Ch2.1A – Quadratic Functions Polynomial function of x with degree n:

1. Ch2.1A – Quadratic Functions Polynomial function of x with degree n: f(x) = anxn + an-1xn-1 + ... 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-1 + ... 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-1 + ... 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) vertex (minimum)

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

5. Ex1) How does each graph compare to y = x2? a) f(x) = b) g(x) = 2x2 c) h(x) = –x2 + 1 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 p165 13-21odd

9. Ch2.1A p165 13-21odd

10. Ch2.1B – Finding Quadratic Functions f(x) = a(x – h)2 + k 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 p166 14-22 even, 31-35 odd

12. Ch2.1B p166 14-22 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) = 0.0014x2 – 0.1529x + 5.855 5 < x < 100 What income level corresponds to the minimum percent? Ch2.1C p167+ 32,34,36,53,55,57,59

16. Ch2.1C p167+ 32,34,36,53,55,57,59

17. 53. Find the max # units that produces a max revenue given by R = 900x – 0.1x2where 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 – 9.31 5 < x < 40 a) graph b) estimate # board feet in 16 in diameter log c) Estdiam 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. Leading Coefficient Test (An attempt to see where a graph is going.) f(x) = anxn + an-1xn-1 + . .. a2x2 + a1x + a0 When n is even: (an > 1) (an < 1) When n is odd: (an > 1) (an < 1)

23. 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 p177 1-4,17-26

24. Ch2.2A p177 1-4,17-26

27. Ch2.2B – Zeros • f(x) = anxn + an-1xn-1 + . .. 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

28. 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 p178 35 – 55 odd

29. Ch2.2B p178 35 – 55 odd

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

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

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

33. Synthetic Division • Going down, add terms. Going diagonally multiply by the zero. • Ex4) Divide x4 – 10x2 – 2x + 4 by (x + 3) • Ex5) Divide

34. 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

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

38. Ch2.3B – Rational Zero Test • f(x) = anxn + an-1xn-1 + . .. 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.

39. Ex2) Find all the zeros of f(x) = x3 – 10x2 + 27x – 22 • Ch2.3B p192 51 – 60 all

40. Ch2.3B p192 51 – 60 all • HW#55) Find all the zeros of f(x) = x3 + x2 – 4x – 4

41. Ch2.3B p192 51 – 60 all • HW#60) Find all the zeros of f(x) = 4x4 – 17x2 + 4

42. Ch2.3B p192 51 – 60 all

44. Ch2.3C p192 8-16even, 24-30even,61-69odd • 8) Divide 5x2 – 17x – 12 by (x – 4)

45. Ch2.3C p192 8-16even, 24-30even,61-69odd • 16) Divide x3 – 9 by (x2 + 1)

46. Ch2.3C p192 8-16even, 24-30even,61-69odd • 24) Synthetic Divide 9x3 – 16x – 18x2 +32 by (x – 2)

47. Ch2.3C p192 8-16even, 24-30even,61-69odd • 30) Synthetic Divide –3x4by (x + 2)

48. Ch2.3C p192 8-16even, 24-30even,61-69odd • 61) Zeros: 32x3 – 52x2 + 17x + 3

49. Ch2.3C p192 8-16even, 24-30even,61-69odd • 69) Zeros: 2x4 – 11x3 – 6x2 + 64x + 32 = 0

50. Ch2.3C p192 8-16even, 24-30even,61-69odd • 8,16,24,30,61,69 in class