2.4. Families of Polynomial Functions

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1 2. Families of Polnomial Functions Crstal pieces for a large chandelier are to be cut according to the design shown. The graph shows how the design is created using polnomial functions. What do all the functions have in common? How are the different? How can ou determine the equations that are used to create the design? CONNECTIONS Go to mcgrawhill.ca/links/ functions and follow the links for this chapter. At the linked site, click on laser-etched math models to see some interesting laser-etched crstals based on mathematical functions In this section, ou will determine equations for a famil of polnomial functions from a set of zeros. Given additional information, ou will determine an equation for a particular member of the famil. 2. Families of Polnomial Functions MHR 113

2 Investigate How are polnomial functions with the same zeros related? Tools graphing calculator 1. a) Eamine each set of parabolas and the corresponding functions. Set A Set B ii) i) iii) v) iv) vi) i) ( 1)( 2) iv) ( 1)( 2) ii) 2( 1)( 2) v) ( 1)( 2) iii) _ 1 2 ( 1)( 2) vi) _ 1 ( 1)( 2) 2 b) Reflect How are the graphs of the functions in part a) similar and how are the different? 2. Reflect Describe the relationship between the graphs of functions of the form k( 1)( 2), where k. Wh do ou think this is called a famil of functions? 3. a) Eamine the following functions. How are the similar? How are the different? i) ( 1)( 3)( 2) ii) ( 1)( 3)( 2) iii) ( 1)( 3)( 2) iv) 2( 1)( 3)( 2) b) Reflect Predict how the graphs of the functions in part a) will be similar and how the will be different.. a) Use a graphing calculator to graph the functions in step 3 on the same set of aes. b) Eamine the graphs. Was our prediction accurate? If not, eplain how it should be changed. 5. Reflect Describe the relationship between the graphs of polnomial functions of the form k( r)( s)( t), where k. Wh is it appropriate to call this a famil of polnomial functions? 11 MHR Advanced Functions Chapter 2

3 A famil of functions is a set of functions that have the same characteristics. Polnomial functions with the same zeros are said to belong to the same famil. The graphs of polnomial functions that belong to the same famil have the same -intercepts but have different -intercepts (unless zero is one of the -intercepts). An equation for the famil of polnomial functions with zeros a 1, a 2, a 3,..., a n is k( a 1 )( a 2 )( a 3 )... ( a n ), where k, k. Eample 1 Represent a Famil of Functions Algebraicall The zeros of a famil of quadratic functions are 2 and. a) Determine an equation for this famil of functions. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil that passes through the point (1, ). Solution a) The factor associated with 2 is 2 and the factor associated with is 3. An equation for this famil is k( 2)( 3), where k. b) Use an two values for k to write two members of the famil. For k, ( 2)( 3). For k, ( 2)( 3). c) To find the member whose graph passes through (1, ), substitute 1 and into the equation and solve for k. k(1 2)(1 3) k( 1)() k k 1 The equation is ( 2)( 3). Eample 2 Determine an Equation for a Famil of Cubic Functions Given Integral Zeros The zeros of a famil of cubic functions are, 1, and 3. a) Determine an equation for this famil. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil whose graph has a -intercept of 15. d) Sketch graphs of the functions in parts b) and c). 2. Families of Polnomial Functions MHR 115

4 Solution a) Since the zeros are, 1, and 3, then 2, 1, and 3 are factors of the famil of cubic functions. An equation for this famil is k( 2) ( 1)( 3), where k. b) Use an two values for k to write two members of the famil. For k 2, 2( 2)( 1)( 3). For k 1, ( 2)( 1)( 3). c) Since the -intercept is 15, substitute and 15 into k( 2)( 1)( 3). 15 k( 2)( 1)( 3) 15 6k k.5 The equation is.5( 2)( 1)( 3). d) From part a), the three functions have zeros, or -intercepts,, 1, and 3. From part c), the -intercept of.5( 2)( 1)( 3) is 15. Substitute to determine the -intercepts of the functions from part b). 2( 2)( 1)( 3) ( 2)( 1)( 3) 2( 2)( 1)( 3) ( 2)( 1)( 3) The -intercept of The -intercept of 2( 2)( 1)( 3) is. ( 2)( 1)( 3) is. To sketch a graph of the functions, plot the common -intercepts. Plot the -intercept for each function. The cubic function 2( 2)( 1)( 3) has a positive leading coefficient, so its graph will etend from quadrant 3 to quadrant 1. The cubic functions ( 2)( 1)( 3) and.5( 2)( 1)( 3) have negative leading coefficients, so their graphs will etend from quadrant 2 to quadrant MHR Advanced Functions Chapter 2

5 Eample 3 Determine an Equation for a Famil of Quartic Functions Given Irrational Zeros a) Determine a simplified equation for the famil of quartic functions with zeros 1 and 2 3. b) Determine an equation for the member of the famil whose graph passes through the point (2, 1). Solution a) The zeros are 1, 1, 2 3, and 2 3. So, ( 1), ( 1), ( 2 3 ), and ( 2 3 ) are factors of the famil of quartic functions. An equation for this famil is k( 1)( 1)( 2 3 )( 2 3 ) k( 1)( 1)[( 2) 3 ][( 2) 3 ] k( 2 1)[( 2) 2 ( 3 ) 2 ] k( 2 1)( 2 3) k( 2 1)( 2 1) k( ) k( 3 1) b) To find the member whose graph passes through (2, 1), substitute 2 and 1 into the equation and solve for k. 1 k[(2) (2) 3 (2) 1] 1 9k k The equation is ( 3 1), or 3 2. CONNECTIONS Each pair of factors has the difference of squares pattern: (a b)(a b) a 2 b 2. Eample Determine an Equation for a Quartic Function From a Graph Determine an equation for the quartic function represented b this graph Families of Polnomial Functions MHR 117

6 Solution From the graph, the -intercepts are, _ 1, 1, and 2. 2 The corresponding factors are 3, 2 1, 1, and 2. An equation for the famil of polnomial functions with these zeros is k( 3)(2 1)( 1)( 2). Select a point that the graph passes through, such as ( 1, ). Substitute 1 and into the equation to solve for k. k[( 1) 3][2( 1) 1][( 1) 1][( 1) 2] k(2)( 1)()() k k.5 The equation is.5( 3)(2 1)( 1)( 2). << >> KEY CONCEPTS A famil of functions is a set of functions with the same characteristics. Polnomial functions with graphs that have the same -intercepts belong to the same famil. A famil of polnomial functions with zeros a 1, a 2, a 3,..., a n can be represented b an equation of the form k( a 1 )( a 2 )( a 3 )... ( a n ), where k, k. An equation for a particular member of a famil of polnomial functions can be determined if a point on the graph is known. Communicate Your Understanding C1 C2 C3 C How man polnomial functions can have the same -intercepts? Eplain. What information is required to determine an equation for a famil of polnomial functions? What information is required to determine an equation for a particular member of a famil of polnomial functions? Describe how the graphs of the members of a famil of polnomial functions are the same and how the are different. 11 MHR Advanced Functions Chapter 2

7 A Practise For help with question 1, refer to Eample The zeros of a quadratic function are 7 and. a) Determine an equation for the famil of quadratic functions with these zeros. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil that passes through the point (2, 1). C 2 6 For help with questions 2 to, refer to Eample Eamine the following functions. Which function does not belong to the same famil? Eplain. D 2 A 1.5( )( 5)( 2) 16 B 1.5( 2)( 5)( ) C 1.5( 2)( )( 2) D 3( 5)( 2)( ) 3. The graphs of four polnomial functions are given. Which graphs represent functions that belong to the same famil? Eplain. A Which of the following polnomial functions belong to the same families? Eplain. Sketch a graph of the functions in each famil to verif our answer. A f() ( 2)( 1)( 3) B h() ( 2)( 1)( 3) B C g() 3( 2)( 1)( 3) D p().( 3)( 1)( 2) E r() _ 2 ( 1)( 2)( 3) 5 F q() 3 ( 1)( 3)( 2) Families of Polnomial Functions MHR 119

8 For help with question 5, refer to Eample Write an equation for a famil of polnomial functions with each set of zeros. a) 5, 2, 3 b) 1, 6, c), 1, 9 d) 7,, 2, 5 For help with question 6, refer to Eample. 6. Determine an equation for the function that corresponds to each graph in question 3. B Connect and Appl 7. a) Determine an equation for the famil of cubic functions with zeros,, and 2. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil whose graph passes through the point (, ). d) Sketch a graph of the functions in parts b) and c).. a) Determine an equation for the famil of cubic functions with zeros, 1, and _ 1 2. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil whose graph has a -intercept of 6. d) Sketch a graph of the functions in parts b) and c). 9. a) Determine an equation for the famil of quartic functions with zeros, 1, 2, and 3. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil whose graph has a -intercept of. d) Sketch a graph of the functions in parts b) and c). 1. a) Determine an equation for the famil of quartic functions with zeros _ 5 2, 1, _ 7 2, and 3. b) Write equations for two functions that belong to this famil. c) Determine an equation for the member of the famil whose graph passes through the point (, 25). d) Sketch a graph of the functions in parts b) and c). 11 a) Determine an equation, in simplified form, for the famil of cubic functions with zeros 1 2 and _ 1 2. b) Determine an equation for the member of the famil whose graph passes through the point (3, 35).. a) Determine an equation, in simplified form, for the famil of quartic functions with zeros 3 (order 2) and 3. b) Determine an equation for the member of the famil whose graph passes through the point (1, 2). 13. a) Determine an equation, in simplified form, for the famil of quartic functions with zeros 1 5 and 2 2. b) Determine an equation for the member of the famil whose graph has a -intercept of Determine an equation for the cubic function represented b this graph MHR Advanced Functions Chapter 2

9 15. Determine an equation for the quartic function represented b this graph Chapter Problem Clear plastic sheets that measure cm b 6 cm are to be used to construct gift boes for Best of U personal care products. The boes are formed b folding the sheets along the dotted lines, as shown in the diagram. 1 cm cm 16. Determine an equation for the quartic function represented b this graph Use Technolog Are the functions in each set a famil? Justif our answer. Set A (3 1)(2 1)( 3)( 2) 2(3 1)(2 1)( 3)( 2) 1 3(3 1)(2 1)( 3)( 2) 2 (3 1)(2 1)( 3)( 2) 3 Set B (3 1)(2 1)( 3)( 2) (3 1)( 2)( 3)( 2) 3(3 1)(1 2)( 3)( 2) (3 1)(2 1)( 3)(6 3) 6 1 a) Epress the volume of one of the boes as a function of. b) Determine possible dimensions of the bo if the volume of each bo is to be 23 cm 3. c) How does the volume function in part a) change if the height of the bo is doubled? tripled? Describe the famil of functions formed b multipling the height b a constant. d) Sketch graphs of two members of this famil on the same coordinate grid. 19. The graph represents a section of the track of a rollercoaster. Write an equation for the famil of functions that models the section of the track. Vertical Height (m) 16 Height of a Rollercoaster Horizontal Distance (m) 2. Families of Polnomial Functions MHR 1

10 2. An open-top bo is to be constructed from a piece of cardboard b cutting congruent squares from the corners and then folding up the sides. The dimensions of the cardboard are shown. 36 cm 2 cm Achievement Check 21. a) The design for the crstal pieces of the chandelier at the beginning of this section is shown below. Determine an equation for the famil of functions used to create the design. b) Find equations for the members of the famil that make up the design. c) Create a design of our own. Write equations for the famil of functions and the members used in our design. a) Epress the volume of the bo as a function of. b) Write an equation to represent a bo with volume that is Reasoning and Proving Representing Selecting Tools Problem Solving Connecting Reflecting Communicating i) twice the volume of the bo represented b the function in part a) 3 ii) three times the volume of the bo represented b the function in part a) c) How are the equations in part b) related to the one in part a)? d) Sketch graphs of all three functions on the same coordinate grid. e) Determine possible dimensions of a bo with volume 12 cm C Etend and Challenge 22. a) Write an equation for a famil of even functions with four -intercepts, two of which are _ 2 and 5. 3 b) What is the least degree this famil of functions can have? c) Determine an equation for the member of this famil that passes through the point ( 1, 96). d) Determine an equation for the member of this famil that is a reflection in the -ais of the function in part c). 23. Refer to question 19. Design our own rollercoaster track using a polnomial function of degree si or higher. Sketch a graph of our rollercoaster. 2. Math Contest Two concentric circles have radii 9 cm and 15 cm. Determine the length of a chord of the larger circle that is tangent to the smaller circle. 25. Math Contest Given a function g() such that g( 2 2) 5 2 3, determine g( 2 1). 2 MHR Advanced Functions Chapter 2