1. Section 2.1 Complex Numbers

2. The Imaginary Unit i

6. Example Express as a multiple of i:

7. Operations with Complex Numbers

9. Example Perform the indicated operation:

11. Example Perform the indicated operation:

12. Complex Conjugates and Division

14. Using complex conjugates to divide complex numbers

15. Example Divide and express the result in standard form:

16. Roots of Negative Numbers

19. Example Perform the indicated operations and write the result in standard form:

20. Example Perform the indicated operations and write the result in standard form:

21. Section 2.2 Quadratic Functions

22. Graphs of Quadratic Functions

23. Graphs of Quadratic Functions Parabolas Minimum Vertex Axis of symmetry Maximum

25. Graphing Quadratic Functions in Standard Form

27. Seeing the Transformations

28. Example Graph the quadratic function f(x) = - (x+2) 2 + 4.

29. Graphing Quadratic Functions in the Form f(x)=ax 2 =bx+c

32. Using the form f(x)=ax 2 +bx+c a>0 so parabola has a minimum, opens up

33. Example Find the vertex of the function f(x)=-x 2 -3x+7

35. Example Graph the function f(x)= - x 2 - 3x + 7. Use the graph to identify the domain and range.

36. Minimum and Maximum Values of Quadratic Functions

38. Example For the function f(x)= - 3x 2 + 2x - 5 Without graphing determine whether it has a minimum or maximum and find it. Identify the function’s domain and range.

39. Graphing Calculator – Finding the Minimum or Maximum Input the equation into Y= Go to 2 nd Trace to get Calculate. Choose #4 for Maximum or #3 for Minimum. Move your cursor to the left (left bound) of the relative minimum or maximum point that you want to know the vertex for. Press Enter. Then move your cursor to the other side of the vertex – the right side of the vertex when it asks for the right bound. Press Enter. When it asks to guess, you can or simply press Enter. The next screen will show you the coordinates of the maximum or minimum.

40. Section 2.3 Polynomial Functions and Their Graphs

41. Smooth, Continuous Graphs

42. Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system.

43. Notice the breaks and lack of smooth curves.

44. End Behavior of Polynomial Functions

46. Odd-degree polynomial functions have graphs with opposite behavior at each end. Even-degree polynomial functions have graphs with the same behavior at each end.

47. Example Use the Leading Coefficient Test to determine the end behavior of the graph of f(x)= - 3x 3 - 4x + 7

48. Zeros of Polynomial Functions

49. If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. These values of x are the roots, or solutions, of the polynomial equation f(x)=0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function.

50. Find all zeros of f(x)= x 3 +4x 2 - 3x - 12

51. Example Find all zeros of x 3 +2x 2 - 4x-8=0

52. Multiplicity of x-Intercepts

55. Graphing Calculator- Finding the Zeros x3+2x 2 - 4x-8=0 One of the zeros The other zero Other zero One zero of the function The x-intercepts are the zeros of the function. To find the zeros, press 2 nd Trace then #2. The zero -2 has multiplicity of 2.

56. Example Find the zeros of f(x)=(x- 3) 2 (x-1) 3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. Continued on the next slide.

57. Example Now graph this function on your calculator. f(x)=(x- 3) 2 (x-1) 3

58. The Intermediate Value Theorem

60. Show that the function y=x 3 - x+5 has a zero between - 2 and -1.

61. Example Show that the polynomial function f(x)=x 3 - 2x+9 has a real zero between - 3 and - 2.

62. Section 2.4 Dividing Polynomials; Remainder and Factor Theorems

63. Long Division of Polynomials and The Division Algorithm

64. Dividing Polynomials Using Synthetic Division

65. The Factor Theorem

66. Solve the equation 2x 3 -3x 2 -11x+6=0 given that 3 is a zero of f(x)=2x 3 -3x 2 -11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor. Another factor

67. Example Solve the equation 5x 2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x 2 + 9x - 2

68. Section 2.5 Zeros of Polynomial Functions

69. The Rational Zero Theorem

71. Example List all possible rational zeros of f(x)=x 3 -3x 2 -4x+12 Find one of the zeros of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function? How can the graph below help you find the zeros?

74. The Fundamental Theorem of Algebra

76. The Linear Factorization Theorem

78. Example Find a fourth-degree polynomial function f(x) with real coefficients that has -1,2 and i as zeros and such that f(1)=- 4

79. Descartes’s Rule of Signs

82. Example For f(x)=x 3 - 3x 2 - x+3 how many positive and negative zeros are there? What are the zeros of the function?