Online Homework Hints and Help Extra Practice

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1 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 30 Lesson 4 Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL 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.) COMMON CORE 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 30

2 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 () = ( - ) 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 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) 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 > 3. Houghton Mifflin Harcourt Publishing Compan 11. ƒ () = ( + ) ( - 1) 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 Lesson 4 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 0 3 Strategic Thinking MP. Reasoning 1 3 Strategic Thinking MP.4 Modeling 303 Lesson 5.4

3 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 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. Houghton Mifflin Harcourt Publishing Compan Module Lesson 4 Graphing Polnomial Functions 304

4 INTEGRATE MATHEMATICAL 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) Houghton Mifflin Harcourt Publishing Compan 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 Lesson Lesson 5.4

5 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. Houghton Mifflin Harcourt Publishing Compan Module Lesson 4 Graphing Polnomial Functions 306

6 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 () = ( + ) ( - ). 1. 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, and 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. 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. Houghton Mifflin Harcourt Publishing Compan - 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 Lesson Lesson 5.4