5.2 Graphing Polynomial Functions

Transcription

1 Locker LESSON 5. Graphing Polnomial Functions Common Core Math Standards The student is epected to: F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing end behavior. Also A-SSE.1a, A-APR.3, F-IF.4 Mathematical Practices MP.7 Using Structure Language Objective Work with a partner or small group to match function tpes to their equations or definitions. Name Class Date 5. Graphing Polnomial Functions Essential Question: How do ou sketch the graph of a polnomial function in intercept form? Eplore 1 Investigating the End Behavior of the Graphs of Simple Polnomial Functions Linear, quadratic, and cubic functions belong to a more general class of functions called polnomial functions, which are categorized b their degree. Linear functions are polnomial functions of degree 1, quadratic functions are polnomial functions of degree, and cubic functions are polnomial functions of degree 3. In general, a polnomial function of degree n has the standard form p () = a n n + a n-1 n a + a 1 + a 0, where a n, a n-1,..., a, a 1, and a 0 are real numbers called the coefficients of the epressions a n n, a n-1 n - 1,..., a, a 1, and a 0, which are the terms of the polnomial function. (Note that the constant term, a 0, appears to have no power of associated with it, but since 0 = 1, ou can write a 0 as a 0 0 and treat a 0 as the coefficient of the term.) A polnomial function of degree 4 is called a quartic function, while a polnomial function of degree 5 is called a quintic function. After degree 5, polnomial functions are generall referred to b their degree, as in a sith-degree polnomial function. A Use a graphing calculator to graph the polnomial functions ƒ () =, ƒ () =, ƒ () = 3, ƒ () = 4, ƒ () = 5, and ƒ () = 6. Then use the graph of each function to determine the function s domain, range, and end behavior. (Use interval notation for the domain and range.) Resource Locker ENGAGE Essential Question: How do ou sketch the graph of a polnomial function in intercept form? Determine the end behavior based on the degree and whether or not the function includes a negative constant factor. Identif and plot the -intercepts. Use the factors to determine the sign of the function s values on the intervals determined b the -intercepts. Sketch the graph b showing where it crosses the -ais, where it is tangent to the -ais, and roughl where turning points occur. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo, asking the students to list the known quantities for the sheet of cardboard and the unknown variables for the bo. Then preview the Lesson Performance Task. Function Domain Range End Behavior f () = As, f (). As -, f () -. f () = [0, ) As, f (). As -, f (). f () = 3 As, f (). As -, f () -. f () = 4 [0, ) As, f (). As -, f (). f () = 5 As, f (). As -, f () -. f () = 6 [0, ) As, f (). As -, f (). Module 5 49 Lesson Name Class Date 5. Graphing Polnomial Functions Essential Question: How do ou sketch the graph of a polnomial function in intercept form? F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing end behavior. Also A-SSE.1a, A-APR.3, F-IF.4 Eplore 1 Investigating the End Behavior of the Graphs of Simple Polnomial Functions Linear, quadratic, and cubic functions belong to a more general class of functions called polnomial functions, which are categorized b their degree. Linear functions are polnomial functions of degree 1, quadratic functions are polnomial functions of degree, and cubic functions are polnomial functions of degree 3. In general, a polnomial function of degree n has the standard form p () = a n n + a n-1 n a + a 1 + a 0, where a n, a n-1,..., a, a 1, and a 0 are real numbers called the coefficients of the epressions a n n, a n-1 n - 1,..., a, a 1, and a 0, which are the terms of the polnomial function. (Note that the constant term, a 0, appears to have no power of associated with it, but since 0 = 1, ou can write a 0 as a 0 0 and treat a 0 as the coefficient of the term.) A polnomial function of degree 4 is called a quartic function, while a polnomial function of degree 5 is called a quintic function. After degree 5, polnomial functions are generall referred to b their degree, as in a sith-degree polnomial function. Use a graphing calculator to graph the polnomial functions ƒ () =, ƒ () =, ƒ () = 3, ƒ () = 4, ƒ () = 5, and ƒ () = 6. Then use the graph of each function to determine the function s domain, range, and end behavior. (Use interval notation for the domain and range.) f () = f () = 3 f () = 4 f () = 5 f () = 6 f () = As, f (). As -, f (). As, f (). As -, f (). Resource Function Domain Range End Behavior - [0, ) - [0, ) - [0, ) As, f (). As -, f (). As, f (). As -, f (). As, f (). As -, f (). As, f (). As -, f (). Module 5 49 Lesson HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 49 Lesson 5.

2 B Use a graphing calculator to graph the polnomial functions ƒ () = -, ƒ () = -, ƒ () = - 3, ƒ () = - 4, ƒ () = - 5, and ƒ () = - 6. Then use the graph of each function to determine the function s domain, range, and end behavior. (Use interval notation for the domain and range.) Function Domain Range End Behavior f () = - As, f () -. As -, f (). EXPLORE 1 Investigating the End Behavior of the Graphs of Simple Polnomial Functions Reflect f () = - f () = - 3 f () = - 4 f () = - 5 f () = - 6 As, f () -. As -, f () -. As, f () -. As -, f (). As, f () -. As -, f () -. As, f () -. As -, f (). As, f () -. As -, f () How can ou generalize the results of this Eplore for ƒ () = n and ƒ () = - n where n is positive whole number? The domains of the functions f () = n and f () = - n are both. The ranges depend on whether n is even or odd. If n is odd, the ranges of f () = n and f () = - n are both. If n is even, the range of f() = n is [0, ) while the range of f () = - n is (-, 0]. The end behavior of f () = - n is alwas the opposite of the end behavior of f () = n. (-, 0] (-, 0] (-, 0] INTEGRATE TECHNOLOGY Students have the option of completing the graphing calculator activit either in the book or online. QUESTIONING STRATEGIES Suppose f() as -. What does this statement tell ou about the graph of f() = n and the direction in which ou are moving on the -ais? What can ou sa about the value of n? The statement sas that as ou move to the left along the negative -ais so decreases without bound, the graph of f () = n rises without bound. This is true of the graph of f () = n when n is even. Does an function of the form f() = n have end behavior where f() - as? Eplain. No; this would mean that as ou move right along the positive -ais, the graph of f () = n falls without bound, which is not true of the graph of f () = n for all values of n. Module 5 50 Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP.7, which calls for students to look for and make use of structure. Students analze some of the attributes of polnomial functions. Specificall, the eamine domain, range, intercepts, turning points, and end behavior. The also investigate the -intercepts of graphs of polnomial functions, and how the relate to the factored form of the related polnomial epression. Students also analze polnomial functions in real-world contets. Graphing Polnomial Functions 50

3 EXPLORE Investigating the -intercepts and Turning Points of the Graphs of Polnomial Functions CONNECT VOCABULARY Relate turning point to driving a car north and then turning the car to drive south. Eplain that the turning point is the point at which the car reverses direction. Eplore Investigating the -intercepts and Turning Points of the Graphs of Polnomial Functions The cubic function ƒ () = 3 has three factors, all of which happen to be. One or more of the s can be replaced with other linear factors in, such as -, without changing the fact that the function is cubic. In general, a polnomial function of the form p () = a ( - 1 ) ( - )... ( - n ) where a, 1,,..., and n are real numbers (that are not necessaril distinct) has degree n where n is the number of variable factors. The graph of p () = a ( - 1 ) ( - )... ( - n ) has 1,,..., and n as its -intercepts, which is wh the polnomial is said to be in intercept form. Since the graph of p() intersects the -ais onl at its -intercepts, the graph must move awa from and then move back toward the -ais between each pair of successive -intercepts, which means that the graph has a turning point between those -intercepts. Also, instead of crossing the -ais at an -intercept, the graph can be tangent to the -ais, and the point of tangenc becomes a turning point because the graph must move toward the -ais and then awa from it near the point of tangenc. The -coordinate of each turning point is a maimum or minimum value of the function at least near that turning point. A maimum or minimum value is called global or absolute if the function never takes on a value that is greater than the maimum or less than the minimum. A local maimum or local minimum, also called a relative maimum or relative minimum, is a maimum or minimum within some interval around the turning point that need not be (but ma be) a global maimum or global minimum. A Use a graphing calculator to graph the cubic functions ƒ () = 3, ƒ () = ( - ), and ƒ () = ( - ) ( + ). Then use the graph of each function to answer the questions in the table. Function f () = 3 f () = ( - ) f () = ( - ) ( + ) How man distinct factors does f () have? 1 3 What are the graph s -intercepts? 0 0, 0,, - Is the graph tangent to the -ais or does it cross the -ais at each -intercept? How man turning points does the graph have? Crosses at = 0 Tangent at = 0; crosses at = Crosses at all three -intercepts How man global maimum values? How man local maimum values that are not global? How man global minimum values? How man local minimum values that are not global? 0 No maimum values No minimum values No global maimum values, but one local maimum value No global minimum values, but one local minimum value No global maimum values, but one local maimum value No global minimum values, but one local minimum value Module 5 51 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Instruct one student in each pair to sketch the graph of a polnomial function in intercept form while the other gives verbal instructions for each step. Then have the student who sketched the graph write the steps that were followed. Have students switch roles and repeat the eercise using a different polnomial function in intercept form. Encourage students to displa and use a sign table for their graphs. 51 Lesson 5.

4 B Use a graphing calculator to graph the quartic functions ƒ () = 4, ƒ () = 3 ( - ), ƒ () = ( - ) ( + ), and ƒ () = ( - ) ( + ) ( + 3).Then use the graph of each function to answer the questions in the table. Function f () = 4 f () = 3 ( - ) How man distinct factors? What are the -intercepts? Tangent to or cross the -ais at -intercepts? How man turning points? How man global maimum values? How man local maimum values that are not global? How man global minimum values? How man local minimum values that are not global? 1 0 Tangent at = 0 1 No maimum values One global minimum value and no local minimum values 0, Crosses at both -intercepts No maimum values One global minimum value and no local minimum values f () = ( - ) ( + ) 3 0,, - Tangent at = 0; crosses at = and = - 3 No global maimum values, but one local maimum value One global minimum value (which occurs twice) and no local minimum values f () = ( - ) ( + ) ( + 3) 4 0,, -, -3 Crosses at all four -intercepts 3 No global maimum values, but one local maimum value One global minimum value and one local minimum value QUESTIONING STRATEGIES How is the number of variable factors of a polnomial function related to the degree of the polnomial? The are equal. Where are the turning points of a polnomial function located? How are the local maima and minima related to the turning points? Between consecutive pairs of -intercepts; the local maima and minima occur at the turning points. Reflect. What determines how man -intercepts the graph of a polnomial function in intercept form has? Each distinct factor produces one -intercept. 3. What determines whether the graph of a polnomial function in intercept form crosses the -ais or is tangent to it at an -intercept? If the factor that produces the -intercept is raised to an odd power, the graph will cross the -ais. If the factor is raised to an even power, the graph will be tangent to the -ais. 4. Suppose ou introduced a factor of -1 into each of the quartic functions in Step B. (For instance, ƒ () = 4 becomes ƒ () = - 4.) How would our answers to the questions about the functions and their graphs change? Since each function s graph would be reflected across the -ais, a maimum value would become a minimum value and vice versa. This would change the answers to the questions about global and local maimum and minimum values. For instance, f () = 4 has no maimum values, one global minimum value, and no local minimum values that are not global, whereas f () = - 4 would have one global maimum value, no local maimum values that are not global, and no minimum values. Module 5 5 Lesson DIFFERENTIATE INSTRUCTION Visual Clues When discussing local maima and minima, have students cover irrelevant parts of the graph with a sheet of paper to help them focus on the local etreme value of interest. Cognitive Strategies Suggest that all students create reference cards that show the general shapes of polnomial functions of degrees through 5. Students can think of the cards as a vocabular of graphs and use them when graphing polnomial functions. Graphing Polnomial Functions 5

5 EXPLAIN 1 Sketching the Graph of Polnomial Functions in Intercept Form AVOID COMMON ERRORS To determine end behavior, students can multipl all factors, but the ma make errors in the comple calculation. Point out that end behavior is determined b the term with the greatest degree. If a function is written in factored form, then the highest-degree term is the product of all the first terms of the factors, as long as an repeated factors are considered individuall. So, if f() = (3 + 1) ( - 1) = (3 + 1)(3 + 1)( - 1), then the leading term is (3)(3)() = 9 3, and the polnomial has the same end behavior as f() = 3. Eplain 1 Sketching the Graph of Polnomial Functions in Intercept Form Given a polnomial function in intercept form, ou can sketch the function s graph b using the end behavior, the -intercepts, and the sign of the function values on intervals determined b the -intercepts. The sign of the function values tells ou whether the graph is above or below the -ais on a particular interval. You can find the sign of the function values b determining the sign of each factor and recognizing what the sign of the product of those factors is. Eample 1 Sketch the graph of the polnomial function. ƒ () = ( + ) ( 3) Identif the end behavior. For the function p () = a ( 1 ) ( )... ( n ), the end behavior is determined b whether the degree n is even or odd and whether the constant factor a is positive or negative. For the given function f(), the degree is 3 and the constant factor a, which is 1, is positive, so ƒ() has the following end behavior: As, ƒ(). As -, ƒ() -. Identif the graph s -intercepts, and then use the sign of ƒ() on intervals determined b the -intercepts to find where the graph is above the -ais and where it s below the -ais. The -intercepts are = 0, = -, and = 3. These three -intercepts divide the -ais into four intervals: < -, - < < 0, 0 < < 3, and > 3. Interval Sign of the Constant Factor Sign of Sign of + Sign of - 3 Sign of f () = ( +) ( - 3) < < < < < > So, the graph of ƒ () is above the -ais on the intervals - < < 0 and > 3, and it s below the -ais on the intervals < - and 0 < < 3. Sketch the graph. While ou should be precise about where the graph crosses the -ais, ou do not need to be precise about the -coordinates of points on the graph that aren t on the -ais. Your sketch should simpl show where the graph lies above the -ais and where it lies below the -ais Module 5 53 Lesson LANGUAGE SUPPORT Connect Vocabular Distribute note cards to students. Have half the students write the names of the tpes of functions addressed in this lesson, such as cubic, quartic, quadratic, and linear, writing one name per card. Have the other half of the students write a definition (a function with a degree of 3, for eample, or an equation) on each card. Place the name cards in one pile and the definition cards in another. Students shuffle the cards, place them face down, and take turns turning over one card from each pile. If the have a match, the must eplain to the rest of the group wh it is a match. 53 Lesson 5.

6 B ƒ () = ( 4) ( 1) ( + 1) ( + ) Identif the end behavior. As, ƒ() -. As -, ƒ() -. Identif the graph s -intercepts, and then use the sign of ƒ () on intervals determined b the -intercepts to find where the graph is above the -ais and where it s below the -ais. The -intercepts are = -, = -1, = 1, = 4. Interval Sign of the Constant Factor Sign of - 4 Sign of - 1 Sign of + 1 Sign of + < Sign of f () = - ( - 4) ( - 1) ( + 1) ( + ) - QUESTIONING STRATEGIES When sketching the graph of a polnomial function in intercept form, how do ou know when the graph is tangent to the -ais? When the same zero values occur an even number of times in the factorization of the polnomial, the graph of the function is tangent to the -ais at that value. What are ou finding when ou find f (0)? the graph s -intercept - < < < < < < > So, the graph of ƒ () is above the -ais on the intervals - < < -1 and 1 < < 4, and it s below the -ais on the intervals < -, -1 < < 1, and > Sketch the graph. Module 5 54 Lesson Graphing Polnomial Functions 54

7 EXPLAIN Modeling with a Polnomial Function INTEGRATE TECHNOLOGY On a graphing calculator, students can enter a polnomial function in standard or modified intercept form. For eample, a cubic function can be entered using the general form f () = ( a 1 + b 1 )(a + b )(a 3 + b 3 ), which is not quite intercept form. Your Turn Sketch the graph of the polnomial function. 5. ƒ () = - ( - 4) As, f () -. As -, f (). The -intercepts are = 0 and = 4. Interval Sign of the Constant Factor Sign of Sign of - 4 Sign of f () = ( - 4) < < < > So, the graph of f () is above the -ais on the intervals < 0 and 0 < < 4, and it s below the -ais on the interval > Eplain Modeling with a Polnomial Function You can use cubic functions to model real-world situations. For eample, ou find the volume of a bo (a rectangular prism) b multipling the length, width, and height. If each dimension of the bo is given in terms of, then the volume is a cubic function of. Eample To create an open-top bo out of a sheet of cardboard that is 9 inches long and 5 inches wide, ou make a square flap of side length inches in each corner b cutting along one of the flap s sides and folding along the other side. (In the first diagram, a solid line segment in the interior of the rectangle indicates a cut, while a dashed line segment indicates a fold.) After ou fold up the four sides of the bo (see the second diagram), ou glue each flap to the side it overlaps. To the nearest tenth, find the value of that maimizes the volume of the bo. 9 in. 5 in. Module 5 55 Lesson 55 Lesson 5.

8 Analze Information Identif the important information. A square flap of side length inches is made in each corner of a rectangular sheet of cardboard. The sheet of cardboard measures 9 inches b 5 inches. Formulate a Plan Find the dimensions of the bo once the flaps have been made and the sides have been folded up. Create a volume function for the bo, graph the function on a graphing calculator, and use the graph to find the value of that maimizes the volume. Solve 1. Write epressions for the dimensions of the bo. Length of bo: 9 - Width of bo: 5 - Height of bo: QUESTIONING STRATEGIES Wh do the constraints on for length and width not simpl require that be nonnegative? The constraints on describe those values of that make epressions for length and width nonnegative values. How can ou generalize from this situation to finding the domain for an volume function? Since length, height, and width will alwas be nonnegative, the domain of a volume function will alwas require that the independent variable take on onl those values that make each dimension nonnegative.. Write the volume function and determine its domain. V () = (9 - )(5 - ) Because the length, width, and height of the bo must all be positive, the volume function s domain is determined b the following three constraints: 9 - > 0, or < > 0, or <.5 > 0 Taken together, these constraints give a domain of 0 < < Use a graphing calculator to graph the volume function on its domain. Adjust the viewing window so ou can see the maimum. From the graphing calculator s CALC menu, select 4: maimum to locate the point where the maimum value occurs. So, V () 1.0 when 1.0, which means that the bo has a maimum volume of about 1 cubic inches when square flaps with a side length of 1 inch are made in the corners of the sheet of cardboard. Module 5 56 Lesson Graphing Polnomial Functions 56

9 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Before students graph an polnomial functions, encourage them to predict how the graph should look based on the degree and end behavior of the function. If the function is in intercept form, encourage students to predict the number and location of the turning points. Justif and Evaluate Making square flaps with a side length of 1 inch means that the bo will be 7 inches long, 3 inches wide, and 1 inch high, so the volume is 1 cubic inches. As a check on this result, consider making square flaps with a side length of 0.9 inch and 1.1 inches: V (0.9) = (9-1.8) (5-1.8) (0.9) = V (1.1) = (9 -.) (5 -.) (1.1) = Both volumes are slightl less than 1 cubic inches, which suggests that 1 cubic inches is the maimum volume. Reflect Discussion Although the volume function has three constraints on its domain, the domain involves onl two of them. Wh? All three inequalities must be satisfied simultaneousl. An -value that satisfies <.5 also satisfies < 4.5, so the constraint < 4.5 has no impact on the domain. QUESTIONING STRATEGIES What is the degree of an volume function? Eplain. Three; geometricall, volume is related to a figure with three dimensions, so it will be represented b a function of degree 3. SUMMARIZE THE LESSON What are some of the ke attributes of a polnomial function of degree n that ou can determine from the graph of the function? You can determine the -intercepts and therefore the real zeros of the function, and the approimate location of the turning points. You can also determine the maimum and minimum values, the end behavior, and whether the degree of the leading coefficient is even or odd, and positive or negative. Your Turn 7. To create an open-top bo out of a sheet of cardboard that is 5 inches long and 13 inches wide, ou make a square flap of side length inches in each corner b 13 in. cutting along one of the flap s sides and folding along the other. (In the diagram, a solid line segment in the interior of the rectangle indicates a cut, while a dashed line segment indicates a fold.) Once ou fold up the four sides of the 5 in. bo, ou glue each flap to the side it overlaps. To the nearest tenth, find the value of that maimizes the volume of the bo. The length of the bo is 5 -, the width is 13 -, and the height is. So, the volume function is V () = (5 - ) (13 - ) with a domain of 0 < < 6.5 determined b the constraints that all three dimensions of the bo must be nonnegative. Maimum volume is about 40 cubic inches when square flaps with a side length of.7 inches are made in the corners. Elaborate 8. Compare and contrast the domain, range, and end behavior of ƒ () = n when n is even and when n is odd. The domains are alwas the same, but when n is even, the range contains onl numbers greater than or equal to 0, whereas when n is odd, the range is all real numbers. When n is even, f () approaches as approaches both - and, whereas when n is odd, f () approaches - as approaches - and f () approaches as approaches. 9. Essential Question Check-In For a polnomial function in intercept form, wh is the constant factor important when graphing the function? The sign of the constant factor has an impact on the end behavior of the function. It also has an impact on whether the -coordinates of turning points represent maimum or minimum values. Module 5 57 Lesson 57 Lesson 5.

10 Evaluate: Homework and Practice Use a graphing calculator to graph the polnomial function. Then use the graph to determine the function s domain, range, and end behavior. (Use interval notation for the domain and range.) 1. ƒ () = 7. ƒ () = - 9 Domain: Range: End behavior: As, f () +. As -, f () ƒ () = ƒ () = - 8 Domain: Range: [0, ) End behavior: As, f (). As -, f (). Domain: Range: Use a graphing calculator to graph the function. Then use the graph to determine the number of turning points and the number and tpe (global, or local but not global) of an maimum or minimum values. 5. ƒ () = ( + 1) ( + 3) 6. ƒ () = ( + 1) ( - 1) ( - ) The graph has two turning points. The function has one local maimum value and one local minimum value. Online Homework Hints and Help Etra Practice End behavior: As, f () -. As -, f (). Domain: (-, ) Range: (-, 0] End behavior: As, f () -. As -, f () -. The graph has three turning points. The function has one local maimum value, one global minimum value, and one local minimum value. EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore 1 Investigating the End Behavior of the Graphs of Simple Polnomial Functions Eplore Investigating the -intercepts and Turning Points of the Graphs of Polnomial Functions Eample 1 Sketching the Graph of Polnomial Functions in Intercept Form Eample Modeling with a Polnomial Function Practice Eercises 1 4 Eercises 5 8 Eercises 9 11 Eercises ƒ () = -( - ) The graph has two turning points. The function has one local maimum value and one local minimum value ƒ () = -( - 1) ( + ) The graph has one turning point. The function has one global maimum value. Module 5 58 Lesson INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Students should recognize that the characteristics of the epression that defines a polnomial function determine the nature, and thus the attributes, of the graph of the function. Have students use grid paper and/or their graphing calculators to eplore how different tpes of polnomial functions produce graphs with similar and differing attributes. Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1-8 Skills/Concepts MP. Reasoning 9 11 Skills/Concepts MP.4 Modeling Strategic Thinking MP.5 Using Tools Strategic Thinking MP.4 Modeling 18 Skills/Concepts MP. Reasoning 19 3 Strategic Thinking MP.3 Logic Graphing Polnomial Functions 58

11 AVOID COMMON ERRORS Students ma have difficult identifing the leading coefficient and therefore the epected end behavior of a polnomial function. Emphasize that students must either write the polnomial in standard form with the highest power first, or multipl the variable terms of the factors of the polnomial, if the polnomial is epressed in intercept form. Sketch the graph the polnomial function. 9. ƒ () = ( - ) -4-0 As, f (). As -, f () -. The -intercepts are = 0 and =. Interval Sign of f () = ( - ) < 0-0 < < - > + So, the graph of f () is above the -ais on the interval >, and it s below the -ais on the intervals < 0 and 0 < <. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Suggest that students make a flow chart illustrating the process of graphing a factorable polnomial function. The can choose one polnomial function from this lesson and describe how each step would be applied. 10. ƒ () = - ( + 1) ( - ) ( - 3) -4-0 As, f () -. As -, f (). The -intercepts are = -1, = and = 3. Interval Sign of f () = - ( + 1) ( - ) ( - 3) < < < - < < 3 + > 3 - So, the graph of f () is above the -ais on the intervals < -1 and < < 3, and it s below the -ais on the intervals -1 < < and > ƒ () = ( + ) ( - 1) -4-0 As, f (). As -, f (). The -intercepts are = -, = 0, and = 1. Interval Sign of f () = ( + ) ( - 1) < < < < < 1 - > 1 + So, the graph of f () is above the -ais on the intervals < -, - < < 0, and > 1, it s below the -ais on the intervals 0 < < 1. Module 5 59 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 0 3 Strategic Thinking MP. Reasoning 1 3 Strategic Thinking MP.4 Modeling 59 Lesson 5.

12 1. To create an open-top bo out of a sheet of cardboard that is 6 inches long and 3 inches wide, ou make a square flap of side length inches in each corner b cutting along one of the flap s sides and folding along the other. Once ou fold up the four sides of the bo, ou glue each flap to the side it overlaps. To the nearest tenth, find the value of that maimizes the volume of the bo. The length of the bo is 6 -, the width is 3 -, and the height is. So, the volume function is V () = (6 - ) (3 - ) with a domain of 0 < < 1.5. Maimum volume is 5. cubic inches when square flaps with a side length of 0.6 inch are made in the corners. 13. The template shows how to create a bo from a square sheet of cardboard that has a side length of 36 inches. In the template, solid line segments indicate cuts, dashed line segments indicate folds, and graed rectangles indicate pieces removed. The vertical strip that is inches wide on the left side of the template is a flap that will be glued to the side of the bo that it overlaps when the bo is folded up. The horizontal strips that are inches wide at the top and bottom of the template are also flaps that will overlap to form the top and bottom of the bo when the bo is folded up. Write a volume function for the bo in terms of onl. (You will need to determine a relationship between and first.) Then, to the nearest tenth, find the dimensions of the bo with maimum volume. 6 in. in. 36 in. To find the relationship between and, use that fact that = 36, so + = 17, or = Then the dimensions of the bo are, 17 -, and 36 - ( _ ), or The volume function is V () = (17 - ) (36 - ). The domain of the function is determine b the constraints > 0; 17 - > 0, or < 17; and 36 - > 0, or < 36. So, the domain of the function is 0 < < 17. Using the graphing calculator to locate the graph s highest point on the interval (0, 17), ou find that the bo has a maimum volume of about 03 cubic inches when the dimensions of the bo are 7.3 inches, 9.7 inches, and 8.7 inches. 3 in. 36 in. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Suggest that students work in small groups to discuss wh the sign of the leading coefficient affects the end behavior of a polnomial function, and how the degree of the polnomial is related to the end behavior. Module 5 60 Lesson Graphing Polnomial Functions 60

13 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Students ma have difficult when asked to graph a polnomial function. This ma be because the do not have an overall sense of how the graph should look, or because the do not graph enough points between the zeros to accuratel represent the function. Suggest that the determine the end behavior before starting a graph, and then choose at least two points between each zero to sketch the graph. Write a cubic function in intercept form for the given graph, whose -intercepts are integers. Assume that the constant factor a is either 1 or The -intercepts are = -3, = -1, and = 1. The related factors are + 3, + 1, and - 1. Since there are three factors and the function is cubic, each factor must be raised to the first power. So, the general function is f () = a ( + 3) ( + 1) ( -1) for some constant factor a. Given the function s end behavior, a must be positive. So, the specific function with a = 1 is f () = ( + 3) ( + 1) ( -1) Write a quartic function in intercept form for the given graph, whose -intercepts are integers. Assume that the constant factor a is either 1 or -1. The -intercepts are = - and = 3. The related factors are + and - 3. Since there are onl two factors and the function is cubic, one of the factors must be squared. Given that the graph is tangent to the -ais at = -, the factor + must be squared. So, the general function is f () = a ( + ) ( - 3) for some constant factor a. Given the function s end behavior, a must be negative. So, the specific function with a = -1 is f () = - ( + ) ( - 3) The -intercepts are = -3, = 0, =, and = 4. The related factors are + 3,, -, and - 4. Since there are four factors and the function is quartic, each factor must be raised to the first power. So, the general function is f () = a ( + 3) ( - ) ( - 4) for some constant factor a. Given the function s end behavior, a must be positive. So, the specific function with a = 1 is f () = ( + 3) ( - ) ( - 4) The -intercepts are = - and = 3. The related factors are + and - 3. Since there are onl two factors and the function is quartic, one or both factors must be raised to a power other than 1. Given that the graph is tangent to the -ais at both = - and = 3, both the factor + and the factor - 3 must be squared. So, the general function is f () = a ( + ) ( - 3) for some constant factor a. Given the function s end behavior, a must be negative. So, the specific function with a = -1 is f () = - ( + ) ( - 3). Module 5 61 Lesson 61 Lesson 5.

14 18. Multiple Response Select all statements that appl to the graph of ƒ () = ( - 1) ( + ). A. The -intercepts are = 1 and = -. B. The -intercepts are = -1 and =. C. The graph crosses the -ais at = 1 and is tangent to the -ais at = -. D. The graph crosses the -ais at = -1 and is tangent to the -ais at =. E. The graph is tangent to the -ais at = 1 and crosses the -ais at = -. F. The graph is tangent to the -ais at = -1 and crosses the -ais at =. AVOID COMMON ERRORS Students ma write the wrong power for a factor of a polnomial function. Eplain that if the know the multiplicit of a zero, the automaticall know the power of the corresponding factor. If the are told the function is cubic or quartic, then the know that the power is 3 or 4. G. A local, but not global, minimum occurs on the interval - < < 1, and a local, but not global, maimum occurs at = 1. H. A local, but not global, maimum occurs on the interval - < < 1, and a local, but not global, minimum occurs at = 1. I. A local, but not global, minimum occurs on the interval -1 < <, and a local, but not global, maimum occurs at =. J. A local, but not global, maimum occurs on the interval -1 < <, and a local, but not global, minimum occurs at =. H.O.T. Focus on Higher Order Thinking 19. Eplain the Error A student was asked to sketch the graph of the function ƒ () = ( - 3). Describe what the student did wrong. Then sketch the correct graph The student sketched the graph so that it crosses the -ais at = 0 and is tangent to the -ais at = 3. Instead, the graph should be tangent to the -ais at = 0 and cross the -ais at = 3. Module 5 6 Lesson Graphing Polnomial Functions 6

15 CONNECT VOCABULARY To help students remember the words associated with the attributes of a polnomial function, have them make note cards for each attribute, including cards for domain, range, -intercepts, turning points, maima and minima, and end behavior. Ask them to write a description of the attribute, show an eample polnomial function and its graph, and list an other attribute that this particular graph ma also have. Make sure the use the proper notation for end behavior in their descriptions. Then have students make a poster showing the graph of a polnomial of degree three or higher with all of the applicable attributes included as labels on the graph. 0. Make a Prediction Knowing the characteristics of the graphs of cubic and quartic functions in intercept form, sketch the graph of the quintic function ƒ () = ( + ) ( - ) Represent Real-World Situations A rectangular piece of sheet metal is rolled and riveted to form a circular tube that is open at both ends, as shown. The sheet metal has a perimeter of 36 inches. Each of the two sides of the rectangle that form the two ends of the tube has a length of inches, but the tube has a circumference of - 1 inches because an overlap of 1 inch is needed for the rivets. Write a volume function for the tube in terms of. Then, to the nearest tenth, find the value of that maimizes the volume of the tube. 4 PEER-TO-PEER DISCUSSION Have students work in pairs. Instruct one student in each pair to sketch the graph of a cubic function of the form f () = ( + b 1 ) ( + b ) ( + b 3 ), while the other gives verbal instructions for each step. Then have the student who sketched the graph write the steps that were followed and determine the attributes of the graph. Have students switch roles and repeat the eercise using a quartic function of the form f ( ) = ( + b 1 ) ( + b ) ( + b 3 ) ( + b 4 ). JOURNAL Have students describe how the attributes of a polnomial function are determined from the function written in intercept form, and how information about these attributes is helpful in drawing the graph of the function. - 1 Given that represents one dimension of the rectangle, let represent the other dimension. Since the perimeter of the rectangle is 36 inches, ou know that + = 36, so + = 18, and = Since - 1 represents the circumference of the tube, ou know that πr = - 1 where r is the radius of the tube, so r = - 1. Since the π tube is a clinder with radius r and height, the volume function is V () = π r ( - 1) = π ( 4 π )(18 - ) = 1 π ( - 1) (18 - ). The domain of the function is determined b the constraints > 0; - 1 > 0, or > 1; and 18 - > 0, or < 18. So, the domain of the function is 1 < < 18. Using the graphing calculator to locate the graph s highest point, ou find that the tube has a maimum volume of about 57.9 cubic inches when the length of the sides of the rectangle that form the ends of the tube is 1.3 inches. Module 5 63 Lesson 63 Lesson 5.

16 Lesson Performance Task The template shows how to create a bo with a lid from a sheet of card stock that is 10 inches wide and 4 inches long. In the template, solid line segments indicate cuts, and dashed line segments indicate folds. The square flaps, each with a side length of inches, are glued to the sides the overlap when the bo is folded up. The bo has a bottom and four upright sides. The lid, which is attached to one of the upright sides, has three upright sides of its own. Assume that the three sides of the lid can be tucked inside the bo when the lid is closed. 4 in. 10 in. a. Write a polnomial function that represents the volume of the bo, and state its domain. b. Use a graphing calculator to find the value of that will produce the bo with maimum volume. What are the dimensions of that bo? a. The volume function is V () = (10 - ) ( 1_ (4-3) ), or V () = (10 - ) (1-3_ ). The domain of the function is determined b the constraints > 0; 10 - > 0, or < 5; and 1-3_ > 0, or < 8. So, the domain of the function is 0 < < 5. b. Using a graphing calculator to graph the function and locate the graph s highest point on the interval (0, 5), ou find that the bo has a maimum volume of 108 cubic inches when = so that the dimensions of the bo are inches, 6 inches, and 9 inches. CONNECT VOCABULARY Some students ma not be familiar with the term card stock. Have a student volunteer describe card stock (a sturd paper used to make cards) and compare it to other materials such as cardboard or paper. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Ask students how the can determine the length, width, and height of the bo from the twodimensional diagram. Have them draw a 3D picture of the final bo, labeling the appropriate edge and writing formulas for the other two sides. Ask students what represents on the final bo. the length of one side of the bo, or the height of a short, flat bo QUESTIONING STRATEGIES Wh do we use the interval (0, 5) to find the function maimum? Those are the possible values of the length. Wh can t -values between 5 and 6 be used to find a function maimum? V() is negative between these values, and a volume cannot be negative. What would a bo look like if were ver close to 0? It would appear to be a piece of cardboard with no height. Module 5 64 Lesson EXTENSION ACTIVITY The bo in this Performance Task was an open bo, that is, it had no top. Ask students to research was to create a closed bo from a sheet of cardstock. Have them find boes in their dail lives that the can disassemble to two-dimensional form. Ask students how, based on the resulting two-dimensional geometr, the would set up an equation for the volume of the bo, and whether the would be able to determine a maimum volume based on that equation. What happens to the bo when is ver close to 5? The length of the narrow end of the bo shrinks to 0, and there is no flap to fold up. 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. Graphing Polnomial Functions 64