1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 12 Precalculus Module 3 Rational and Exponential Functions

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Participant Poll Classroom teacher Math trainer or coach Principal or school leader District representative / leader Other

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives Participants will understand complex solutions to polynomial equations. Participants will understand function composition and function inverses. Participants will enrich their knowledge and experience in order to implement Module 3 with confidence and success.

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Today Module 3 Overview Module 3 Topic A – Polynomial Functions and the Fund. Theorem of Algebra Tomorrow Module 3, Topic B – Rational Functions and Composition of Functions Module 3, Topic C – Inverse Functions Lunch Module 4, Topic A – Trigonometric Functions Module 4, Topic B – Trigonometry and Triangles Module 4, Topic C – Inverse Trigonometric Functions Discussion for Implementation

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 3: Rational and Exponential Functions Module Overview 3 Topics 21 Lessons 25 days Topic A – Polynomial Functions and the Fundamental Theorem of Algebra Mid-Module Assessment (After Topic A) Topic B – Rational Functions and Composition of Functions Topic C – Inverse Functions End of Module Assessment (After Topic C)

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 3: Rational and Exponential Functions Standards Addressed in this presentation: (+) N-CN.C.9:Know the FTA; show valid for quadratic polynomials (+) A-APR.C.5:Know and apply the binomial theorem for (x + y) n (+) G-GPE.A.3: Derive equations of ellipses and hyperbolas (+) F-IF.C.7d:Graph rational functions (+) F-BF-A.1c:Compose functions (+) F-BF.B.4b:Verify inverses by composition (+) F-BF.B.4c:Read values of the inverse from a graph or table (+) F-BF-B.4d:Restrict the domain to produce an invertible function Others in the module: N-CN.C.8, G-GMD.A.2, A-APR.D.7, F-IF.C.9, F-BF-B.5

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 3: Rational and Exponential Functions Topic A Solving polynomial equations, and allowing complex number solutions. Square roots of complex numbers and roots of unity. Expanding binomial expressions using Pascal’s triangle. Understanding and graphing ellipses and hyperbolas. Topic B Rational functions and their graphs: end behavior and asymptotes. Function composition. Topic C Inverse functions. Restricting the domain of a function to find an inverse.

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Polynomial Functions and the Fundamental Theorem of Algebra Topic Overview 9 lessons (1-9). Review FTA and establish for quadratic functions. Explore square roots of complex numbers. Consider roots of unity. Know and be able to use the binomial theorem. Understand the meaning of curves in the complex plane. See ellipses and hyperbolas as curves in the complex plane and in the Cartesian plane. Compute volumes by Cavalieri’s principle.

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: The Fundamental Theorem of Algebra Opening Exercise: How many solutions are there to the equation x 2 = 1? Question: Can you prove it?

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: The Fundamental Theorem of Algebra Fundamental Theorem of Algebra: 1.Every polynomial function of degree n ≥ 1 with real or complex coefficients has at least one real or complex zero. 2.Every polynomial of degree n ≥ 1 with real or complex coefficients can be factored into n linear terms with real or complex coefficients.

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: The Fundamental Theorem of Algebra Implications of the FTA: If we allow complex numbers to be used as solutions and coefficients, then: 1.Every polynomial equation of degree 1 or higher has a solution. 2.Any polynomial of degree 1 or higher will factor completely into linear terms.

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: The Fundamental Theorem of Algebra A moment of proof: Prove that the FTA holds in the case when n = 2. That is, prove the following: 1.For a function f(x) = ax 2 + bx + c, there is at least one value of z so that f(z) = 0. 2.Every quadratic expression ax 2 + bx + c factors into linear terms.

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 2: Does Every Complex Number Have a Square Root? Let’s first think about this geometrically. Recall: The modulus of the complex number z = x + iy is the distance from the point (x,y) to the origin: The argument of the complex number z = x + yi is the amount that the positive x-axis is rotated to align with the ray from the origin to the point (x,y), between 0 and 2 . We often denote the modulus of a complex number by r and the argument by . Knowing the modulus and the argument uniquely defines the complex number z.

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 2: Does Every Complex Number have a Square Root? Question: What does squaring a complex number do to its modulus and its argument? Question: How can we find the square root of a given complex number?

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 2: Does Every Complex Number have a Square Root? Question: How many square roots does a nonzero complex number have? Describe them!

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 2: Does Every Complex Number have a Square Root? The square root of a complex number z = x + yi is another number p + qi: 1.Then (p + qi) 2 = x + yi, so p 2 - q 2 + 2pqi = x + iy. 2.Equating real and imaginary parts gives p 2 - q 2 = x and 2pq = y. 3.Solve the resulting system of equations. 4.Thus, to find a square root p + qi of z = x + yi, we solve the two equations p 2 - q 2 = x and 2pq = y.

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 2: Does Every Complex Number have a Square Root? Work through Exercise 8a in Lesson 2, using the algebraic process to find the square roots of z = 5 + 12i.

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Roots of Unity Question: How many square roots of 1 are there? Question: How many cube roots of 1 are there? Work through the Exploratory Challenge in Lesson 3 to find out.

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Roots of Unity For positive integers n, the n th roots of unity are the solutions to the equation z n = 1 in the complex numbers. Question: How many n th roots of unity do you expect there to be? Explain how you know.

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Roots of Unity Work through Exercises 1-4 in Lesson 3 to explore: the square roots of unity the fourth roots of unity the sixth roots of unity the fifth roots of unity Describe the roots of unity in both rectangular and polar form.

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Roots of Unity Plot the n th roots of unity in the complex plane for n = 4 and n = 6. Make a conjecture: Describe the location of the n th roots of unity in the complex plane.

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: The Binomial Theorem We start Lesson 4 by asking students to verify directly that 1 + i and 1 - i are both solutions to the polynomial equation The conclusion drawn: the process of evaluating this polynomial at a complex value of z is extremely tedious.

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 4: The Binomial Theorem Pascal’s Triangle The configuration of numbers below is known as Pascal’s triangle. This triangle is easy to generate, and surprisingly helpful when expanding binomial expressions.

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 4: The Binomial Theorem The oldest known image of Pascal’s Triangle is from 13 th century China, where it is still called Yang Hui’s triangle. It was known in China, Persia and India as early as the 11 th century, AD.

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: The Binomial Theorem The entries in Pascal’s triangle can be found recursively, as we have seen, but can also be found directly using binomial coefficients. The binomial coefficients are the entries in row n, position k of Pascal’s triangle, where we start counting with 0.

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: The Binomial Theorem Back to the original problem – how can we speed up the process of expanding (u + v) n ? Work through Exercises 8-11 in Lesson 4.

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: The Binomial Theorem The Binomial Theorem: For any expressions u and v, That is, the coefficients of the expanded binomial (u + v) n are exactly the numbers in Row n of Pascal’s triangle.

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 5: The Binomial Theorem Work through Exercises 1 and 2 in Lesson 5.

29. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 5: The Binomial Theorem Work through the Exit Ticket from Lesson 5.

30. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 6: Curves in the Complex Plane Question: What is the graph of the equation

31. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 6: Curves in the Complex Plane Work through Exercises 1 and 2 in Lesson 6.

32. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 6: Curves in the Complex Plane The anatomy of an ellipse: Vocabulary: Ellipse, vertices, major axis, semi-major axis, minor axis, semi-minor axis, vertices center.

33. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 6: Curves in the Complex Plane Work through Example 2 in Lesson 6.

34. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 6: Curves in the Complex Plane In the coordinate plane, ellipses centered at the origin can be represented by the equation where |a| is half of the length of the horizontal axis and |b| is half the length of the vertical axis. The points on an ellipse centered at the origin can be written in polar form as

35. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: Curves from Geometry Watch the video of the Whispering Gallery in the National Statuary Hall in the U.S. Capitol. http://www.youtube.com/watch?v=FX6rUU_7kkhttp://www.youtube.com/watch?v=FX6rUU_7kk.

36. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: Curves from Geometry The docent demonstrates one geometric property of an ellipse: Every ray emanating from one focus of the ellipse reflects off the curve in such a way that it travels to the other focus of the ellipse.

37. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: Curves from Geometry Thus, we can describe an ellipse as a set of points P in the plane so that the sum of the distances from P to two fixed points is constant. The general equation for an ellipse centered at (h,k) is

38. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: Curves from Geometry Do the Exercise: Points F and G are located at (0,3) and (0,-3). Let P(x, y) be a point so that PF + PG = 8. Use this information to show that the equation of the ellipse is

39. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions When a satellite moves in a closed orbit around a planet, it follows an elliptical path. however, if the satellite is moving fast enough, it will overcome the gravitational attraction of the planet and break out of its closed orbit. The minimum velocity required for a satellite to escape the closed orbit is called the escape velocity. The velocity of the satellite determines the shape of its orbit. Lesson 8: Curves from Geometry

40. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What happens to the graph if we change the equation to ? Lesson 8: Curves from Geometry

41. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The geometric property of hyperbolas & ellipses: Given two points F and G in the plane, An ellipse is the set of all points P in the plane so that the sum of the distances PF + PG is constant. A hyperbola is the set of all points in the plane so that the difference of the distances PF – PG is constant. Lesson 8: Curves from Geometry

42. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 8: Curves from Geometry Work through Exercises 1-4, in which you will derive an equation of a hyperbola given the foci, and graph the resulting equation.

43. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 8: Curves from Geometry

44. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic A Let’s Review What major concepts and/or themes did you notice in Topic A? How do you think those topics and themes will apply as we move into studying rational expressions and graphing rational functions?

45. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Rational Functions and Composition of Functions Topic Overview 8 lessons (10 - 17). Rational expressions and rational functions. Graphing rational functions: End behavior and asymptotes. Transformations of rational functions. Function composition.

46. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 14: Graphing Rational Functions Do Exercises 1-10, working in pairs. Summarize the key features of the graph of a rational function.

47. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 14: Graphing Rational Functions Key features of the graph of a rational function: x-intercepts y-intercepts end behavior vertical asymptotes horizontal asymptotes

48. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 16: Function Composition Function composition is introduced via an example:

49. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 16: Function Composition Work through Example 1, which presents the ideas of function composition before formally introducing composition notation.

50. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 16: Function Composition

51. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 16: Function Composition Work through Example 2, in which students interpret standard composition notation in order to determine which compositions make sense.

52. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 16: Function Composition Work through Problem 2 on the Exit Ticket for Lesson 16.

53. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 17: Solving Problems by Function Composition Student outcomes: write equations that represent functional relationships and use the equations to compose functions. analyze the domains and ranges of functions and function compositions represented by equations, and solve problems by composing functions.

54. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 17: Solving Problems by Function Composition Work through Example 2 in Lesson 17.

55. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic B Let’s Review What major concepts and/or themes did you notice in Topic B? How have the topics and themes of Topic B prepared you (and students) to understand the upcoming work in Topic C on Inverse Functions?

56. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Inverse Functions Topic Overview 4 lessons (18-21). Inverse functions. Restricting the domain. Inverse of logarithmic and exponential functions. Logarithmic and exponential problem solving.

57. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 18: Inverse Functions Work through Exercises 1-10 in Lesson 18.

58. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 18: Inverse Functions

59. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 18: Inverse Functions Work through Exercises 11-13 in Lesson 18.

60. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 18: Inverse Functions

61. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 18: Inverse Functions Work through Problems 3a-d in the Problem Set.

62. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 19: Restricting the Domain A function is said to be invertible if its inverse is also a function. Look at Exercise 7 and determine which of the functions are invertible. Continue to Exercise 8, which is the first instance where students need to restrict the domain of a function to make it invertible.

63. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 21: Logarithmic and Exponential Problem Solving In this lesson, students use radiocarbon dating to estimate the time of death for wooly mammoth specimens found in Siberia. Look through the Exploratory Challenge for this lesson. These exercises require number sense and quantitative reasoning in addition to knowledge of inverses of exponential functions.

64. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic C Let’s Review What major concepts and/or themes did you notice in Topic C? How do those topics and themes rely on topics studied earlier in the module?

65. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes in Module 3: Rational and Exponential Functions Working in the complex plane: Solving polynomial equations, graphing functions, converting equations for ellipses and hyperbolas between complex and real form. Understanding rational expressions and graphing rational functions. Composing functions and finding inverse functions.

66. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Biggest Takeaway What is your biggest takeaway from this study of Module 3? Given your role, how can you support successful implementation at your school?