghton Mifflin Harcourt Publishing Compan Eplore Representing an Interval on a Number Line An interval is a part of a number line without an breaks. A finite interval has two endpoints, which ma or ma not be included in the interval. An infinite interval is unbounded at one or both ends. Suppose an interval consists of all real numbers greater than or equal to 1. You can use the inequalit 1 to represent the interval. You can also use set notation and interval notation, as shown in the table. Description of Interval Tpe of Interval Inequalit Set Notation Interval notation All real numbers from a to b, Finite a b { a b} [a, b] including a and b All real numbers greater than a Infinite > a { > a} (a, + ) All real numbers less than or Infinite a { a} (, a] equal to a For set notation, the vertical bar means such that, so ou read { 1} as the set of real numbers such that is greater than or equal to 1. For interval notation, do the following: Use a square bracket to indicate that an interval includes an endpoint and a parenthesis to indicate that an interval doesn t include an endpoint. For an interval that is unbounded at its positive end, use the smbol for positive infinit, +. For an interval that unbounded at its negative end, use the smbol for negative infinit, -. Alwas use a parenthesis with positive or negative infinit. So, ou can write the interval 1 as [1, + ). Complete the table b writing the finite interval shown on each number line as an inequalit, using set notation, and using interval notation. Finite Interval Inequalit Set Notation Interval Notation Resource { 3 } [ 3, ] 3 < { 3 < } ( 3, ] Module 1 5 Lesson 1 HARDCOVER PAGES 5 1 Turn to these pages to find this lesson in the hardcover student edition. 5 Lesson 1.1

2 B Complete the table b writing the infinite interval shown on each number line as an inequalit, using set notation, and using interval notation. Infinite Interval Inequalit > EXPLORE Representing an Interval on a Number Line Set Notation Interval Notation Reflect { } (, ] { > } (, + ) INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online. 1. Consider the interval shown on the number line. a. Represent the interval using interval notation. b. What numbers are in this interval?. What do the intervals [, 5], [, 5), and (, 5) have in common? What makes them different? All three intervals contain all the numbers between and 5. The interval [, 5] also includes the endpoints and 5, the interval [, 5) includes onl the endpoint, and the interval (, 5) does not contain either endpoint. 3. Discussion The smbol represents the union of two sets. What do ou think the notation (-, ) (, + ) represents? All real numbers ecept Eplain Identifing a Function s Domain, Range and End Behavior from its Graph Recall that the domain of a function f is the set of input values, and the range is the set of output values ƒ(). The end behavior of a function describes what happens to the ƒ()-values as the -values either increase without bound (approach positive infinit) or decrease without bound (approach negative infinit). For instance, consider the graph of a linear function shown. From the graph, ou can make the following observations. (, + ) All real numbers - - Module 1 Lesson 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to use the language of mathematics to communicate precisel and to attend to precision. Students learn to describe intervals using inequalities, set notation, and interval notation. The also learn how to use mathematical notation to describe end behavior of a function. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Draw students attention to the use of braces, parentheses, and brackets in the various representations. Make sure students can use the smbols correctl, and can eplain the significance of the smbols in each tpe of notation. EXPLAIN 1 Identifing a Function s Domain, Range, and End Behavior from its Graph AVOID COMMON ERRORS Some students ma incorrectl identif the end behavior of a function that increases over the interval (-, ) as As -, ƒ (). Help students to see that for this part of the description, the must consider the behavior of the function as the values of decrease (the behavior of the graph as observed from right to left), and not whether the function itself is an increasing or decreasing function. Domain, Range, and End Behavior

3 QUESTIONING STRATEGIES Is it possible that a linear function with the domain {all real numbers} could have a range that is not {all real numbers}? Eplain. Yes; the function could be a constant function, such as f () =. The domain is {all real numbers}, but the range is {}. Statement of End Behavior As the -values increase without bound, the f()-values also increase without bound. As the -values decrease without bound, the f()-values also decrease without bound. Eample 1 Smbolic Form of Statement As +, f() +. As -, f() -. Write the domain and the range of the function as an inequalit, using set notation, and using interval notation. Also describe the end behavior of the function. The graph of the quadratic function f() = is shown. Domain: Range: Inequalit: < < + Set notation: { < < + } Interval notation: (, + ) End behavior: - - Inequalit: Set notation: { } As +, f() +. As, f() +. Interval notation: [, + ) Houghton Mifflin Harcourt Publishing Compan The graph of the eponential function ƒ() = is shown. Domain: Inequalit: Set notation: Interval notation: < < + { < < + } (, + ) Range: Inequalit: Set notation: Interval notation: > { > } (, + ) End behavior: As +, f() +. As +, f(). - - Module 1 7 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs to draw the graph of a linear function with a restricted domain and keep the graph hidden from their partner. Have students describe the function s domain and range using one of the notation forms from the lesson, then echange descriptions and tr to draw each other s graph. Have students compare their graphs (which ma differ) and discuss the results. 7 Lesson 1.1

4 Reflect. Wh is the end behavior of a quadratic function different from the end behavior of a linear function? Unlike the graph of a linear function, the graph of a quadratic function has a turning point (the verte), which changes the direction of the graph. 5. In Part B, the ƒ()-values decrease as the -values decrease. So, wh can t ou sa that f() as? The f()-values do not decrease without bound. The instead approach. Your Turn Write the domain and the range of the function as an inequalit, using set notation, and using interval notation. Also describe the end behavior of the function.. The graph of the quadratic function ƒ() = is shown. Domain: < < +, { < < + }, (, + ) Range:, { }, (, ] End behavior: As +, f() - ; As -, f(). - - EXPLAIN Graphing a Linear Function on a Restricted Domain INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that the graph of a function represents the set of ordered pairs produced b the function. Help them to see that when the are using a graph to identif the range of a function, the are looking to identif the -values of those ordered pairs. Eplain Graphing a Linear Function on a Restricted Domain Unless otherwise stated, a function is assumed to have a domain consisting of all real numbers for which the function is defined. Man functions such as linear, quadratic, and eponential functions are defined all real numbers, so their domain, when written in interval notation, is (-, + ). Another wa to write the set of real numbers is R. Sometimes a function ma have a restricted domain. If the rule for a function and its restricted domain are given, ou can draw its graph and then identif its range. Eample For the given function and domain, draw the graph and identif the range using the same notation as the given domain. ƒ() = 3 + with domain [, ] Since ƒ() = 3 + is a linear function, the graph is a line segment with endpoints at (, ƒ()), or (, -1), and (, ƒ()), or (, 5). The endpoints are included in the graph. The range is [-1, 5]. - - Houghton Mifflin Harcourt Publishing Compan Module 1 8 Lesson 1 DIFFERENTIATE INSTRUCTION Cognitive Strategies If students have a difficult time consistentl identifing the domain and range of functions, encourage them to use the phrase depends on instead of is a function of. For eample, The distance traveled b a car depends on the amount of gas in the tank. Help them to see that the elements of the range depend on the elements of the domain. Domain, Range, and End Behavior 8

5 QUESTIONING STRATEGIES If a linear function has a restricted domain, must the range consist of a finite number of elements? Eplain. No. If the domain is restricted to an interval (or intervals), as opposed to a finite number of elements, the range could consist of infinitel man values. For eample, the range of the function f () = 3 with domain, 5 is, 15, an interval containing infinitel man numbers. B ƒ() = - - with domain { > -3} Since ƒ() = - - is a linear function, the graph is a ra with its endpoint at (-3, ƒ(-3)), or (-3, 1). The endpoint is not included in the graph. The range is { < 1}. Reflect 7. In Part A, how does the graph change if the domain is (, ) instead of [, ]? The graph no longer includes the endpoints of the segment. - - If the domain of a linear function consists of n elements, how man elements would there be in the range? Eplain. One, if the function is a constant function, or n if it is not. In a non-constant linear function, each element of the domain is paired with a different element of the range. 8. In Part B, what is the end behavior as increases without bound? Wh can t ou talk about the end behavior as decreases without bound? As +, f() -. Because the domain does not include values of that are less than or equal to -3, the values of cannot decrease without bound. Your Turn For the given function and domain, draw the graph and identif the range using the same notation as the given domain. 9. ƒ() = - 1_ + with domain - < 1. f() = _ -1 with domain (, 3] 3 Houghton Mifflin Harcourt Publishing Compan The range is 1 < 5. The range is (, 1]. Module 1 9 Lesson 1 LANGUAGE SUPPORT Connect Vocabular Have students work in pairs. Instruct one student to verball describe the domain, range, and end behavior of a function without using those three terms. Have the other student fill in a graphic organizer with boes titled Domain, Range, and End behavior, and write in the appropriate values under each bo. Have students switch roles and repeat the eercise using a different function. 9 Lesson 1.1

6 Eplain 3 Modeling with a Linear Function Recall that when a real-world situation involves a constant rate of change, a linear function is a reasonable model for the situation. The situation ma require restricting the function s domain. Eample 3 Write a function that models the given situation. Determine a domain from the situation, graph the function using that domain, and identif the range. Joce jogs at a rate of 1 mile ever 1 minutes for a total of minutes. (Use inequalities for the domain and range of the function that models this situation.) Joce s jogging rate is.1 mi/min. Her jogging distance d (in miles) at an time t (in minutes) is modeled b d(t) =.1t. Since she jogs for minutes, the domain is restricted to the interval t. Distance (mi) 3 1 d 1 3 Time (min) t EXPLAIN 3 Modeling with a Linear Function AVOID COMMON ERRORS Some students ma erroneousl identif the domain of a function that represents a real-world situation as an interval, when in fact the domain consists onl of specific numbers within the interval, such as integers or multiples of a particular rational number. Help students to avoid this error b encouraging them to ask themselves whether an number within the interval could be a realistic input value for the situation described b the function. The range is d. A candle inches high burns at a rate of 1 inch ever hours for 5 hours. (Use interval notation for the domain and range of the function that models this situation.) The candle s burning rate is -.5 in./h. The candle s height h (in inches) at an time t (in hours) is modeled b h(t) = -.5t. Since the candle burns for 5 hours, the domain is restricted to the interval, 5. The range is [3.5, ]. Height (in.) h t Time (h) Houghton Mifflin Harcourt Publishing Compan Image Credits: Bounce/ UpperCut Images/Gett Images INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Encourage students to check their work b considering the reasonableness of the range of the function. Have them evaluate whether the values in the range seem realistic for the given situation. Module 1 1 Lesson 1 Domain, Range, and End Behavior 1

7 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss different strategies for determining the range of a function from a graph of the function. Have students describe methods the use, and illustrate their methods using graphs of different tpes of functions, including those with restricted domains. QUESTIONING STRATEGIES Is it possible that a real-world situation can be modeled b a function whose domain consists of both positive and negative real numbers? If no, eplain wh not. If es, give an eample. Yes. Possible eample: a function that describes the relationship between air temperature in degrees Fahrenheit and dew point, at a given level of humidit. The domain consists of all possible air temperatures in degrees Fahrenheit. SUMMARIZE THE LESSON How do ou identif and represent the domain, range, and end behavior of a function? To identif the domain, find the values of for which the function is defined. To find the range, find the values of f () for each value of in the domain. There are different was of representing the domain and range, including using inequalities, set notation, and interval notation. To find the end-behavior, consider what happens to the values of the range as the values of the domain increase or decrease without bound. Houghton Mifflin Harcourt Publishing Compan Reflect 11. In Part A, suppose Joce jogs for onl 3 minutes. The domain is t 3 instead of t. A. How does the domain change? B. How does the graph change? C. How does the range change? Your Turn 1. While standing on a moving walkwa at an airport, ou are carried forward 5 feet ever 15 seconds for 1 minute. Write a function that models this situation. Determine the domain from the situation, graph the function, and identif the range. Use set notation for the domain and range. Elaborate The graph s right endpoint is (3, 3) instead of (, ). The range is d 3 instead of d. The walkwa s rate of motion is 5 ft/s. The distance d (in feet) 3 ou travel at an time t (in seconds) is modeled b d(t) = 5 3 t. Domain: {t t }; range: {d d 1}. 13. If a and b are real numbers such that a < b, use interval notation to write four different intervals having a and b as endpoints. Describe what numbers each interval includes. [a, b]: This interval includes all real numbers between a and b, including both a and b. 1. What impact does restricting the domain of a linear function have on the graph of the function? If the domain is bounded at both ends, the graph of the linear function is a line segment rather than a line. If the domain is bounded at onl one end, the graph of the linear function is a ra rather than a line. 15. Essential Question Check-In How does slope determine the end behavior of a linear function with an unrestricted domain? If slope is positive, the f()-values increase without bound as the -values increase without bound, and the f()-values decrease without bound as the -values decrease without bound. If the slope is negative, the end behavior reverses: The f()-values decrease without bound as the -values increase without bound, and the f()-values increase without bound as the -values decrease without bound. Distance on walkwa (ft) d Time (s) [a, b): This interval includes all real numbers between a and b, including a but not b. (a, b]: This interval includes all real numbers between a and b, including b but not a. (a, b): This interval includes all real numbers between a and b but does not include a or b. Module 1 11 Lesson 1 8 t 11 Lesson 1.1

8 Evaluate: Homework and Practice 1. Write the interval shown on the number line as an inequalit, using set notation, and using interval notation.. Write the interval (5, 1] as an inequalit and using set notation. 5 < 1; { 5 < 1} Online Homework Hints and Help Etra Practice EVALUATE , { 5}, [5, + ) 3. Write the interval -5 < 3 using set notation and interval notation. { -5 < 3}, [-5, 3). Write the interval { -3 < < 5} as an inequalit and using interval notation. Write the domain and the range of the function as an inequalit, using set notation, and using interval notation. Also describe the end behavior of the function or eplain wh there is no end behavior. 5. The graph of the quadratic function f () = + is shown. - Domain: < < +, { < < + }, (, + ) Range: < +, { < + },, + ) End behavior: As +, f() + ; As -, f() < < 5, (-3, 5). The graph of the eponential function f () = 3 is shown Domain: < < +, { < < + }, (, + ) Range: < < +, { < < + }, (, + ) End behavior: As +, f() + ; As -, f(). Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Representing an Interval on a Number Line Eample 1 Identifing a Function s Domain, Range, and End Behavior from its Graph Eample Graphing a Linear Function on a Restricted Domain Eample 3 Modeling with a Linear Function VISUAL CUES Practice Eercises 1 Eercises 5 8 Eercises 9 1 Eercises 11 1 Some students ma benefit from labeling the endpoints of an interval as included or not included, as indicated b the closed circle or open circle on the graph. The will then be sure to use the appropriate smbols when describing the interval using the different tpes of notation. Module 1 1 Lesson 1 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices CONNECT VOCABULARY Relate end behavior to the shape of the graph of different functions. Have students use words to describe the end behavior of each function b looking at the graph. For eample, a function ƒ () appears to rise for positive -values and fall for negative -values Recall of Information MP. Reasoning Recall of Information MP. Modeling 11 1 Recall of Information MP. Precision 1 Skills/Concepts MP. Modeling 13 Skills/Concepts MP.3 Logic 1 3 Strategic Thinking MP. Precision Domain, Range, and End Behavior 1

9 MULTIPLE REPRESENTATIONS When analzing the graph of a function, students ma find it easier to first describe the end behavior of the function in words. The then can translate their verbal descriptions into algebraic notation, making sure that the smbols accuratel reflect their descriptions. 7. The graph of the linear function g () = - is shown The graph of a function is shown. - - KINESTHETIC EXPERIENCE To help students correctl identif end behavior, suggest that the use a finger to trace along the graph of a function, moving first from left to right, as, and then from right to left, as -. Help them to match their observations of the behavior of the graph to its correct description and notation. Houghton Mifflin Harcourt Publishing Compan Domain: < < +, { < < + }, (, + ) Range: < < +, { < < + }, (, + ) End behavior: As +, f() + ; As -, f() -. Domain:, { }, [, ] Range:, { }, [, ] There is no end behavior because the domain is bounded at both ends. For the given function and domain, draw the graph and identif the range using the same notation as the given domain. 9. f() = + 5 with domain [ 3, ] 1. f() = _ with domain { > } Since f () = is a linear function, the graph is a line segment with endpoints at (-3, f (-3)), or ( 3, 8), and (, f()), or (, 3). The endpoints are included in the graph. The range is [3, 8]. Since f () = _ is a linear function, the graph is a ra with its endpoint at (-, f (-)), or (, ). The endpoint is not included in the graph. The range is { > - }. Module 1 13 Lesson 1 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 15 Skills/Concepts MP. Modeling 13 Lesson 1.1

10 Write a function that models the given situation. Determine the domain from the situation, graph the function using that domain, and identif the range. 11. A bicclist travels at a constant speed of 1 miles per hour for a total of 5 minutes. (Use set notation for the domain and range of the function that models this situation.) The bicclist s speed is 1 mi/h. The distance traveled d (in miles) at an time t (in hours) is modeled b d(t) = 1t. Since the bicclist travels for 5 minutes, or.75 hour, the domain is restricted to the interval {t t.75}. The range is {d d 9}. 1. An elevator in a tall building starts at a floor of the building that is 9 meters above the ground. The elevator descends meters ever.5 second for seconds. (Use an inequalit for the domain and range of the function that models this situation.) The elevator s rate of motion is m/s. The elevator s height h (in meters) at an time t (in seconds) is modeled b h(t) = 9 - t. Since the elevator descends for seconds, the domain is restricted to the interval t. The range is h 9. H.O.T. Focus on Higher Order Thinking 13. Eplain the Error Cameron sells tickets at a movie theater. On Frida night, she worked from p.m. to 1 p.m. and sold about 5 tickets ever hour. Cameron sas that the number of tickets, n, she has sold at an time t (in hours) can be modeled b the function n(t) = 5t, where the domain is t 1 and the range is n 5. Is Cameron s function, along with the domain and range, correct? Eplain. Cameron s function is correct, but the domain and range are incorrect. Cameron worked for a total of hours, so the domain of the function should be t. After hours, Cameron has sold 5 = 15 tickets. So, the range of the function should be n 15. Distance traveled (mi) d Height above ground (m) h Time (h) Time (s) t t Houghton Mifflin Harcourt Publishing Compan Image Credits: Caro/ Alam QUESTIONING STRATEGIES When is the graph of a linear function with a restricted domain a line segment? When is it a ra? It is a line segment when the domain is a closed interval. It is a ra when the domain is restricted to real numbers greater than or equal to a number, or less than or equal to a number. AVOID COMMON ERRORS When attempting to model a real-world situation, students sometimes confuse the dependent and independent variables, thus reversing the domain and range. Help them to understand that the domain consists of the values represented b the independent variable, and that the range values are a function of the values of the domain. Module 1 1 Lesson 1 Domain, Range, and End Behavior 1

11 PEER-TO-PEER DISCUSSION Is it possible that a linear function with the domain {all real numbers} could have a range that is not {all real numbers}? Eplain. Yes; the function could be a constant function, such as f () =. The domain is {all real numbers}, but the range is {}. You ma want to have students work on the modeling questions in pairs, thus providing them with an opportunit to discuss each situation with a partner, and decide how to best model the situation using a function. JOURNAL Have students describe how to identif the domain and range of a function given its graph, or given the situation it models. 1. Multi-Step The graph of the cubic function ƒ() = 3 is shown. a. What are the domain, range, and end behavior of the function? (Write the domain and range as an inequalit, using set notation, and using interval notation.) b. How is the range of the function affected if the domain is restricted to [, ]? (Write the range as an inequalit, using set notation, and using interval notation.) c. Graph the function with the restricted domain. a. Domain: < < +, { < < + }, (, + ) Range: < < +, { < < + }, (, + ) End behavior: As +, f() + ; As -, f() -. b. Restricted range: -, { +}, [-, ] 15. Represent Real-World Situations The John James Audubon Bridge is a cablestaed bridge in Louisiana that opened in 11. The height from the bridge deck to the top of the tower where a particular cable is anchored is about 5 feet, and the length of that cable is about 1 feet. Draw the cable on a coordinate plane, letting the -ais represent the bridge deck and the -ais represent the tower. (Onl use positive values of and.) Write a linear function whose graph models the cable. Identif the domain and range, writing each as an inequalit, using set notation, and using interval notation. c Houghton Mifflin Harcourt Publishing Compan Since the cable, the bridge deck, and the tower form a right triangle, find the length of the leg that lies on the -ais b using the Pthagorean Theorem. a + b = c 5 + b = 1 b slope: m = = The -intercept is 5. So, the linear function is = Height above bridge deck (ft) Distance from tower (ft) Domain:, 191, { 191}, [, 191] Range: 5, { 5}, [, 5] Module 1 15 Lesson 1 15 Lesson 1.1

12 Lesson Performance Task QUESTIONING STRATEGIES The fuel efficienc for a 7 passenger car was 31. mi/gal. For the same model of car, the fuel efficienc increased to 35. mi/gal in 1. The gas tank for this car holds 1 gallons of gas. a. Write and graph a linear function that models the distance that each car can travel for a given amount of gas (up to one tankful). b. Write the domain and range of each function using interval notation. c. Write and simplif a function ƒ (g) that represents the difference in the distance that the 1 car can travel and the distance that the 7 car can travel on the same amount of gas. Interpret this function using the graphs of the functions from part a. Also find and interpret ƒ (1). d. Write the domain and range of the difference function using set notation. Distance traveled (mi) d a. For both cars, let g be the amount of gas (in gallons) that each car uses, and let d be the distance (in miles) that each car travels. For the 7 car, the linear model is d 7 (g) = 31.g. For the 1 car, the linear model is d 1 (g) = 35.g. b. The domain for the 7 model is [, 1], and the range is [, 99.]. The domain for the 1 model is [, 1], and the range is [, 59.]. 1 c. The difference function is f(g) = d 1 (g) d 7 (g) = 35.g 31.g =.g. This function gives the vertical distance between the graphs of d 1 (g) and d 7 (g). For instance, when g = 1, the vertical distance between the graphs is f(1) =. 1 = 7., which means the 1 car can travel 7. miles farther on a tankful of gas than the 7 car. d. The domain is {g g 1}, and the range is {f(g) f(g) 7.} Gas used (gal) 1 g Houghton Mifflin Harcourt Publishing Compan Wh must restrictions be placed on the domain? The tank onl holds at most 1 gallons of gas and at least gallons of gas. A graph with a negative number of gallons of gas or more than 1 gallons of gas would not make sense for this situation. Wh does the domain contain all of the points in the interval and not just the integer values? The amount of gas used is continuous. There is a distance traveled for an value of g in the domain. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Encourage students to relate the ranges that the wrote for the difference functions to their graphs. Have them determine whether the values in the range make sense in the real-world situation. Then have students eplain whether it makes sense that as the number of gallons used increases, the difference in miles traveled b the two cars also increases. Module 1 1 Lesson 1 EXTENSION ACTIVITY Have students research the average fuel costs per gallon in 7 and in 1. Have students use the data to create a new graph representing the distance each car could travel for a given amount of mone (up to the cost of a tpical full tank). Ask students to describe the aspects of their graphs. 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. points: Student does not demonstrate understanding of the problem. Domain, Range, and End Behavior 1