Representing Polynomials Graphically

Transcription

1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Polnomials Graphicall Mathematics Assessment Resource Service Universit of Nottingham & UC Berkele For more details, visit: 15 MARS, Shell Center, Universit of Nottingham Ma be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Representing Polnomials Graphicall MATHEMATICAL GOALS This lesson unit is intended to help ou assess how well students are able to translate between graphs and algebraic representations of polnomials. In particular, this unit aims to help ou identif and assist students who have difficulties: Recognizing the connection between the zeros of polnomials when suitable factorizations are available and graphs of the functions are defined b polnomials. Recognizing the connection between transformations of the graphs and transformations of the functions obtained b replacing f() b f( + k), f() + k, -f(), f(-). COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: A-SSE: Interpret the structure of epressions. A-APR: Understand the relationship between zeros and factors of polnomials. F-IF: Analze functions using different representations. F-BF: Build new functions from eisting functions. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices and 7: 1. Make sense of problems and persevere in solving them.. Reason abstractl and quantitativel. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategicall.. Attend to precision. 7. Look for and make use of structure.. Look for and epress regularit in repeated reasoning. INTRODUCTION The lesson unit is structured in the following wa: Before the lesson, students attempt the assessment task individuall. You then review their work and formulate questions that will help them improve their solutions. During the lesson, students work collaborativel in pairs or threes, matching functions to their graphs and creating new eamples. Throughout their work students justif and eplain their decisions to peers. During a whole-class discussion, students eplain their reasoning. Finall, students improve their solutions to the initial task and complete a second, similar task. MATERIALS REQUIRED Each student will need a mini-whiteboard, pen, and eraser, and a cop of Cubic Graphs and Their Equations and Cubic Graphs and Their Equations (revisited). Each small group of students will need cut-up cards Cubic Graphs, Cubic Functions and Statements to Discuss: True or False?, a large sheet of poster paper, and a glue stick. You ma want to enlarge the cards and/or cop them onto transparencies to be used on an overhead projector to support the whole-class discussion. TIME NEEDED minutes before the lesson, a 9-minute lesson (or two 5-minute lessons), and minutes in a follow-up lesson. Timings are approimate and will depend on the needs of our students. Teacher guide Representing Polnomials Graphicall T-1

3 "# $# %# &# '#!(#!%#!)#!&#!*# '# *# &# )# %# (#!&#!%#!$#!"# "# $# %# &# '#!(#!%#!)#!&#!*# '# *# &# )# %# (#!&#!%#!$#!"# "# $# %# &# '#!(#!%#!)#!&#!*# '# *# &# )# %# (#!&#!%#!$#!"# BEFORE THE LESSON Assessment task: Cubic Graphs and Their Equations ( minutes) Have students do this task, in class or for homework, a da or more before the formative assessment lesson. This will give ou an opportunit to assess the work and identif students who have misconceptions or need other forms of help. You should then be able to target our help more effectivel in the net lesson. Give each student a cop of Cubic Graphs and Their Equations. Spend minutes working individuall, answering these questions. Show all our work on the sheet. Make sure ou eplain our answers clearl. It is important that, as far as possible, students answer the questions without assistance. Students who sit together often produce similar (iii) Horizontall translate = ( +1)(! ) through + units: answers and then when the come to work (iv) Verticall translate = ( +1)(! ) together on similar tasks, the have little to through +3 units: discuss. For this reason, we suggest that when students do the assessment task, ou ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual places. Eperience has shown that this produces more profitable discussions. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and difficulties. The purpose of this is to forewarn ou of the issues that will arise during the lesson, so that ou ma prepare carefull. We suggest that ou do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores and distracts their attention from how the ma improve their mathematics. Instead, help students to make further progress b summarizing their difficulties as a list of questions. Some suggestions for these are given in the Common issues table on the net page. We suggest that ou make a list of our own questions, based on our students work, using the ideas on the following page. We recommend ou: write one or two questions on each student s work, or Cubic Graphs and Their Equations 1. Write down an equation of a cubic function that would give a graph like the one shown here. It crosses the -ais at (-3, ), (, ), and (5, ).. Write down an equation of a cubic function that would give a graph like the one shown here. It crosses the -ais at (, -). 3. On the aes, sketch a graph of the function = ( +1)(! ). You do not need to plot it accuratel! Show where the graph crosses the - and -aes.. Write down the equation of the graph ou get after ou: (i) Reflect = ( +1)(! ) over the -ais: (ii) Reflect = ( +1)(! ) over the -ais: give each student a printed version of our list of questions and highlight appropriate questions for each student. If ou do not have time to do this, ou could select a few questions that will be of help to the majorit of students and write these questions on the board when ou return the work to the students in the follow-up lesson. Teacher guide Representing Polnomials Graphicall T-!" -3!! 5!!" -!!"

4 Common issues: Writes an equation that is not cubic (Q1, ) Writes a cubic equation, but gets the signs wrong (Q1, ) For eample: The student writes (Q1): = ( 3)( + )( + 5). Writes an incorrect equation (Q) For eample: The student correctl takes account of the -intercept, but not the - intercepts: = 3. Or: The student correctl takes account of the -intercepts but ignores the -intercepts: = ( + 3)( + )( ). Plots the graph rather than sketches it (Q3) Incorrectl sketches the graph (Q3) For eample: The student sketches a graph with three roots. Or: The student sketches a graph with - intercepts at = 1 and =. Or: When sketching the graph the student does not use an approimate scale. Attempts to draw a reflection or translation of the graph and then fit a function to it (Q) Suggested questions and prompts: Does our equation give a straight or curved graph? How do ou know? From our equation, when = (or = ), what must (or ) be? Does this agree with the graph? Will there be positive or negative signs inside the parentheses? How do the number of roots equate to the number of parentheses in the function? What happens to the values when is reall large? How can this be represented in the function? Where does the equation tell ou that the graph crosses the -ais? What value for do ou get when =? Does this correspond with the graph? Will this cross the -ais in approimatel the right places? Will this cross the -ais in approimatel the right places? Can ou get an idea of the shape of the graph without plotting points? Can ou get an idea of the shape of the graph b looking at where the graph crosses the aes? What happens when is ver large? What does the function tell ou about the number of roots? To help sketch the graph, what numbers can ou substitute into the function? Check our graph is correct. What are the values for when is zero? Check ou have placed our - and - intercepts in approimatel the correct position for the scale ou've used. What happens to the point (3, ) after it is: (i) Reflected over the -ais? (ii) Reflected over the -ais? (iii) Translated + units horizontall? (iv) Translated +3 verticall? So what happens to ever value ( value) in the function? After the translation, where will the roots be? Teacher guide Representing Polnomials Graphicall T-3

5 SUGGESTED LESSON OUTLINE In this lesson students should not use a graphing calculator. Introduction: sketching and interpreting graphs of polnomials (5 minutes) Give each student a mini-whiteboard, pen, and eraser. Maimize participation in the discussion b asking all students to show solutions on their mini-whiteboards. Throughout this introduction encourage students to justif their responses. Tr not to correct responses, but encourage students to challenge each others eplanations. Show me a sketch of the graph of =, marking just the - and -intercepts. Now do the same for = 5 1. Ensure that students know how to find the -intercept b putting equal to zero and the -intercept(s) b equating to zero. Repeat this for a few quadratic graphs. Show me a sketch of the graph of = ( - 3)( + ), marking the - and -intercepts. Some students ma sketch a graph with one negative and one positive -intercept, but position them incorrectl. This ma be because the assume the intercepts are at = -3 and =, or the do not appl an approimate scale correctl. To find the -intercept some students ma multipl out the parentheses, whilst others will substitute = into the equation. You ma also find some students sketch graphs with two -intercepts and one -intercept. Some students ma figure out the coordinates of the verte. This would improve the accurac of their sketch, but it ma also derail the goals of this lesson, as it is a ver time consuming activit. Throughout this introduction our task is to investigate students strategies and solutions. Displa Slide P-1 of the projector resource. Show me a possible equation that fits (Graph A / Graph B / Graph C / Graph D). How can ou check our answer? Graph A Graph B Graph C Graph D Graph A could be = ( + 1)( - ) because it might intersect the -ais at about (-1, ) and (+, ). Graph B could be = ( - 1)( - 3) because it might intersect the -ais at about (+1, ) and (+3, ). Graph C could be = -( - 1)( + 5) because it might intersect the -ais at about (+1, ) and (-5, ) and when is large (sa 1), then is negative. Graph D could be = ( + 3) because it onl intersects the -ais at one point, so it must have a repeated root. Teacher guide Representing Polnomials Graphicall T-

6 In this wa, students should learn to pa particular attention to the intercepts and the sign of when is ver large (or ver small). For each correct suggestion, ask students to state where the graph crosses the -ais and give their reasons. Students should now be in a good position to eplore cubics. Ask students to sketch = ( 1)( )( 3) on their mini-whiteboards, using the same ideas (as the used with the quadratic functions) of finding the intercepts first. Check that everone understands b asking them to eplain their graphs. Repeat this using additional eamples using both + and inside the parentheses. Collaborative small-group work ( minutes) Organize students into groups of two or three. For each group, provide a cut-up cop of the Cubic Graphs and Cubic Functions cards and a large sheet of paper for making a poster. Do not distribute the glue sticks et. Displa Slide P- of the projector resource and eplain how students are to work together: Projector Resources Working Together 1. Take turns to match a function to its graph.. As ou do this, label the graph to show the intercepts on the - and -aes. 3. If ou match two cards, eplain how ou came to our decision.. If ou don't agree or understand, ask our partner to eplain their reasoning. 5. You all need to agree on and eplain the matching of ever card.. You ma find that more than one function will fit some graphs! 7. If ou have functions left over, sketch graphs on the blank cards to match these functions. Representing Polnomials P- The purpose of this structured work is to encourage students to engage with each other s eplanations and take responsibilit for each other s understanding. Darren matched these cards. Marla, can ou eplain Darren s thinking? If ou find the student is unable to answer this question, ask them to discuss the work further. Eplain that ou will return in a few minutes to ask a similar question. During small group work, tr to listen and support students thinking and reasoning. Note difficulties that emerge for more than one group. Are students matching a function to a graph or a graph to a function? Do students thoughtfull substitute values for into a function? Do students use the zeros of the functions effectivel? Are students able to use ver large or ver small values of to discriminate between = f and = f? Are students multipling out the parentheses? And if so, wh? Do students find the coordinates of the verte? And if so, wh? (The coordinates of the verte are not needed for this task.) Do students attend carefull to the signs in the function? Do students recognize repeated roots in a function? Teacher guide Representing Polnomials Graphicall T-5

7 Are students making assumptions? For eample, do students assume that if there is one positive -intercept and one negative -intercept the verte will lie on the -ais? You will be able to use this information in the whole-class discussion. Tr not to make suggestions that move students towards a particular strateg. Instead, ask questions to help students to reason together. Encourage students to use an efficient method. You ma want to use the questions in the Common issues table to help address misconceptions. It is not essential that everone completes this activit, but it is important that everone has matched the si graph cards to some functions. Function cards 7 and 11 are the same function, but there is no graph card for this function. Students that are struggling with sketching a graph for function card 7 could be encouraged to look at how it relates to the other functions given (i.e. function 7 is a translation of function /graph F). The ma also choose to emplo an algebraic method to help with this. Once ou are satisfied students have matched enough cards, give them a glue stick and ask them to glue their cards onto the paper to make a poster. Etending the lesson over two das If ou are taking two das to complete the unit ou might want to end the first lesson here. Before the end of the lesson, make sure that students have glued onto their poster paper all the cards the have matched so far. Then, at the start of the second da, allow the students time to familiarize themselves with their posters, before the share their work with another group. Sharing work (15 minutes) When students have completed the task, ask them to check their work against that of a neighboring group: Check to see which matches are different from our own. If there are differences, ask for an eplanation. If ou still don't agree, eplain our own thinking. You ma then need to consider whether to make an changes to our own work. It is important that everone in both groups understands the math. You are responsible for each other s learning. Whole-class discussion (3 minutes) To promote an interactive discussion between all students ask them to write solutions on their miniwhiteboards. Organize a whole-class discussion about the different strategies used to match the cards. The intention is that ou work with the students to make the math of the lesson eplicit. First select a set of cards that most groups matched correctl. This approach ma encourage good eplanations. Then select one or two cards that most groups found difficult to match. Once one group has justified their choice for a particular match, ask other students to contribute ideas of alternative approaches and their views on which reasoning method was easier to follow. The intention is that ou focus on getting students to understand and share their reasoning, not just checking that everone produced the right answers. Use our knowledge of the students individual and group work to call on a wide range of students for contributions. How did ou decide to match this card? Can someone else put that into their own words? Teacher guide Representing Polnomials Graphicall T-

8 # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# You ma want to draw on the questions in the Common issues table to support our own questioning. You ma also choose to begin to generalize the learning b asking students to pair up two sets of cards that have some similar graphical or/and algebraic features. After a few minutes working individuall on the task, the are to discuss it with a partner. Once the have completed the task, ask each group of students to report back on their pair of statements, giving eamples and full eplanations. This should reinforce the important lessons learned. You ma want to support the discussion with questions from the Common issues table. What do these two functions have in common? What are their differences? How are these similarities and differences represented in the graph? Show me two graphs that are reflections of each other. Describe the reflection. How is this reflection represented in the function? Show me two graphs that represent a translation of one graph onto another. Describe the translation. How is this translation represented in the function? Follow-up suggestions b prompting students to be specific and if appropriate, move towards more general representations. Below are four possible answers: Graph A Function Graph E Function 3 # # # # " #!5#!#!3#!#!1# # 1# # 3# # 5#!# = ( +1)( + )( + ) # " # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!# = ( +1)( + )( + )!#!#!#!#!#!# Graph A is a reflection of Graph E over the -ais. How can ou tell from the equation that this is a reflection? [The negative sign in front of the function.] If Graph A is represented b = f(), then how can we represent Graph E?[ = -f().] Graph C Function # # " Graph F Function " # # #!5#!#!3#!#!1# # 1# # 3# # 5#!# = ( 1) ( ) = ( +1) ( + )!#!#!# Graph C is a reflection of Graph F over the -ais. How can ou tell from the equation that this is a reflection? [Replace b - in the equation for Graph C and simplif, noting that (- 1) = (-1)( + 1) (-1)( + 1) = ( + 1) and that (- ) = -( + ).] If Graph C is represented b = f(), then how can we represent Graph F? [ = f(-).] Teacher guide Representing Polnomials Graphicall T-7

9 # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# " # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# " Graph A Function Graph B Function 1 # # " " # # #!5#!#!3#!#!1# # 1# # 3# # 5#!# = ( +1)( + )( + ) = ( 1)( + )!#!#!# Graph A is a horizontal translation of Graph B. How far does Graph A have to translate horizontall to get Graph B? [+ units.] How can ou tell? [The middle root on Graph A is at = -, on Graph B it is at the origin.] How can we transform the equation for Graph A to get the equation for Graph B? [Replace each instance of in the equation b.] If Graph A is represented b = f(), then how can we represent Graph B?[ = f( - ).] Graph C Function Graph D Function # " # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!# = ( 1) ( ) # " # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# = ( 3) Function 9 = ( 1) ( )+!# Graphs D is a vertical translation of Graph C. How far does Graph C have to verticall translate to get Graph D? [+ units.] How can ou tell? [The -intercept on Graph C is at = -, on Graph D it is at the origin.] How can we transform the equation for Graph C to get the equation for Graph D?[Simpl add onto the end of the function.] If Graph C is represented b = f(), then how can we represent Graph D?[ = f() +.] (Optional) Etension activit: small-group work 1 minutes) If students have done well, ou ma wish to give each pair/group of students a cop of Statements to Discuss: True or False. This sheet attempts to further develop and etend students understanding. Eplain to students what the need to do: Choose one or two pairs of statements. Decide which statement is true and which is false. Be prepared to justif our answers. Give students 5 minutes to work on each pair of statements. You ma wish to allocate particular statements to specific students to provide an appropriate challenge. A1 B1 C1 D1 E1 f () = f () = ( + ) is a factor of f () A possible equation for this graph is: = ( +1)( ) If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) = f () = ( ) ( 7) g() = ( ) (7 ) f () is a reflection of g() over the ais f () = g( + ) g() is a horizontal translation of f () A B C D E f () = f () = ( ) is a factor of f () A possible equation for this graph is: = ( 1)( + ) If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) could be an number but zero. f () = ( ) g() = ( ) f () is a reflection of g() over the ais f () = g() + g() is a horizontal translation of f () Teacher guide Representing Polnomials Graphicall T

10 Follow-up lesson: reviewing the assessment task ( minutes) Return to the students their original assessment Cubic Graphs and Their Equations. If ou have not added questions to individual pieces of work, then write our list of questions on the board. Students should select from this list onl those questions the think are appropriate to their own work. Read through our original solution to the task and the questions I have written (on our script/on the board). Spend a few minutes answering these questions and revise our response. You ma want to make notes on our mini-whiteboard. To help students see their own progress, ask them to complete the task using a different color pen or give them a second blank cop of the task. Now give students a cop of the task Cubic Graphs and Their Equations (revisited). Use what ou have learned to complete the new assessment task. Show all our work on the sheet and make sure ou eplain our answers reall clearl. If students struggled with the original assessment task, ou ma feel it more appropriate for them to just spend their time revisiting Cubic Graphs and Their Equations rather than attempting Cubic Graphs and Their Equations (revisited) as well. Teacher guide Representing Polnomials Graphicall T-9

11 SOLUTIONS Assessment task: Cubic Graphs and Their Equations 1. A suitable cubic function will take the form: = A( + 3)( )( 5) where A is an real positive number.. A suitable cubic function will take the form: = A( + a)( + b)( c) where A is an positive real number, and a, b, c are positive numbers, such that a > b > c and Aabc =. Specific eamples are: = ( + 3)( + )( 1), or = ( + )( + 3)( ) 3. A suitable sketch will show the -ais intercepts at ( 1, ) and (, ), the -ais intercept at (, 1) and be of the following shape: # " # # #!# #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#. (i) When reflected in the -ais, the equation will become: = ( +1)( ) (ii) When reflected in the -ais, the equation will become: = ( +1)( ) = ( +1)( + ) (iii) When horizontall translated + units, the equation will become: = (( )+1)(( ) ) = ( 1)( ) (iv) When verticall translated +3 units, the equation will become: = ( +1)( ) + 3 Teacher guide Representing Polnomials Graphicall T-1

12 Lesson task: The correct matching of the graphs and functions is shown below: Teacher guide Representing Polnomials Graphicall T-11

13 # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# # # # # #!5#!#!3#!#!1# # 1# # 3# # 5#!#!#!#!# Etension activit: Statements to Discuss: True or False? A1 A f () = f () = ( + ) is a factor of f () f () = f () = ( ) is a factor of f () A1 is false; A is true. If f() =, ( - ) must be a factor of f(). B1 B " A possible equation for this graph is: " A possible equation for this graph is: = ( +1)( ) = ( 1)( + ) B1 is true; B is false. There are also other possible equations for the function, such as = ( +1)( ). B is clearl false as the repeated root must be positive. C1 If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) = C If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) could be an number but zero C1 is true; C is false. The conditions mean that the function must take the form: f () = A( 1)( 3)( ), where A is a non-zero constant. Now f (5) = A, so it can take an value apart from zero. D1 f () = ( ) ( 7) g() = ( ) (7 ) f () is a reflection of g() over the ais D1 is false; D is true. D1 is a reflection over the -ais. D f () = ( ) g() = ( ) f () is a reflection of g() over the ais E1 E f () = g( + ) g() is a horizontal translation of f () f () = g() + g() is a horizontal translation of f () E1 is true; E is false. E is a vertical translation. Teacher guide Representing Polnomials Graphicall T-1

14 %)3 3 3,3 &3 )3 '/3 ',3 '(3 '&3 '%3 )3 %3 &3 (3,3 /3 '&3 ',3 '3 '3 '%)3 Assessment task: Cubic Graphs and Their Equations (revisited) 1. A suitable cubic function will take the form: = A ( ) or = A( ) where A is an real number, positive or negative.. A suitable sketch will show the -ais intercepts at (, ), (1, ), and (3, ) the -ais intercept at (, ) and be of the following shape:! When reflected over the -ais, the equation will become: = ( + )( 3)( 1) = ( )( + 3)( +1) 3. A suitable cubic function will take the form: = A( + a)( + b)( + c) where A is an real positive number, and a, b, c are positive numbers, such that a > b > c and Aabc =. Specific eamples are: = ( + )( + )( +1), or = ( + )( + )( +1). (i) A vertical translation through + units. (ii) A reflection over the -ais. (iii) A horizontal translation through 3 units = (( + 3) )(( + 3)+3) = ( +1)( + ) Teacher guide Representing Polnomials Graphicall T-13

15 Cubic Graphs and Their Equations 1. Write down an equation of a cubic function that would give a graph like the one shown here. It crosses the -ais at (-3, ), (, ), and (5, ) Write down an equation of a cubic function that would give a graph like the one shown here. It crosses the -ais at (, -) On the aes, sketch a graph of the function = ( +1)( - ). You do not need to plot it accuratel! Show where the graph crosses the - and -aes Write down the equation of the graph ou get after ou: (i) Reflect = ( +1)( - ) over the -ais: (ii) Reflect = ( +1)( - ) over the -ais: (iii) Horizontall translate = ( +1)( - ) through + units: (iv) Verticall translate = ( +1)( - ) through +3 units: Student materials Representing Polnomials Graphicall S-1 15 MARS, Shell Center, Universit of Nottingham

16 Cubic Graphs Graph A Graph B Graph C Graph D Graph E Graph F Student materials Representing Polnomials Graphicall S- 15 MARS, Shell Center, Universit of Nottingham

17 Cubic Graphs (continued) Complete these graphs for the remaining functions: Graph G Graph H Graph I Graph J Student materials Representing Polnomials Graphicall S-3 15 MARS, Shell Center, Universit of Nottingham

18 Cubic Functions 1 = ( -1)( + ) = ( +1)( + )( + ) 3 = -( +1)( + )( + ) = ( -1) ( - ) 5 = -( -1) ( - ) = ( - 3) 7 = -( +1) ( + )+ = -( +1) ( + ) 9 = ( -1) ( - )+ 1 = ( -1)( - )( - ) 11 = -( + 3) Student materials Representing Polnomials Graphicall S- 15 MARS, Shell Center, Universit of Nottingham

19 Statements to Discuss: True or False? A1 f () = f () = Þ ( + ) is a factor of f () A f () = f () = Þ ( - ) is a factor of f () B1 A possible equation for this graph is: = ( +1)( - ) B A possible equation for this graph is: = ( -1)( + ) C1 If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) = C If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) could be an number but zero. D1 D f () = ( - ) ( - 7) g() = ( - ) (7 - ) f () is a reflection of g() over the ais f () = ( - ) g() = -(- - ) f () is a reflection of g() over the ais E1 f () = g( + ) Þ g() is a horizontal translation of f () E f () = g() + Þ g() is a horizontal translation of f () Student materials Representing Polnomials Graphicall S-5 15 MARS, Shell Center, Universit of Nottingham

20 +/ 3 +' 3 +,3 +&3 +%3 (3 %3 &3, 3 ' 3 / ' 3 &3 ( 3 +&3 +' Cubic Graphs and Their Equations (revisited) 1. A cubic function has just two -intercepts, one at = and the other at =. Write down at least three possible equations with these intercepts:. On the aes, sketch a graph of the function = ( + )( - 3)( -1) You do not need to plot it accuratel! Show where the graph crosses the - and -aes. Write down the equation of the graph after ou reflect it over the -ais Write down an equation of a cubic function that would give a graph like the one shown here. It crosses the -ais at (, ).!. The function = ( - )( + 3) is transformed to: (i) = ( - )( + 3) +. Describe in words how the function has been transformed. (ii) = -( - )( + 3). Describe in words how the function has been transformed. (iii) = ( +1)( + ). Describe in words how the function has been transformed. Student materials Representing Polnomials Graphicall S- 15 MARS, Shell Center, Universit of Nottingham

21 Show me an equation that fits Graph A Graph B Graph C Graph D P-1 Projector Resources Representing Polnomials Graphicall

22 Working Together 1. Take turns to match a function to its graph.. As ou do this, label the graph to show the intercepts on the - and -aes. 3. If ou match two cards, eplain how ou came to our decision.. If ou don't agree or understand, ask our partner to eplain their reasoning. 5. You all need to agree on and eplain the matching of ever card.. You ma find that more than one function will fit some graphs! 7. If ou have functions left over, sketch graphs on the blank cards to match these functions. P- Projector Resources Representing Polnomials Graphicall

23 True or False? A1 f () = f () = ( + ) is a factor of f () A f () = f () = ( ) is a factor of f () P-3 Projector Resources Representing Polnomials Graphicall

24 True or False? B1 B1 "# $# %# &# '#!(#!%#!)#!&#!*# '# *# &# )# %# (#!&#!%#!$#!"#!" A possible equation for this graph is: = ( +1)(! ) B "# $# %# &# '#!(#!%#!)#!&#!*# '# *# &# )# %# (#!&#!%#!$#!"#!" A possible equation for this graph is: = (!1)( + ) P- Projector Resources Representing Polnomials Graphicall

25 True or False? C1 If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) = C If f () is a cubic function and f (1) =, f (3) = and f () =, then f (5) could be an number but zero. P-5 Projector Resources Representing Polnomials Graphicall

26 True or False? D1 f () = ( ) ( 7) g() = ( ) (7 ) f () is a reflection of g() over the ais D f () = ( ) g() = ( ) f () is a reflection of g() over the ais Projector Resources Representing Polnomials Graphicall

27 True or False? E1 f () = g( + ) g() is a horizontal translation of f () E f () = g() + g() is a horizontal translation of f () P-7 Projector Resources Representing Polnomials Graphicall

28 Mathematics Assessment Project Classroom Challenges These materials were designed and developed b the Shell Center Team at the Center for Research in Mathematical Education Universit of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, And Noes, and Daniel Pead We are grateful to the man teachers and students, in the UK and the US, who took part in the classroom trials that plaed a critical role in developing these materials The classroom observation teams in the US were led b David Foster, Mar Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) b Alan Schoenfeld at the Universit of California, Berkele, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the Universit of Nottingham Thanks also to Mat Crosier, Anne Flode, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularl grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from 15 MARS, Shell Center, Universit of Nottingham This material ma be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact map.info@mathshell.org if this license does not meet our needs.