MHF 4U Unit 1 Polnomial Functions Outline Da Lesson Title Specific Epectations 1 Average Rate of Change and Secants D1., 1.6, both D1.1A s - Instantaneous Rate of Change and Tangents D1.6, 1.4, 1.7, 1.5, both D1.1A s 4 Solving Problems Involving Average and Instantaneous Rate of Change Numericall and Graphicall 5-6 Characteristics of Polnomial Functions Through Numeric, Graphical, and Algebraic Representations D1.8, 1.1 A1.1, 1., 1., 1.4 7 (Lesson included) Using the Factored Form of a Polnomial Function to Sketch a Graph and Write Equations A1.6, 1.8 8 Transformations of f() = and f() = 4 and Even and Odd Functions A1.7, 1.9 9-10 (Lessons included) Dividing Polnomials, The Remainder Theorem and The Factor Theorem A.1, A.1A 11-1 The Zeros of a Polnomial Function Graphicall and Algebraicall with Applications to Curve Fitting 1-14 Solving Polnomials Inequalities Graphicall, Numericall, and Algebraicall A.,., 4,.6 A.1,.. 15-16 JAZZ DAY 17 SUMMATIVE ASSESSMENT TOTAL DAYS: 17 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 1 of 15
Unit 1: Da 7: In Factored Form Learning Goals: Materials Minds On: 15 Make connections between a polnomial function in factored form and the BLM 1.7.1 - intercepts of its graph BLM 1.7. Sketch the graph of polnomial functions, epressed in factored form using BLM 1.7. Action: 40 the characteristics of polnomial functions Graphing calculators/ Determine the equation of a polnomial given a set of conditions (e.g. Consolidate:0 software/ zeros, end behaviour) and recognize there ma be more than one such sketch: function GSP_gr1_U1D 7.gsp Total=75 min Assessment Opportunities Minds On Individual Eploration Displa sketches of Recall with students that (-1) =(-1) (-1), (-1) esterdas =(-1) (-1) (-1) etc. polnomial functions. Students will complete questions 1, and on BLM 1.7.1. Take up questions Reference to finite and. Possible answer #: the degree of the polnomial is equal to the number differences and its of factors that contain an. Possible answers #: multipl the factored form; connection to the degree would be multipl the variable term of each factor and logicall conclude that this is the beneficial degree of the polnomial. Action! Individual Investigation Students observe the sketches of the graphs generated and displaed in minds on and answer questions #4, 5, 6. Teacher takes up questions with the class or students check their answers in pairs. Make connection between the -intercept and its corresponding co-ordinates ( I,0). Introduce language to describe shape and post on word wall. Possible answer #4: the graph crosses the ais when = 0 in the polnomial function. This occurs when each factor is equal to zero. When these mini equations are solved the -intercepts are evident. Answers #6 respectivel: The graph bounces at the intercept. The graph has an inflection point at the intercept. Pairs Think Pair Share Students complete BLM 1.7. in pairs. Curriculum Epectation/Observation/Mental Note: Circulate to observe pairs are successful in representing the graph of a polnomial function given the equation in factored form. Make a mental note to consolidate misconceptions. Whole Class Discussion then Practice Present a graph of a polnomial function with intercepts and ask: What is a possible equation of this polnomial? How do ou know? What are some possible other equations? Do a few other eamples including some with double roots and inflection points. Then state some characteristics of a polnomial function and ask for an equation that satisfies these characteristics. (e.g. What is a possible equation of the polnomial function of degree that begins in quadrant, ends in quadrant 4, and has intercepts of -1, ½, and 4.) Students complete BLM 1.7.. Word wall: concave up/down, hills, valles, inflection point (informal descriptions in students' own words) If students look onl at factors of the form (-a) the ma have the misconception that the intercept is the opposite of the a value. Use some factors of the form (a-b) and have the students understand that the intercept can be found b solving a-b=0 This assessment for learning will determine if students are read to formulate an equation given the characteristics of a polnomial function. Consolidate Debrief Pairs to Whole Class A answers B Teacher asks the following questions: What are the advantages of a polnomial function being given in factored form? Wh is it possible that people sketching the same graph could have higher or lower hills and valles? How could ou get a more accurate idea of where these turning points are located? Eplain wh the graph the function = ( 4)( 9) is easier or more difficult to graph than = ( + 4)( + 9). How can ou identif the intercept of a function given algebraicall and wh is this helpful to know? A answers B: Ask a series of questions one at a time, to be answered in pairs, then ask for one or more students to answer, alternating A and B partners Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page of 15
Concept Practice Home Activit or Further Classroom Consolidation Assign questions similar to BLM 1.7.1 Ask the question: What information is needed to determine an eact equation for a graph if intercepts are given? Possible answer to be taken up the net da and followed through in lesson : Another point on the function. A-W 11 McG-HR 11 H11 A-W1 (MCT) H1 McG-HR 1 1.1.1 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page of 15
1.7.1 What Role Do Factors Pla? 1. Use technolog (graphing calculator, software, GSP_Gr1_U1D7) to determine the graph of each polnomial function. Sketch the graph, clearl identifing the -intercepts. a) f ()= ( - )( + 1) b) f () = ( - )( + 1)(+) c) f () = - ( - )( + 1)(+) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 -intercepts: c) f () = (+1) = (+1)(+1) -intercepts: d) f () = ( - ) ( + ) -intercepts: f) f () = ( - )( + 1)(+) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 -intercepts: g) f () = (-) -intercepts: h) f () = ( +)( - 1)(-) -intercepts: i) f () = -( - )(+) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 -intercepts: -intercepts: -intercepts:. Compare our graphs with the graphs generated on the previous da and make a conclusion about the degree of a polnomial when it is given in factored form.. Eplain how to determine the degree of a polnomial algebraicall if given in factored form. 4. What connection do ou observe between the factors of the polnomial function and the -intercepts? Wh does this make sense? (hint: all co-ordinates on the ais have = 0). 5. Use our conclusions from #4 to state the -intercepts of each of the following. Check b graphing with technolog, and correct if necessar. f() = (-)(+5)(-1/) -intercepts: does this check? f() = (-)(+5)(-1) -intercepts: does this check? f() = (-)(+5)(-1)(-) -intercepts: does this check? 6. What do ou notice about the graph when the polnomial function has a factor that occurs twice? Three times? Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 4 of 15
1.7. Factoring in our Graphs Draw a sketch of each graph using the properties of polnomial functions. After ou complete each sketch, check with our partner, discuss our strategies and make an corrections needed. a) f ()= ( - 4)( + ) d) f () = -( - 1)( + 4)( ½ ) c) f () = ( - 1)( + 1) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 e) f () = ( -) d) f () = - ( - ) ( + ) f) f () = ( - )( + 1)(+) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 g) f () = (-4) h) f () = -( +) ( - ) i) f () = ( +)( -1)(-)(+ 4) 5 4 1 1 4 5 6 5 4 1 1 4 5 6 5 4 1 1 4 5 6 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 5 of 15
1.7. What's M Polnomial Name? 1. Determine a possible equation for each polnomial function. a) f ()= f) f () = c) f () = g) f () = d) f () = f) f () =. Determine an eample of an equation for a function with the following characteristics: a) Degree, a double root at 4, a root at - b) Degree 4, an inflection point at, a root at 5 c) Degree, roots at ½, ¾, -1 d) Degree, starting in quadrant, ending in quadrant 4, root at - and double root at e) Degree 4, starting in quadrant, ending in quadrant 4, double roots at -10 and 10 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 6 of 15
Unit 1: Da 9: Dividing Polnomials (Da 1) Learning Goals: Minds On: 5 Divide polnomials Eamine remainders of polnomial division and connect to the remainder theorem Make connections between the polnomial function f(), the divisor a, the Action: 50 remainder of the division f()/(-a) and f(a) using technolog Identif the factor theorem as a special case of the remainder theorem Consolidate:0 Factor polnomial epressions in one variable of degree greater than two. MHF4U Materials BLM 1.9.1 BLM 1.9.. Total=75 min Assessment Opportunities Minds On Whole Class Discussion Students should put awa their calculators and attempt the following without the use of calculators remind them of the long division methods for dividing: Divide 789 b 7. result: 11 remainder 5 Divide 1546 b 6. result: 091 remainder 0 Divide 455 b 4. result: 811 remainder Include as man more as desired until students have a clear understanding of the method of long division. Action! Consolidate Debrief Eploration Application Small Groups Eperiment Students will work either as a class or in pairs to complete the activities: Activit 1: Read through the worked eample on BLM 1.9.1 and then do the division questions on page of BLM 1.9.1 Activit : Work time for homework eercises (BLM 1.9. or tetbook work). Small Groups Interview Students will consolidate : Activit 1: Teacher can choose to have students work in pairs or work through the first few questions as a class and then in pairs or individuall for the remainder. As students finish, the teacher can have students complete questions on the board to aid those who might be having difficult. Activit : Complete eercises on BLM 1.9.. Home Activit or Further Classroom Consolidation BLM 1.9. or tetbook work from outlined tet below: Addison Wesle 1 (MCT): Sections.4 A-W 11 McG-HR 11 H11 A-W1 (MCT) H1 McG-HR 1 1., 1.4,.1,..,.,.4 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 7 of 15
1.9.1 Dividing Polnomials Dividing a polnomial b another polnomial is similar to performing a division of numbers using long division. For eample, divide the polnomial + 1 + 9 + 46 b + 9 Solution: 1) 9 + 1 + 9 + + 46 first divide into to get ) ) 4) + 9 + 1 + 9 + 46 now multipl b + 9 to get + 9 + 9 + 9 + 9 + 9 4 + 1 + 9 4 + 1 + 4 + 9 + 46 + 9 + 4 + 9 + 46 + 9 4 + 9 4 + 6 + 1 + 4 + + 9 + 46 then subtract + 9 from + 1 to get 4 bring down the + 9 divide 4 b to get 4 now multipl 4 b + 9 to get 4 + 6 then subtract 4 + 6 from 4 + 9 to get 5) + 9 4 4 + 9 + 6 + 46 bring down the + 46 divide b to get multipl b + 9 to get + 7 then subtract + 7 from + 46 to get 19 + 7 19 Since the remainder has a lower degree than the divisor, the division is now complete. The result can be written as: + 1 + 9 + 46 =( + 9)( + 4 + ) + 19 (NOTE: You could check our answer b multipling out the result.) Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 8 of 15
1.9.1 Dividing Polnomials (Continued) Using the previous eample, complete the polnomial division questions below: 1. 5 10 b. + 7 6 b +. 8 41 b + 4 4. 8 b + 1 [*note: 8 41 = + 0 8 41] 5. 6 + 9 + 7 1 b 1 6. + 4 1 b + 4 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 9 of 15
1.9.1 Dividing Polnomials (Answers) 1. 5 10 b. + 7 6 b + Result: 5 7 10 + 6 7 10 ( )( 7 + 14 4 7) 4 + + + 6 5 5 5 1 7 6 7 15 1 6 1 6 0 Result: ( + )( 5 1) etra ( + )( + )( 4). 8 41 b + 4 4. 8 b + 1 4 1 + 4 8 41 + 1 8 + 4 4 4 8 16 1 41 + 4 4 8 6 1 48 6 7 0 Result: ( + 4)( 4 1) + 7 Result: ( + 1)( ) etra ( + 4)( + )( 6) + 7 etra ( + 1)( + 1)( ) 5. 6 + 9 + 7 1 b 1 6. + 4 1 b + 4 + 1 + 10 1 6 + 9 + 7 1 6 + + 4 + 4 1 6 + 7 + 4 6 1 0 1 0 1 1 0 10 0 Result: etra ( 1)( + 1 + 10) Result: ( + 4)( ) ( 1)( )( 5) etra ( + 4)( + )( ) Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 10 of 15
1.9. Dividing Polnomials Complete the eercises below: 1. Find each quotient and remainder: (a) ( + 6 + 15) ( + ) (b) ( 4 + 1) ( ) (c) ( + ) ( + ) (d) ( + 4 ) ( 4). When a certain polnomial is divided b +, the quotient is + 5 and the remainder is 6. What is the polnomial?. When a certain polnomial is divided b, the quotient is + 4 7 and the remainder is 4. What is the polnomial? 4. Divide: (a) ( + 4 1) ( ) (b) ( + 11 1) ( + 1) (c) ( + 4 ) ( 4) (d) ( + 14 1) ( ) Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 11 of 15
Unit 1: Da 10: Dividing Polnomials (Da ) Learning Goal: Minds On: 5 Divide polnomials Eamine remainders of polnomial division and connect to the remainder theorem Make connections between the polnomial function f(), the divisor a, the Action: 50 remainder of the division f()/(-a) and f(a) using technolog Identif the factor theorem as a special case of the remainder theorem Consolidate:0 Factor polnomial epressions in one variable of degree greater than two. MHF4U Materials BLM 1.10.1 BLM 1.10.. Total=75 min Minds On Whole Class Discussion Students should take out the BLM 1.9.1 from last da: Eamine the results from the left hand (odd) questions and the right hand (even) questions. Using the original polnomial tr substituting the indicated values for the variable in each question: 1. =. = - ( = -/ and = 4). = -4 ( = - and = 6) 4. = -½ ( = -1 and = ) 5. = ½ ( = -/ and = 5) 6. = -4 ( = and = ) Note: at this point (if not previousl completed) it might be suggested to the students that the further factor their previous answers for questions through 6 and then tr the numbers given in parenthesis above Assessment Opportunities Action! After eamining the results tr and come to a generalization of the results this is essentiall the Remainder Theorem (odd questions) and Factor Theorem (even questions) Small Groups Eperiment Students will work either as a class or in pairs to complete the activities: Activit 1: Complete the minds on and then work on the BLM 1.10.1. Activit : Work time for homework eercises BLM 1.10. or tetbook work. Consolidate Small Groups Interview Debrief Students will consolidate : Activit 1: Teacher can choose to have students work in pairs or work through the first couple of questions as a class and then either in pairs or individuall for the remainder. As students finish, the teacher can have students complete questions on the front board to aid those who might be having difficult. Activit : Complete eercises on BLM 1.10.. Eploration Application Home Activit or Further Classroom Consolidation BLM 1.10. or tetbook work from outlined tet below: Addison Wesle 1 (MCT): Sections.5 A-W 11 McG-HR 11 H11 A-W1 (MCT) H1 McG-HR 1 1., 1.4,.1,..,.,.4 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 1 of 15
1.10.1 Remainder Theorem and Factor Theorem Remainder Theorem: When a polnomial f () is divided b a, the remainder is f (a) 1. Find the remainder when + 17 0 is divided b each of the following: (a) 1 (b) (c) (d) +1 (e) + (f) + Factor Theorem: If = a is substituted into a polnomial for, and the remainder is 0, then a is a factor of the polnomial.. Using the above Theorem and our results from question 1 which of the given binomials are factors of + 17 0?. Using the binomials ou determined were factors of + 17 0, complete the division (i.e. divide + 17 0 b our chosen ( a ) and remember to full factor our result in each case. Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 1 of 15
1.10.1 Remainder Theorem and Factor Theorem (Answers) 1. Find the remainder when + 17 0 is divided b each of the following: (a) 1 (b) (c) a = 1 f (1) = (1) + (1) 17(1) 0 a = a = f ( 1) = + 17 0 f ( a) = 6 f ( a) = 0 f ( 1) = 4 (d) +1 (e) + (f) + a = 1 a = a = f ( a) = 1 f ( a) = 0 f ( a) = 6. Using the above Theorem and our results from question 1 which of the given binomials are factors of + 17 0? From results (c) and (e) + are factors. Using the binomials ou determined were factors of + 17 0 complete the division (i.e. divide in each case. + 17 0 b our chosen a ) and remember to full factor our result (c) (e) + + 9 + 10 + 17 0 + + 15 17 0 6 9 9 17 7 10 0 10 0 0 + 4 17 15 0 15 0 0 Result: ( )( + 9 + 10) Result: ( + )( 15) ( )( + 5)( + ) ( + )( + 5)( ) (Note: The results are the same just rearranged.) Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 14 of 15
1.10. Dividing Polnomials Practice Complete the polnomial divisions below: 1. Without using long division, find each remainder: (a) ( + 6 + 8) ( + 1) (b) ( + 4 + 1) ( 4) (c) ( + 6 4 + ) ( + ) (d) ( + 7 11) ( ). Find each remainder: (a) ( + 6) ( + ) (b) ( + 6 4 + ) ( + 1) (c) ( + 1 1) ( ) (d) ( 4 + 4 + ) ( + ). When + k 4 + is divided b + the remainder is 6, find k. 4. When + k 1 is divided b 1 the remainder is, find k. ANSWERS: 1. (a) 4 (b) 44 (c) 7 (d) 7. (a) 0 (b) 11 (c) -5 (d) 6. 6 4. 4 Advanced Functions: MHF4Y Unit 1 Polnomial Functions (Draft August 007) Page 15 of 15