1. Ch2.1A – Quadratic Functions
Polynomial function of x with degree n:
f(x) = anxn + an-1xn a2x2 + a1x + a0
Ex: Quadratic function: f(x) = ax2 + bx + c
Graphs of quadratic functions are: _____________

2. Ch2.1A – Quadratic Functions
Polynomial function of x with degree n:
f(x) = anxn + an-1xn a2x2 + a1x + a0
Ex: Quadratic function: f(x) = ax2 + bx + c
Graphs of quadratic functions are: parabolas!
If a > 0: If a < 0:

3. Ch2.1A – Quadratic Functions
Polynomial function of x with degree n:
f(x) = anxn + an-1xn a2x2 + a1x + a0
Ex: Quadratic function: f(x) = ax2 + bx + c
Graphs of quadratic functions are: parabolas!
If a > 0: If a < 0:
vertex
axis of symmetry (maximum)
(minimum)

4. c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3
Ex1) How does each graph compare to y = x2?
a) f(x) = b) g(x) = 2x2
c) h(x) = –x d) k(x) = (x+2)2 – 3

5. c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3
Ex1) How does each graph compare to y = x2?
a) f(x) = b) g(x) = 2x2
c) h(x) = –x d) k(x) = (x+2)2 – 3
y = ax2
If a > 1 (skinny, up) 0 < a < 1 (wide, up)
If a < –1 (skinny, down) –1 < a < 0 (wide, down)

6. Standard Form of a Quadratic Function:
f(x) = a(x – h)2 + k
Ex2) Describe the graph of f(x) = 2x2 + 8x + 7

7. Ex3) Describe the graph of f(x) = –x2 + 6x – 8

8. HW#14) Describe the graph of f(x) = ½x2 – 4
HW#17) Describe the graph of f(x) = x2 – x + 5/4
HW#20) Describe the graph of f(x) = –x2 – 4x + 1
Ch2.1A p odd

9. Ch2.1A p odd

10. f(x) = a(x – h)2 + k Ch2.1B – Finding Quadratic Functions
Ex4) Find the equation for the parabola that has a vertex at (1,2) and
passes thru (0,0), as shown.

11. f(x) = a(x – h)2 + k
HW#36) Find the equation for the parabola that has a vertex at (-2,-2) and
passes thru (-1,0), as shown.
Ch2.1B p even, odd

12. Ch2.1B p even ,31-35 odd

13. Ch2.1B p even ,31-35 odd

14. Ch2.1C – Quadratic Word Problems
Ex5) The height of a ball thrown can be found using the equation
f(x) = –0.0032x2 + x + 3
where f(x) is the height of the ball and x is the distance from where its thrown.
Find the maximum height.

15. Ex6) The percent of income (P) that families give to charity
varies with income (x) by the following function:
P(x) = x2 – x < x < 100
What income level corresponds to the minimum percent?
Ch2.1C p ,34,36,53,55,57,59

16. Ch2.1C p ,34,36,53,55,57,59

17. R = 900x – 0.1x2 where R is revenue and x is units sold.
Ch2.1C p ,34,36,53,55,57,59
53. Find the max # units that produces a max revenue given by
R = 900x – 0.1x2 where R is revenue and x is units sold.
55. A rancher has 200ft of fencing to enclose corrals.
Determine the max enclosed area. Write a function.
x x
A = (2x).y y
P = (2x) + (2x) + y + y
200 = x + x + x + x + y + y + y

18. 57. The height y of a ball thrown by a child is given by:
x is horiz distance.
a. Graph on calc.
b. How high when leaves childs hand at x = 0?
c. Max height?
d. How far when strikes ground?
59. # Board feet (V) as a function of diameter (x) given by:
V(x) = 0.77x2 – 1.32x – < x < 40
a) graph
b) estimate # board feet in 16 in diameter log
c) Est diam when 500 board feet.

19. Ch2.2A – Polynomial Functions of Higher Degree
Graphs of polynomial functions are always smooth and continuous

20. Types of simple graphs:
y = xn
When n is even: When n is odd:
Exs: Exs:

21. Ex1) Sketch:
a) f(x) = –x5
b) g(x) = x4 +1
c) h(x) = (x+1)4

22. f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0
Leading Coefficient Test (An attempt to see where a graph is going.)
f(x) = anxn + an-1xn a2x2 + a1x + a0
When n is even: (an > 1) (an < 1)
When n is odd: (an > 1) (an < 1)

23. Ch2.2A p177 1-4,17-26 Ex2) Use LCT to determ behavior of graphs:
a) f(x) = –x3 + 4x
b) g(x) = x4 – 5x2 + 4
c) h(x) = x4 – x
Ch2.2A p ,17-26

24. Ch2.2A p ,17-26

25. Ch2.2A p ,17-26

26. Ch2.2A p ,17-26

27. Ch2.2A p ,17-26

28. f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0
Ch2.2B – Zeros
f(x) = anxn + an-1xn a2x2 + a1x + a0
1. Graph has at most n zeros.
2. Has at most n – 1 relative extrema (bumps on the graph).
Ex3) Find all the zeros of f(x) = x3 – x2 – 2x

29. Ex4) Find all the real zeros of f(x) = x5 – 3x3 – x2 – 4x – 1
Ex5) Find the polymonial with the following zeros:
–2, –1, 1, 2
Ch2.2B p – 55 odd

30. Ch2.2B p – 55 odd

31. Ch2.2B p – 55 odd

32. Ex1) Divide f(x) = 6x3 – 19x2 – 4 by (x – 2) then factor completely.
Ch2.3 – More Zeros
Ex1) Divide f(x) = 6x3 – 19x2 – 4 by (x – 2)
then factor completely.

33. Ex2) Divide f(x) = x3 – 1 by (x – 1)

34. Ex3) Divide f(x) = 2x4 + 4x3– 5x2 + 3x – 2 by x2 + 2x – 3

35. Going down, add terms. Going diagonally multiply by the zero.
Synthetic Division
Going down, add terms. Going diagonally multiply by the zero.
Ex4) Divide x4 – 10x2 – 2x by (x + 3)
Ex5) Divide

36. The Remainder Theorem – if u evaluate (divide) a function
for a certain x in the domain, the remainder will equal
the corresponding y from the range.
Ex5) Evaluate f(x) = 3x3 + 8x2 + 5x – 7 at x = –2
Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

37. Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

38. Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

39. Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

40. f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0
Ch2.3B – Rational Zero Test
f(x) = anxn + an-1xn a2x2 + a1x + a0
any factor any factor
of this (q) of this (p)
Possible zeros:
Ex1) Find all the zeros of f(x) = 4x3 + 4x2 – 7x + 2.

41. Ex2) Find all the zeros of f(x) = x3 – 10x2 + 27x – 22
Ch2.3B p – 60 all

42. Ch2.3B p – 60 all
HW#55) Find all the zeros of f(x) = x3 + x2 – 4x – 4

43. Ch2.3B p – 60 all
HW#60) Find all the zeros of f(x) = 4x4 – 17x2 + 4

44. Ch2.3B p – 60 all

45. Ch2.3B p – 60 all

46. Ch2.3C p even, 24-30even,61-69odd
8) Divide 5x2 – 17x – 12 by (x – 4)

47. Ch2.3C p even, 24-30even,61-69odd
16) Divide x3 – 9 by (x2 + 1)

48. Ch2.3C p even, 24-30even,61-69odd
24) Synthetic Divide 9x3 – 16x – 18x2 +32 by (x – 2)

49. Ch2.3C p even, 24-30even,61-69odd
30) Synthetic Divide –3x4 by (x + 2)

50. Ch2.3C p even, 24-30even,61-69odd
61) Zeros: 32x3 – 52x2 + 17x + 3

51. Ch2.3C p even, 24-30even,61-69odd
69) Zeros: 2x4 – 11x3 – 6x2 + 64x + 32 = 0

52. Ch2.3C p even, 24-30even,61-69odd
8,16,24,30,61,69 in class

53. Ch2.3C p even, 24-30even,61-69odd
8,16,24,30,61,69 in class

54. Ch2.3C p even, 24-30even,61-69odd
8,16,24,30,61,69 in class

55. Ch2.3C p even, 24-30even,61-69odd
8,16,24,30,61,69 in class

56. Ch2.4 – Complex Numbers
x2 + 1 = 0

57. x2 + 1 = 0 Real Imaginary Ch2.4 – Complex Numbers
Complex Numbers have the standard form: a + bi
Real Imaginary
Quick Review: Unit Unit
Rational numbers  normal ex: 2.5
Irrational numbers  square roots ex:
Imaginary numbers  negative square roots ex:

58. b) 2i + (–4 – 2i) = c) 3 – (–2 – 3i) + (–5 + i) = b) (2 – i)(4 + 3i) =
Ex1) a) (3 – i) + (2 + 3i) =
b) 2i + (–4 – 2i) =
c) 3 – (–2 – 3i) + (–5 + i) =
Ex2) a) (i)(–3i) =
b) (2 – i)(4 + 3i) =
c) (3 + 2i)(3 – 2i) =
complex conjugates  their product is a real #!
Important for getting I out of the denominator.

59. Ex3)
Ex4)

60. Ex5) Plot complex #’s in the complex plane:
a) 2 + 3i b) –1 + 2i c) 4 + 0i
Imag axis
Real axis
HW#1) Solve for a and b:
a + bi = –10 + 6i
HW#5) Solve:
Ch2.4 p202 1–63odd,67–81odd

61. Ch2.4 p202 1–63odd,67–81odd

62. Ch2.4 p202 1–63odd,67–81odd

63. Ch2.4 p202 1–63odd,67–81odd

64. Ch2.4 p202 1–63odd,67–81odd

65. Ch2.4 p202 1–63odd,67–81odd

66. Ch2.4 p202 1–63odd,67–81odd

67. Ch2.4 p202 1–63odd,67–81odd

68. Ch2.5A – Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n,
it has at least one zero in the complex plane.
Ex1) Write f(x) = x5 + x3 + 2x2 – 12x + 8
as a product of linear factors.
Ch2.5A p210 9 – 21 all

69. HW#9) Write f(x) = x2 + 25 as a product of linear factors.
HW#14) f(y) = y4 – 625

70. HW#15) Write f(z) = z2 – 2z + 2
as a product of linear factors.
HW#20) Write f(s) = 2s3 – 5s2 + 12s – 5
Ch2.5A p210 9 – 21 all

71. Ch2.5A p210 9 – 21 all

72. Ch2.5B – More FTA
If f(x) is a polynomial of degree n,
it has at least one zero in the complex plane.
Ex2) Write a fourth degree polynomial that has –1, +1, and 3i as zeros.

73. Ex3) Find all zeros of f(x) = x3 – 4x2 + 9x – 36 if 3i is a zero.
Ch2.5B p210 23–35odd, 41-43all

74. HW#33) i, –i, 6i, –6i
43) Find all zeros of f(x) = 2x4 – x3 + 7x2 – 4x – 4, r = 2i.
Ch2.5B p210 23–35odd, 41-43all

75. Ch2.5B p210 23–35odd, 41-43all

76. Ch2.5B p210 23–35odd, 41-43all

77. Ch2.5B p210 23–35odd, 41-43all

78. Ch2.6 – Rational Functions and Asymptotes
Ex1) Find the domain of and what happens near
the excluded values of x?

79. Ch2.6 – Rational Functions and Asymptotes
Ex1) Find the domain of and what happens near
the excluded values of x?
For any function f(x):
-If n < m, x axis is a horizontal asymptote
-If n > m, no horizontal asymptote
-If n = m, the line is a horizontal asymptote

80. -If n < m, x axis is a horizontal asymptote
-If n > m, no horizontal asymptote
-If n = m, the line is a horizontal asymptote
Ex2) List the horiz asymptotes:
a) b) c)
Ex3) This non-rational function has 2 horiz asymptotes,
to the left and right of x = 0. Find them algebraically
and graphically.
Ch2.6 p218 1,3,7,11-19odd

81. Ch2.6 p218 1,3,7,11-19odd

82. Ch2.6 p218 1,3,7,11-19odd

83. Ch2.6 p218 1,3,7,11-19odd

84. Ch2.6 p218 1,3,7,11-19odd

85. Ch2.7 – Graphs of Rational Functions
1. y-intercept is the value of f(0).
2. x-intercepts are the zeros of the numerator.
Solve p(x) = 0. (If any.)
3. Vertical asymptotes are the zeros of the denominator.
Solve q(x) = 0. (If any.) (Look for the graph to approach +/– .)
4. Horizontal asymptotes where f(x) increases or decreases
without bound. (Approaches but does not reach some #.)
(Notes from yesterday.)
5. You’ll have to figure out what’s going on everywhere else.
(Don’t forget to take advantage of ur calculator.)
Ex1) Analyze the function

86. Ex1) Analyze the function
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x g(x) 1 -4 3 5

87. Ex2) Analyze the function
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x f(x) 1 10 -1
-10

88. Ex3) Analyze the function
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x f(x)
Ch2.7A p – 23odd, 31,33

89. Ch2.7A p – 23odd, 31,32

90. Ch2.7A p – 23odd, 31,32

91. Ch2.7B – More Graphing
Ex4) Analyze the function
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x f(x)

92. Slant asymptotes
If the degree of the numerator is exactly one more than the denominator,
you get a slant asymptote.
Use long division to find it
Ex4) Graph
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
5. slant asymp:

93. HW#50) Graph
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
5. slant asymp:
Ch2.7B p odd,50

94. Ch2.7B p odd

95. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

96. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

97. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

98. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

99. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

100. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

101. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

102. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

103. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

104. Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd