Planting the Seeds Exploring Cubic Functions

295 Planting the Seeds Exploring Cubic Functions 4.1 LEARNING GOALS In this lesson, you will: Represent cubic functions using words, tables, equations, and graphs. Interpret the key characteristics of the graphs of cubic functions. Analyze cubic functions in terms of their mathematical context and problem context. Connect the characteristics and behaviors of cubic functions to its factors. Compare cubic functions with linear and quadratic functions. Build cubic functions from linear and quadratic functions. KEY TERMS relative maximum relative minimum cubic function multiplicity Carnegie Learning ESSENTIAL IDEAS A cubic function is a function that can be written in the standard form f(x) = a x 3 1 b x 2 1 cx 1 d where a fi 0. Multiple representations such as tables, graphs, and equations are used to represent cubic functions. A relative maximum is the highest point in a particular section of a graph. A relative minimum is the lowest point in a particular section of a graph. Key characteristics are used to interpret the graph of cubic functions. Characteristics and behaviors of cubic functions are related to its factors. Multiplicity is how many times a particular number is a zero for a given polynomial function. The Fundamental Theorem states that a cubic function must have 3 roots. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS (2) Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to: (A) graph the functions f(x) 5 x, f(x) 5 1 x, f(x) 5 x 3, f(x) 5 3 x, f(x) 5 b x, f(x) 5 x, and f(x) 5 log b (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval

(6) Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: (A) analyze the effect on the graphs of f(x) 5 x 3 and f(x) 5 3 x when f(x) is replaced by af(x), f(bx), f(x 2 c), and f(x) 1 d for specific positive and negative real values of a, b, c, and d (7) Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to: (I) write the domain and range of a function in interval notation, inequalities, and set notation 4 Overview The terms cubic function, relative minimum, relative maximum, and multiplicity are defined in this lesson. The standard form of a cubic equation is given. In the first activity, a rectangular sheet of copper is used to create planters if squares are removed from each corner of the sheet and the sides are then folded upward. Students will analyze several sized planters and calculate the volume of each size. They then write a volume function in terms of the height, length, and width and graph the function using a graphing calculator. Using key characteristics, students analyze the graph and conclude that the graph is cubic. Students differentiate the domain and range of the problem situation from the domain and range of the cubic function. Using a graphing calculator and specified volumes, students determine which if any possible sized of planters meet the criteria. The second activity is similar but uses a cylindrical planter. The volume function is written in three forms and students will algebraically and graphically verify their equivalence. A graphing calculator is used throughout this lesson. 295B Chapter 4 Polynomial Functions

Warm Up Simplify each expression and identify its function family. 1. (x 1 4) (10) (x 1 4) (10) 5 10x 1 40 Linear Function 2. (x 2 4) (x 2 5) (x 2 4) (x 2 5) 5 x 2 2 9x 1 20 Quadratic Function 3. (x 1 8) 2 (x 1 8) 2 5 x 2 1 16x 1 64 Quadratic Function 4. (x 2 4) (x 2 5) (x 2 1) (x 2 4) (x 2 5) 5 x 2 + 9x 1 20 ( x 2 1 9x 1 20) (x 2 1) 5 x 3 1 8x 2 1 11x 2 20 Cubic Function Carnegie Learning 4.1 Exploring Cubic Functions 295

4 295D Chapter 4 Polynomial Functions

Planting the Seeds Exploring Cubic Functions 4.1 LEARNING GOALS In this lesson, you will: Represent cubic functions using words, tables, equations, and graphs. Interpret the key characteristics of the graphs of cubic functions. Analyze cubic functions in terms of their mathematical context and problem context. Connect the characteristics and behaviors of cubic functions to its factors. Compare cubic functions with linear and quadratic functions. Build cubic functions from linear and quadratic functions. KEY TERMS relative maximum relative minimum cubic function multiplicity If you have ever been to a 3D movie, you know that it can be quite an interesting experience. Special film technology and wearing funny-looking glasses allow moviegoers to see a third dimension on the screen depth. Three dimensional filmmaking dates as far back as the 1920s. As long as there have been movies, it seems that people have looked for ways to transform the visual experience into three dimensions. However, your brain doesn t really need special technology or silly glasses to experience depth. Think about television, paintings, and photography artists have been making two-dimensional works of art appear as three-dimensional for a long time. Several techniques help the brain perceive depth. An object that is closer is drawn larger than a similarly sized object off in the distance. Similarly, an object in the foreground may be clear and crisp while objects in the background may appear blurry. These techniques subconsciously allow your brain to process depth in two dimensions. Can you think of other techniques artists use to give the illusion of depth? Carnegie Learning 295 4.1 Exploring Cubic Functions 2

4 Problem 1 A planter is constructed from a rectangular sheet of copper. Given the dimensions of the rectangular sheet, students will complete a table of values listing various sizes of planters with respect to the length width, height, and volume. They describe observable patterns, analyze the relationship between the height, length, and width, and write a function to represent the volume of the planter box. Students use a graphing calculator to graph the function, describe the key characteristics of the graph, identify the maximum volume, the domain and range of the function versus the problem situation and conclude that it is cubic. The terms relative maximum and relative minimum are defined. Given specified volumes, students use a graphing calculator to determine possible sized planter boxes that meet the criteria. Grouping Have students complete Questions 1 and 2 with a partner. Then have students share their responses as a class. Guiding Questions Question 1, part (a) How did you determine the height of each planter in the table of values? PROBLEM 1 Business Is Growing The Plant-A-Seed Planter Company produces planter boxes. To make the boxes, a square is cut from each corner of a rectangular copper sheet. The sides are bent to form a rectangular prism without a top. Cutting different sized squares from the corners results in different sized planter boxes. Plant-A-Seed takes sales orders from customers who request a sized planter box. Each rectangular copper sheet is 12 inches by 18 inches. In the diagram, the solid lines indicate where the square corners are cut and the dotted lines represent where the sides are bent for each planter box. h h h h Recall the volume formula V 5 lwh. 18 inches h h h h 12 inches 1. Organize the information about each sized planter box made from a 12 inch by 18 inch copper sheet. a. Complete the table. Include an expression for each planter box s height, width, length, and volume for a square corner side of length h. Square Corner Side Length (inches) Height (inches) Width (inches) It may help to create a model of the planter by cutting squares out of the corners of a sheet of paper and folding. Length (inches) Volume (cubic inches) 0 0 12 18 0 1 1 10 16 160 2 2 8 14 224 3 3 6 12 216 4 4 4 10 160 5 5 2 8 80 6 6 0 6 0 7 7 22 4 256 h h 12 2 2h 18 2 2h h(12 2 2h)(18 2 2h) How did you determine the width of each planter in the table of values? How did you determine the length of each planter in the table of values? How did you determine the volume of each planter in the table of values? How did you determine each of the expressions when the length of the corner side was h inches in the table of values? 296 Chapter 4 Polynomial Functions

Guiding Questions Question 1, part (b) and Question 2 Is the height the same as the side length of each square? How does the increasing height affect the width and length of planter? As the height increases by one inch, what happens to the width and the length? If the volume is 0 cubic inches, what does that mean with respect to the problem situation? What happens to the width of the planter if the size of the square corner is equal to 6 inches? Is the corner square s length subtracted from the length and width of the planter box? How was the table of values useful when writing the function for the volume of the planter box? b. What patterns do you notice in the table? The height is the same as the side length of the square. As the height increases, the width and length decrease. For every inch increase in height, the width and length decrease by 2 inches. The volume starts at 0 cubic inches, increases, and then decreases back to 0 cubic inches. 2. Analyze the relationship between the height, length, and width of each planter box. a. What is the largest sized square corner that can be cut to make a planter box? Explain your reasoning. The size of the square corner must be less than 6 inches. A 6-inch square would result in a width of 0 inches. b. What is the relationship between the size of the corner square and the length and width of each planter box? Twice the corner square s length is subtracted from the length and the width of each planter box. For example, a 1-inch cut corner square results in a length of 18 2 (2 3 1) and a width of 12 2 (2 3 1). c. Write a function V(h) to represent the volume of the planter box in terms of the corner side of length h. V(h) 5 h(12 2 2h)(18 2 2h) Carnegie Learning 600005_A2_TX_Ch04_293-402.indd 297 14/03/ 4.1 Exploring Cubic Functions 2

Grouping Have students complete Questions 3 and 4 with a partner. Then have students share their responses as a class.?3. Louis, Ahmed, and Heidi each used a graphing calculator to analyze the volume function, V(h), and sketched their viewing window. They disagree about the shape of the graph. Louis Ahmed y y Guiding Questions Question 3 What is a complete graph? Did Louis, Ahmed, or Heidi sketch a complete graph? Does Louis s graph have an axis of symmetry? What is the interval in which the graph increases? What is the interval in which the graph decreases? volume height The graph increases and then decreases. It is a parabola. Heidi volume y x volume height The graph lacks a line of symmetry, so it can t be a parabola. height x x 4 I noticed the graph curves back up so it can t be a parabola. Evaluate each student s sketch and rationale to determine who is correct. For the student(s) who is/are not correct, explain why the rationale is not correct. Ahmed and Heidi are correct. Louis is not correct. The graph is not a parabola because it does not have a line of symmetry. Extending the viewing window on the graph also shows that the graph curves back up. 600005_A2_TX_Ch04_293-402.indd 298 298 Chapter 4 Polynomial Functions

Guiding Questions Question 4 Where are the x-intercepts? Where are the y-intercepts? Does the function have a maximum or a minimum point? Where on the graph is the point which describes the maximum volume of a planter box? What is the significance of the x-value at the maximum point of the function? What is the significance of the y-value at the maximum point of the function? Why is the domain of the function different than the domain of the problem situation? Why is the range of the function different than the domain of the problem situation? Does it make sense to have a planter box with the height of 0 inches? 4. Represent the function on a graphing calculator using the window [210, 15] 3 [2400, 400]. a. Describe the key characteristics of the graph? The graph increases until it reaches a peak and then decreases. The x-intercepts are (0, 0) and (6, 0) and (9, 0). The y-intercept is (0, 0). The graph first increases and then begins to decrease at (2.35, 228). Then the graph continues to decrease and finally begins to increase at (7.65, 268.16). b. What is the maximum volume of a planter box? State the dimensions of this planter box. Explain your reasoning. The maximum volume is 228 cubic inches. The dimensions of this planter are 2.35 inches 3 7.30 inches 3 13.30 inches. Graphically this is the highest point between x 5 0 and x 5 6. In this problem you are determining the maximum value graphically, but consider other representations. How will your solution strategy change when using the table or equation? c. Identify the domain of the function V(h). Is the domain the same or different in terms of the context of this problem? Explain your reasoning. The domain of the function is (2`, `). In terms of this problem situation, only the height values (0, 6) make sense. Values outside of this domain result in negative planter box dimensions. d. Identify the range of the function V(h). Is the range the same or different in terms of the context of this problem? Explain your reasoning. The range of the function is (2`, `). The range in terms of this problem situation is (0, 228) because the maximum volume is 228 and it is impossible to have a volume less than 0. e. What do the x-intercepts represent in this problem situation? Do these values make sense in terms of this problem situation? Explain your reasoning. The x-intercepts represent the planter box heights in which the volume is 0 cubic inches. It does not make sense to have a planter box with a height of 0 inches. Carnegie Learning 4.1 Exploring Cubic Functions 2

4 Grouping Ask a student to read the information and discuss the worked example as a class. Complete Question 5 as a class. Guiding Questions for Discuss Phase What is the difference between the maximum point on the graph of a function and a relative maximum point on the graph of a function? What is the difference between the minimum point on the graph of a function and a relative minimum point on the graph of a function? Can the width or the length of the planter box have a negative value? Which values on the volume function result in negative values for the width or length of the planter box? If a horizontal line such as y 5 50 is graphed with the volume function on a coordinate plane, what is the significance of the points of intersection? The key characteristics of a function may be different within a given domain. The function V(h) 5 h(12 2 2h)(18 2 2h) has x-intercepts at x 5 0, x 5 6, and x 5 9. Volume (cubic inches) 200 100 2100 2200 y 0 (2.35, 228) 2 4 6 8 Height (inches) As the input values for height increase, the output values for volume approach infinity. Therefore, the function doesn t have a maximum; however, the point (2.35, 228) is a relative maximum within the domain interval of (0, 6). A relative maximum is the highest point in a particular section of a graph. Similarly, as the values for height decrease, the output values approach negative infinity. Therefore, a relative minimum occurs at (7.65, 268.16). A relative minimum is the lowest point in a particular section of a graph. The function v(h) represents all of the possible volumes for a given height h. A horizontal line is a powerful tool for working backwards to determine the possible values for height when the volume is known. The given volume of a planter box is 100 cubic inches. You can determine the possible heights from the graph of V(h). Volume (cubic inches) 200 100 2100 2200 y 0 2 y = 100 V(h) 4 6 8 Height (inches) x V(h) x Draw a horizontal line at y 5 100. Identify each point where V(h) intersects with y 5 100, or where V(h) 5 100. The first point of intersection is represented using function notation as V(0.54) 5 100. 300 Chapter 4 Polynomial Functions

Carnegie Learning Guiding Questions Question 5 How did you identify the points at which v(h) 5 100 intersected the volume function? What values are represented on the y-axis with respect to the problem situation? What is the unit of measure? What values are represented on the x-axis with respect to the problem situation? What is the unit of measure? Are all three solutions reasonable with respect to the problem situation? Why or why not? Grouping Have students complete Questions 6 through 8 with a partner. Then have students share their responses as a class. Guiding Questions Question 6 How is this problem situation different than the last problem situation? How many times does the horizontal line y 5 175 intersect the volume function? Are all three points of intersection relevant to the problem situation? Why or why not? 5. A customer ordered a particular planter box with a volume of 100 cubic inches, but did not specify the height of the planter box. a. Use a graphing calculator to determine when V(h) 5 100. Then write the intersection points in function notation. What do the intersection points mean in terms of this problem situation? V(0.54) 5 100, V(4.76) 5 100, V(9.70) 5 100, The intersection points are the heights that create a planter box with a volume of 100 cubic inches. b. How many different sized planter boxes can Plant-A-Seed make to fill this order? Explain your reasoning. The graph has 3 solutions, the points of intersection (0.54, 100), (4.76, 100), and (9.70, 100). The first two intersection points lie within the domain of this problem context, indicating that planter boxes with height 4.76 inches and 0.54 inches have a volume of 100 cubic inches. A height of 9.70 inches also results in a volume of 100 cubic inches, but this value does not make sense in this problem situation. This height is not within the domain because it leads to negative values for length and width. 6. A neighborhood beautifying committee would like to purchase a variety of planter boxes with volumes of 175 cubic inches to add to business window sill store fronts. Determine the planter box dimensions that the Plant-A-Seed Company can create for the committee. Show all work and explain your reasoning. The 2 planters with dimensions h 5 3.78, l 5 10.44, w 5 4.44 and h 5 1.15, l 5 15.70, w 5 9.70 have a volume of 175 cubic inches. The function has 3 graphical solutions, but only 2 possible planters make sense in this problem situation. I graphed the volume function and the horizontal line y 5 175. The intersection points are the solutions to this problem. 4.1 Exploring Cubic Functions 3

4 Guiding Questions Questions 7 and 8 If the area of the base of the planter box is 12 square inches, what does this tell you about the length and width of the planter box? What algebraic expressions are used to determine the length and width of the planter box? What equation can be used to determine the length and width of the planter box? When the equation representing the area of the planter box is graphed, what is represented on the x- and y-axis? What is the width of a planter box that has a height of 5 inches? What is the length of a planter box that has a height of 5 inches? How many planter boxes have a height of 5 inches? What is the volume of a planter box that has a height of 5 inches? 7. Plant-A-Seed s intern claims that he can no longer complete the order because he spilled a cup of coffee on the sales ticket. Help Jack complete the order by determining the missing dimensions from the information that is still visible. Explain how you determined possible unknown dimensions of each planter box. Plant-A-Seed Sales Ticket Base Area: 12 square inches Height: Length: Width: Volume: The height of the planter box is 5.21 inches, the length is 7.58 inches, and the width is 1.58 inches. I set up the equation (18 2 2x)(12 2 2x) 5 12. Then I used a graphing calculator to graph y 1 5 (18 2 2x)(12 2 2x) and y 2 5 12 and calculated the intersection points. The intersection points are x 5 5.21 and x 5 9.79. However, the second value is greater than 6, so it doesn t make sense in this problem situation. Finally, I substituted x 5 5.21 back into the expression (18 2 2x)(12 2 2x) to determine the values for the length and width of the planter box. 8. A customer sent the following email: To Whom It May Concern, I would like to purchase several planter boxes, all with a height of 5 inches. Can you make one that holds 100 cubic inches of dirt? Please contact me at your earliest convenience. Thank you, Muriel Jenkins Write a response to this customer, showing all calculations. Dear Ms. Jenkins, Unfortunately, the Plant-A-Seed Company cannot create a planter box that holds 100 cubic inches of dirt and has a height of 5 inches. The height of the planter box determines the other dimensions. The three dimensions then determine the volume. A planter box with a height of 5 inches is 2 inches wide and 8 inches long. It will hold 80 cubic inches of dirt. This is our only planter box available with a height of 5 inches. I hope that this option for planter boxes will work for you. Sincerely, Plant-A-Seed How is the volume function built in this problem? 302 Chapter 4 Polynomial Functions

Problem 3 The volume function from Problem 1 is written in three different forms; the product of three linear functions, the product of a linear function and a quadratic function, and a cubic function in standard form. Students will algebraically and graphically verify the three forms of the volume function are equivalent. Grouping Have students complete Questions 1 and 2 with a partner. Then have students share their responses as a class. PROBLEM 3 Cubic Equivalence Let s consider the volume formula from Problem 1, Business is Growing. 1. Three forms of the volume function V(h) are shown. V(h) 5 h(18 2 2h)(12 2 2h) V(h) 5 h(4h 2 2 60h 1 216) V(h) 5 4h 3 2 60h 2 1 216h The product of three linear functions that represent height, length, and width. The product of a linear function that represent the height and a quadratic function representing the area of the base. A cubic function in standard form. a. Algebraically verify the functions are equivalent. Show all work and explain your reasoning. V(h) 5 h(18 2 2h)(12 2 2h) 5 h(216 2 36h 2 24h 1 4h 2 ) 5 h(216 2 60h 1 4h 2 ) 5 216h 2 60h 2 1 4h 3 5 4h 3 2 60h 2 1 216h V(h) 5 h(4h 2 2 60h 1 216) 5 4h 3 2 60h 2 1 216h V(h) 5 4h 3 2 60h 2 1 216h b. Graphically verify the functions are equivalent. Sketch all three functions and explain your reasoning. Carnegie Learning Guiding Questions Question 1 What algebraic properties were used to show the functions were equivalent? What should happen graphically, if the three functions are equivalent? Do all three forms of the function produce the same graph? 4000 3000 2000 1000 240 230 220 210 0 10 20 30 40 21000 22000 23000 24000 y Graphing each of the functions on the same coordinate plane, I notice that they all produce the same graph. This means that the functions must be equivalent. c. Does the order in which you multiply factors matter? Explain your reasoning. No. The order in which I multiply factors doesn t matter. The properties of integers hold for manipulating expressions algebraically. In this case, the Associative Property of Multiplication holds true. 600005_A2_TX_Ch04_293-402.indd 307 14/03/ x 4.1 Exploring Cubic Functions 3

Guiding Questions Worked Example and Question 2 Is there another way to multiply binomials? If the graph of the equation written in standard form is not the same as the graph of the equation written in factored form, what can you conclude? You can determine the product of the linear factors (x 1 2)(3x 2 2)(4 1 x) using multiplication tables. Step 1: Step 2: Choose 2 of the binomials to multiply. Multiply the product from step 1 with the Then combine like terms. remaining binomial. Then combine like terms.? x 2 3x 3x 2 6x 22 22x 24? 4 x 3x 2 12x 2 3x 3 4x 16x 4x 2 24 216 24x (x 1 2)(3x 2 2)(4 1 x) 5 3x 3 1 16x 2 1 12x 2 16. 2. Analyze the worked example for the multiplication of three binomials. a. Use a graphing calculator to verify graphically that the expression in factored form is equivalent to the product written in standard form. 4 I entered y 1 5 3x 3 1 16x 2 1 12x 2 16 and y 2 5 (x 1 2)(3x 2 2)(4 1 x) in my graphing calculator. Each equation produced the same graph, therefore the expressions are equivalent. b. Will multiplying three linear factors always result in a cubic expression? Explain your reasoning. Yes. A linear factor has an x-term. The product of three first degree terms is a third degree term. 600005_A2_TX_Ch04_293-402.indd 308 308 Chapter 4 Polynomial Functions

Carnegie Learning Grouping Have students complete Questions 3 through 5 with a partner. Then have students share their responses as a class. Guiding Questions Questions 3 through 5 Which factors did you multiply together first? Does it matter? What method did you use to multiply the binomials? What effect does a negative leading term have on the graph of the cubic function? What effect does the constant term have on the graph of the cubic function? Does the function pass through the origin? If the function passes through the origin, does this give you any information about its factor(s)? Does the product of a monomial and two binomials create a cubic equation? If the graphs of two or more functions are different, what can you conclude? If the graphs of two or more functions are the same, do the functions always have the same factors? 3. Determine each product. Show all your work and then use a graphing calculator to verify your product is correct. a. (x 1 2)(23x 1 2)(1 1 2x) (x 1 2)(23x 2 6x 2 1 2 1 4x) (x 1 2)(26x 2 1 x 1 2) (26x 3 1 x 2 1 2x 2 12x 2 1 2x 1 4) 26x 3 2 11x 2 1 4x 1 4 b. (10 1 2x)(5x 1 7)(3x) (50x 1 70 1 10x 2 1 14x)(3x) (10x 2 1 64x 1 70)(3x) 30x 3 1 192x 1 210x 4.1 Exploring Cubic Functions 3

4. Determine the product of the linear and quadratic factors. Then verify graphically that the expressions are equivalent. a. (x 2 6)(2x 2 2 3x 1 1) (2x 3 2 3x 2 1 x 2 12x 2 1 18x 2 6) 2x 3 2 15x 2 1 19x 2 6 b. (x)(x 1 2) 2 x(x 2 1 4x 1 4) x 3 1 4x 2 1 4x 4 310 Chapter 4 Polynomial Functions

5. Max determined the product of three linear factors. Max The function f(x) 5 (x 1 2) 3 is equivalent to f(x) 5 x 3 1 8 a. Explain why Max is incorrect. The product of (x 1 2)(x 1 2)(x 1 2) is x 3 1 6x 2 1 12x 1 8. The functions (x 1 2) 3 and x 3 1 8 produce different graphs which proves that they are not equivalent. b. How many x-intercepts does the function f(x) 5 (x 1 2) 3 have? How many zeros? Explain your reasoning. The function has only one x-intercept, (22, 0) since it crosses the x-axis only once. The function has three zeros. The zero x 5 22 is a multiple root, occurring 3 times. Carnegie Learning 4.1 Exploring Cubic Functions 3