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Guess Who…

Guess Who…. Rules Yes or No Questions ONLY One question per person Whiteboards: Brainstorm questions you will ask to determine who you are Predict: How many questions until you figure out who you are?. Play…. Once you figure out who you are sit back down Erase whiteboards and answer:

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Guess Who…

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  1. Guess Who… • Rules • Yes or No Questions ONLY • One question per person • Whiteboards: Brainstorm questions you will ask to determine who you are • Predict: How many questions until you figure out who you are?

  2. Play… • Once you figure out who you are sit back down • Erase whiteboards and answer: • Did it take more or less questions than you thought? Explain WHY this happened

  3. Guess Who… What is the equation of this polynomial? • Create a PLAN: How would you go about determining the equation?

  4. Unit 3- Polynomials and Quadratics in the Real Number System 3A-I can identify key features of a polynomial (graphically & algebraically) and explain the connection between roots, factors, multiplicity, degree, and the stretch/compress factor in writing equations of and graphing polynomials.

  5. What’s in a Polynomial Name? • Vocabulary • A sum of monomials or a single monomial is considered a polynomial • Polynomials can be written in two forms: • Standard Form-Written with terms in descending order by degree • Factored Form-Written with factors instead of terms • Polynomials are classified by their degree (exponent with the highest value), their terms, or when possible both

  6. B. Classifying Polynomials • By Degree 0=Constant 1=Linear 2=Quadratic 3=Cubic 4=Quartic 2) By Terms 1 term=Monomial 2 terms=Binomial 3 terms=Trinomial 4 or more terms=Polynomial

  7. C. Sketching Polynomials • Use the chart and graphing calculators to sketch each polynomial in the correct “spot”

  8. 1) What is the connection between the degree of the polynomial and the roots of the polynomial? 2) Explain the difference between the graphs of polynomials with even degrees versus odd degrees. 3) Sketch what the graph of f(x)= -8x7 would look like. DEFEND in words that your sketch is “correct”.

  9. D. Examples • Describe everything you possibly can about the polynomial you see

  10. D. Examples • Describe everything you possibly can about the polynomial you see

  11. D. Examples • Describe everything you possibly can about the polynomial you see

  12. GUESS WHO… • Player A describes to Play B • Player B tried to draw the polynomial • Player A describes “missing pieces” needed until Player B’s graph is good • Switch roles

  13. HOMEWORK • Page 318 (1-10) • Write out the end behavior the way we have done in class!!! • Or work on concept map from Unit 2

  14. Guess Who… What is the equation of this polynomial? • Create a PLAN: What ideas do you have now?

  15. Guess Who… What is the equation of this polynomial? • Create a PLAN: What ideas do you have now? • Together: What do you think we NEED to find the equation?

  16. Model Card Sample

  17. Model Card Must Address These Issues…

  18. Exit Post-It On Homework PICK ONE and sketch the polynomial f(x)=-5x3+7x2-10x f(x)=-8x4+5x3-2x2+7x-1 Write the end behavior of the polynomial

  19. II. Polynomial Equations  Graphs • Definitions • Roots=X-Intercepts=Zeros=Solutions • Polynomials can be written in two forms: standard form and factored form • Factors are the “pieces” used to build/break down a polynomial function

  20. Relationship between Roots and Factors of a Polynomial

  21. Relationship between Roots and Factors of a Polynomial • EXPLORE with your graphing calculator • ANSWER (on your paper): What pattern do you notice between the function in factored form and the roots of the function? Keep Going After you have answered the question on the paper…

  22. Relationship between Roots and Factors of a Polynomial • EXPLORE with your graphing calculator • ANSWER: What pattern do you notice between the function in factored form and the roots of the function? • TOGETHER (whiteboards): Based on your pattern… • If a root of a polynomial was M what would a factor of the polynomial be? • If a factor of a polynomial was (x+c) what would a root of the polynomial be? • If a factor of a polynomial was (Ax+c) what would a root of the polynomial be? • EXPLAIN YOUR REASONING!!!

  23. Relationship between Roots and Factors of a Polynomial • If a polynomial has a root (x-intercept) of M, then the polynomial has a factor of (X-M)

  24. C. Key Features of Polynomials • ??? • ??? • ??? • ??? • ??? • ??? • ??? • Stretch/Compress Factor (A) • Multiplicity

  25. D. Relationship between ODD and EVEN Multiplicity Factors • EXPLORE with your graphing calculator • ANSWER (on your paper): What pattern do you notice between the function in factored form and the GRAPH of the function? Keep Going After you have answered the question on the paper…

  26. Homework • Level A Pg. 323 (13, 14, 16-18) • Level B Pg. 323 (19-20, 47-50) • Level C Pg. 353 (15, 37, 39, 58) • EVERYONE PICK TWO OF Pg. 323 (57, 59, 61, or 64)

  27. D. Relationship between ODD and EVEN Multiplicity Factors • EXPLORE with your graphing calculator • ANSWER (on your paper): What pattern do you notice between the function in factored form and the GRAPH of the function? • TOGETHER (without a calculator): Sketch a graph of the following • g(c)=8(c+2)4(c-9)3 • h(t)=-5(t-10)6(t-8)2 • A polynomial function with roots at x=-4 with a multiplicity of 3, x=0 with a multiplicity of 5, and x=7 with a multiplicity of 6 • EXPLAIN YOUR REASONING!!!

  28. D. Relationship between ODD and EVEN Multiplicity Factors • When factors/roots appear more than once in a polynomial function this is called multiplicity • EVEN Multiplicities will look like… • ODD Multiplicities will look like…

  29. iv. Sketch the graph of g(c)=8(c+2)4(c-9)3

  30. V. Sketch the graph of h(t)=-5(t-10)6(t-8)2

  31. vi. Sketch the graph of a polynomial function with roots at x=-4 with a multiplicity of 3, x=0 with a multiplicity of 5, and x=7 with a multiplicity of 6

  32. Homework • Level A Pg. 323 (21-28) Write in FACTORED FORM • Level B Pg. 323 (29, 30, 33-36, 51-53) • Level C Pg. 353 (15, 37, 39, 58) • EVERYONE PICK THE OTHER TWO OF Pg. 323 (57, 59, 61, or 64)

  33. III. The Stretch/Compress Factor A. Sketch a graph of the following three functions and explain IN WORDS the difference you see in the graphs f(x)= (x-3)2(x+1)3 f(x)=8(x-3)2(x+1)3 f(x)= ¼ (x-3)2(x+1)3

  34. The polynomial is B. The Stretch/compress factor is the “a” in factored and standard form of a polynomial. The polynomial is The polynomial is When A is…

  35. C. Examples-Write the equation in factored form for the function below (0, -120)

  36. (1, -10.5)

  37. (4.781, -4.034)

  38. Sideways • Level A Equation in Factored form when stretch compress factor is a=1 • Level B Equation in Factored form when stretch compress factor is a≠1 • FIND Answer then move SIDEWAYS

  39. Guess Who… What is the equation of this polynomial? • Create a PLAN: What ideas do you have now? • Write down everything you can about this polynomial (analyze the polynomial)

  40. Guess Who… What is the equation of this polynomial? • Create a PLAN: What ideas do you have now? • In teams on big whiteboards set up Model Card (make sense of problem and step to solving)

  41. Guess Who… • Sort the polynomials into categories • No polynomial can be on it’s own • Name your categories

  42. Guess Who… • Sort the polynomials into categories • No polynomial can be on it’s own • Name your categories • Write each polynomial onto a post-it • Together sort the polynomials into at least four categoriesNAME each category • No polynomial can “fit” into more than one category

  43. Guess Who… • Sort the polynomials into categories • No polynomial can be on it’s own • Name your categories • Write each polynomial onto a post-it • Together sort the polynomials into at least four categoriesNAME each category • No polynomial can “fit” into more than one category • Record your new thinking

  44. Learning Target 3B I can create equivalent expressions/functions of polynomials by applying various factoring techniques and explain the connections between the algebraic and graphical form of the polynomial.

  45. NOTES FROM SUB DAY GO HERE • Converting Factored Form into Standard Form One last example Convert f(x)=-3(x-3)2(x+1) to standard form

  46. II. Factoring Polynomials PRIOR: In Algebra I you learn how to factor all of the following: • x2-10x+21 • 6x3-12x2+36x • 49b2-c2 • 3x2+4x-4

  47. II. Factoring Polynomials PRIOR: In Algebra I you learn how to factor all of the following • x2-10x+21 • 6x3-12x2+36x • 49b2-c2 • 3x2+4x-4 • 16y2-1 • 4m2x+7m2y-8x-14y • 6x4y2+8x2y-8 Can you factor each polynomial? Can you name the factoring method you are using? Try the level C Algebra I factoring questions 5-7

  48. Homework • Watch a video OR read about factoring: • Factoring greatest common factor from polynomials • Factoring difference of squares • Factoring quadratic trinomials when a=1 • Factoring quadratic trinomials when a≠1 • Pick TWO that you need to review Bring in EVIDENCE of having watched the video or print and annotate the page from on-line

  49. II. Factoring Polynomials • Special Factoring Patterns • Sum of Cubes a3+b3=(a+b)(a2-ab+b2) • Difference of Cubes a3-b3=(a-b)(a2+ab+b2) • Difference of Squares a2-b2=(a-b)(a+b) THINK: 13= 43= 73= 103= 23= 53= 83= 113= 33= 63= 93= 123=

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