1. Algebra 2 Trig A Final Review 2007

2. #1 Hyperbola  Center (0, 0)  a = 8, b = 7, c =  Vertices: (+8, 0)  Foci: (, 0)  Slopes of asymptotes: +7/8 Hyperbola  Center (0, 0)  a = 8, b = 7, c =  Vertices: (+8, 0)  Foci: (, 0)  Slopes of asymptotes: +7/8

3. #2 y 2 = 121 - x 2  Circle: x 2 + y 2 = 121  Center: (0, 0)  Radius = 11  Circle: x 2 + y 2 = 121  Center: (0, 0)  Radius = 11

4. #3 y = 2(x - 2) 2 + 1  Parabola  Center/Vertex: (2, 1)  AOS: x = 2  DOO: up  Focus: (2, 9/8)  Directrix: y = 7/8  Parabola  Center/Vertex: (2, 1)  AOS: x = 2  DOO: up  Focus: (2, 9/8)  Directrix: y = 7/8

5. #4 6x 2 + 16y 2 = 96  Ellipse:  Center: (0, 0)  a = 4, b =, c =  M vertices: (±4, 0)  Foci: (, 0)  LMA = 8  lma =  Ellipse:  Center: (0, 0)  a = 4, b =, c =  M vertices: (±4, 0)  Foci: (, 0)  LMA = 8  lma =

6. #5 x 2 - 2x + y - 8 = 0  Parabola: y = -(x - 1) 2 + 9  Center/Vertex: (1, 9)  AOS: x = 1  DOO: down  Focus: (1, 8 3/4)  Directrix: y = 9 1/4  Parabola: y = -(x - 1) 2 + 9  Center/Vertex: (1, 9)  AOS: x = 1  DOO: down  Focus: (1, 8 3/4)  Directrix: y = 9 1/4

7. #6 x 2 = 2x + y 2 - 4y + 7  Hyperbola  Center: (1, 2)  a = 2, b = 2, c =  Vertices: (3, 2), (-1, 2)  Foci: (1±, 2)  Slopes of Asymptotes: ±1  Hyperbola  Center: (1, 2)  a = 2, b = 2, c =  Vertices: (3, 2), (-1, 2)  Foci: (1±, 2)  Slopes of Asymptotes: ±1

8. #7 x 2 +4y 2 + 2x - 24y + 33 = 0  Ellipse  Center: (-1, 3)  a = 2, b = 1, c =  Vertices:(-3, 3),(1, 3)  Foci:  LMA = 4  lma = 2  Ellipse  Center: (-1, 3)  a = 2, b = 1, c =  Vertices:(-3, 3),(1, 3)  Foci:  LMA = 4  lma = 2

9. #8 x 2 + y 2 = x + 2  Circle  Center: (1/2, 0)  Radius= 3/2  Circle  Center: (1/2, 0)  Radius= 3/2

10. #9 Find f(x) + g(x)  f(x) = x 2 -x+3 g(x) = x+8  f(x)+g(x) = (x 2 -x+3) + (x+8)  f(x)+g(x) = x 2 + 11  f(x) = x 2 -x+3 g(x) = x+8  f(x)+g(x) = (x 2 -x+3) + (x+8)  f(x)+g(x) = x 2 + 11

11. #10 Find f(x) - h(x)  f(x) = x 2 -x+3 g(x) = x+8  f(x) - h(x) = (x 2 - x + 3) - (3x 2 +1)  f(x) - h(x) = x 2 - x + 3 - 3x 2 - 1  f(x) - h(x) = -2x 2 - x + 2  f(x) = x 2 -x+3 g(x) = x+8  f(x) - h(x) = (x 2 - x + 3) - (3x 2 +1)  f(x) - h(x) = x 2 - x + 3 - 3x 2 - 1  f(x) - h(x) = -2x 2 - x + 2

12. #11 Find f(g(x))  f(x) = x 2 -x+3 g(x) = x+8  f(x) = x 2 - x + 3  f(g(x)) =(x+8) 2 - (x+8) + 3  f(g(x)) = x 2 + 16x +64 - x - 8 + 3  f(g(x)) = x 2 +15x + 59  f(x) = x 2 -x+3 g(x) = x+8  f(x) = x 2 - x + 3  f(g(x)) =(x+8) 2 - (x+8) + 3  f(g(x)) = x 2 + 16x +64 - x - 8 + 3  f(g(x)) = x 2 +15x + 59

13. #12 Find f(h(x))  f(x) = x 2 -x+3 h(x) = 3x 2 +1  f(x) = x 2 - x + 3  f(h(x)) = (3x 2 +1) 2 - (3x 2 +1) + 3  f(h(x)) = 9x 4 +6x 2 +1-3x 2 -1+3  f(h(x)) = 9x 4 +3x 2 +3  f(x) = x 2 -x+3 h(x) = 3x 2 +1  f(x) = x 2 - x + 3  f(h(x)) = (3x 2 +1) 2 - (3x 2 +1) + 3  f(h(x)) = 9x 4 +6x 2 +1-3x 2 -1+3  f(h(x)) = 9x 4 +3x 2 +3

14. #13 Find g(f(x))  g(x) = x+8 f(x) = x 2 -x+3  g(x) = x + 8  g(f(x)) = (x 2 - x + 3) + 8  g(f(x)) = x 2 - x + 11  g(x) = x+8 f(x) = x 2 -x+3  g(x) = x + 8  g(f(x)) = (x 2 - x + 3) + 8  g(f(x)) = x 2 - x + 11

15. #14 Find h(f(x))  h(x) = 3x 2 +1 f(x) = x 2 -x+3  h(x) = 3x 2 + 1  h(f(x))= 3(x 2 - x + 3) 2 + 1  h(f(x))= 3(x 4 -2x 3 +4x 2 -3x+9)+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+27+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+28  h(x) = 3x 2 +1 f(x) = x 2 -x+3  h(x) = 3x 2 + 1  h(f(x))= 3(x 2 - x + 3) 2 + 1  h(f(x))= 3(x 4 -2x 3 +4x 2 -3x+9)+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+27+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+28

16. #15 Find h(g(x))  h(x) = 3x 2 +1 g(x) = x+8  h(x) = 3x 2 + 1  h(g(x)) = 3(x + 8) 2 + 1  h(g(x)) = 3(x 2 + 16x + 64)+1  h(g(x)) = 3x 2 + 48x + 192 + 1  h(g(x)) = 3x 2 + 48x + 193  h(x) = 3x 2 +1 g(x) = x+8  h(x) = 3x 2 + 1  h(g(x)) = 3(x + 8) 2 + 1  h(g(x)) = 3(x 2 + 16x + 64)+1  h(g(x)) = 3x 2 + 48x + 192 + 1  h(g(x)) = 3x 2 + 48x + 193

17. #16 Find f(-3)  f(x) = x 2 - x + 3  f(-3) = (-3) 2 - (-3) + 3  f(-3) = 9 + 3 + 3  f(-3) = 15  f(x) = x 2 - x + 3  f(-3) = (-3) 2 - (-3) + 3  f(-3) = 9 + 3 + 3  f(-3) = 15

18. #17 Find h(f(4))  h(x) = 3x 2 +1 f(x) = x 2 -x+3  f(4) = (4) 2 - (4) + 3  f(4) = 15  h(x) = 3x 2 + 1  h(15) = 3(15) 2 + 1  h(f(4)) = 676  h(x) = 3x 2 +1 f(x) = x 2 -x+3  f(4) = (4) 2 - (4) + 3  f(4) = 15  h(x) = 3x 2 + 1  h(15) = 3(15) 2 + 1  h(f(4)) = 676

19. #18 Find g(h(2))  g(x) = x+8 h(x) = 3x 2 +1  h(2) = 3(2) 2 + 1  h(2) = 3(4) + 1  h(2) = 13  g(13) = 13 + 8  g(h(2)) = 21  g(x) = x+8 h(x) = 3x 2 +1  h(2) = 3(2) 2 + 1  h(2) = 3(4) + 1  h(2) = 13  g(13) = 13 + 8  g(h(2)) = 21

20. #19 Inverse of f(x) = 4x + 5  y = 4x + 5  x = 4y + 5  x - 5 = 4y  x/4 - 5/4 = y  y = 4x + 5  x = 4y + 5  x - 5 = 4y  x/4 - 5/4 = y

21. #20 Inverse of g(x) = 3x 2 - 12  y = 3x 2 - 12  x = 3y 2 - 12  x + 12 = 3y 2  x/3 + 4 = y 2  y = 3x 2 - 12  x = 3y 2 - 12  x + 12 = 3y 2  x/3 + 4 = y 2

22. #21 f(x)=1/2x+2 g(x)=2x-4  f(g(x))=1/2(2x - 4) + 2  f(g(x)) = x - 2 + 2  f(g(x)) = x  f(g(x))=1/2(2x - 4) + 2  f(g(x)) = x - 2 + 2  f(g(x)) = x

23. #22 f(x) = 3x-9 g(x) = -3x+9  f(x) = 3x-9  y = 3x - 9  x = 3y - 9  x + 9 = 3y  x/3 + 3 = y  Not equal to g(x)  f(x) = 3x-9  y = 3x - 9  x = 3y - 9  x + 9 = 3y  x/3 + 3 = y  Not equal to g(x)

24. #23 {(1,3),(1,-1),(1,-3),(1,1)}  {(3,1),(-1,1),(-3,1),(1,1)}  Domain: 3, -1, -3, 1  Unique x - coordinates  {(3,1),(-1,1),(-3,1),(1,1)}  Domain: 3, -1, -3, 1  Unique x - coordinates

25. #24 Simplify  Simplify:

26. #25Simplify  Simplify

27. #26Simplify  Simplify:

28. #27Simplify  Simplify:

29. #28Absolute value equation  Solve:

30. #29Absolute Value Inequality  Solve:

31. #30 Find f(-5)  If f(x) = 4x 3 - x + 1  f(-5) = 4(-5) 3 - (-5) +1  f(-5) = -500 + 5 + 1  f(-5) = -494  If f(x) = 4x 3 - x + 1  f(-5) = 4(-5) 3 - (-5) +1  f(-5) = -500 + 5 + 1  f(-5) = -494

32. #31Do the math  (8x 3 + 2x 2 + 3x)÷(2x + 3)

33. #32Simplify  Simplify:

34. #33Factor: 27a 3 + 125b 3  Factor:27a 3 + 125b 3  (3a + 5b)(9a 2 - 15ab + 25b 2 )  Factor:27a 3 + 125b 3  (3a + 5b)(9a 2 - 15ab + 25b 2 )

35. #34 Factor: 9x 2 - 12x + 4  Factor: 9x 2 - 12x + 4  (3x -2) 2  Factor: 9x 2 - 12x + 4  (3x -2) 2

36. #35Factor: 7y - 12x + 4xy - 21  Factor: 7y - 12x + 4xy - 21  7y - 21 + 4xy - 12x  7(y - 3) + 4x(y - 3)  (y - 3)(7 + 4x)  Factor: 7y - 12x + 4xy - 21  7y - 21 + 4xy - 12x  7(y - 3) + 4x(y - 3)  (y - 3)(7 + 4x)

37. #36Factor: 15a 3 b - 5a 2 b 2 - 10ab 3  Factor: 15a 3 b - 5a 2 b 2 - 10ab 3  5ab(3a 2 - ab - 2b 2 )  5ab(3a 2 - 3ab +2ab - 2b 2 )  5ab[3a(a - b) + 2b(a - b)]  5ab(a - b)(3a + 2b)  Factor: 15a 3 b - 5a 2 b 2 - 10ab 3  5ab(3a 2 - ab - 2b 2 )  5ab(3a 2 - 3ab +2ab - 2b 2 )  5ab[3a(a - b) + 2b(a - b)]  5ab(a - b)(3a + 2b)

38. #37Simplify:  Simplify:

39. #38 Simplify:  Simplify:

40. #39Simplify:  Simplify:

41. #40Solve:  Solve:

42. #41Solve:x 2 + 441 = 0  Solve: x 2 + 441 =0  x 2 = -441  x =  x = ±21i  Solve: x 2 + 441 =0  x 2 = -441  x =  x = ±21i

43. #42Simplify: (9 - 3i) - (3 + 5i)  (9 - 3i) - (3 + 5i)  9 - 3 - 3i - 5i  6 - 8i  (9 - 3i) - (3 + 5i)  9 - 3 - 3i - 5i  6 - 8i

44. #43Simplify: (5 + 4i)(3 - 7i)  Simplify: (5 + 4i)(3 - 7i)  (5 + 4i)(3 - 7i)  15 - 35i + 12i - 28i 2  15 - 23i - 28(-1)  15 - 23i + 28  43 - 23i  Simplify: (5 + 4i)(3 - 7i)  (5 + 4i)(3 - 7i)  15 - 35i + 12i - 28i 2  15 - 23i - 28(-1)  15 - 23i + 28  43 - 23i

45. #44Simplify:  Simplify:

46. #45Simplify: (7 - 3i)(7 + 3i)  Simplify: (7 - 3i)(7 + 3i)  49 + 21i - 21i - 9i 2  49 - 9(-1)  49 + 9  58  Simplify: (7 - 3i)(7 + 3i)  49 + 21i - 21i - 9i 2  49 - 9(-1)  49 + 9  58

47. #46Simplify: i 10 i 21 i 30  Simplify: i 10 i 21 i 30  i 10+21+30 =i 61 = i 4(15)+1 = i 1 = i  Simplify: i 10 i 21 i 30  i 10+21+30 =i 61 = i 4(15)+1 = i 1 = i

48. #47Simplify  Simplify:

49. #48Solve: x 2 + 5x + 13 = 0  x 2 + 5x + 13 = 0

50. #49Solve: 6x 2 + 7x = 3  Solve:

51. #50Solve:2x 2 + 3x - 13 = 0  Solve:

52. #51 Word Problem  h(t) = -16t 2 + 10t + 50  h(1) = -16(1) 2 + 10(1) + 50  h(1) = 44 feet  0 = -16t 2 + 10t + 50  h(t) = -16t 2 + 10t + 50  h(1) = -16(1) 2 + 10(1) + 50  h(1) = 44 feet  0 = -16t 2 + 10t + 50

53. #52 Simplify, combine like terms  (4b 4 + 6b 2 - 3b + 5) - (2b 3 + 3b - 2)  4b 4 - 2b 3 + 6b 2 - 6b + 7  (4b 4 + 6b 2 - 3b + 5) - (2b 3 + 3b - 2)  4b 4 - 2b 3 + 6b 2 - 6b + 7

54. #53 Simplify, remove parentheses  (y + 2)(y 2 - 4y + 1)  y 3 - 4y 2 + y + 2y 2 - 8y + 2  y 3 - 2y 2 - 7y + 2  (y + 2)(y 2 - 4y + 1)  y 3 - 4y 2 + y + 2y 2 - 8y + 2  y 3 - 2y 2 - 7y + 2

55. #54Do the arithmetic  (2x 3 - 3x 2 + 4x - 5) ÷ (x - 2)  Synthetic Division  2 2 -3 4 -5  4 2 12  2 1 6 7  2x 2 + x + 6x + 7/(x-2)  (2x 3 - 3x 2 + 4x - 5) ÷ (x - 2)  Synthetic Division  2 2 -3 4 -5  4 2 12  2 1 6 7  2x 2 + x + 6x + 7/(x-2)

56. #55 Factor  64x 2 y 2 - 25z 2  (8xy - 5z)(8xy + 5z)  64x 2 y 2 - 25z 2  (8xy - 5z)(8xy + 5z)

57. #56Find the zeros  y = 3x 2 + 5x + 2  0 = 3x 2 + 5x + 2  0 = (3x 2 + 3x) + (2x + 2)  0 = 3x(x + 1) + 2(x + 1)  0 = (x + 1)(3x + 2)  x = -1, -2/3  y = 3x 2 + 5x + 2  0 = 3x 2 + 5x + 2  0 = (3x 2 + 3x) + (2x + 2)  0 = 3x(x + 1) + 2(x + 1)  0 = (x + 1)(3x + 2)  x = -1, -2/3

58. #57Find the max or min of #56  y = 3x 2 + 5x + 2  DOO: up, therefore a minimum  x = -b/2a x = -5/2(3) = -5/6  y = 3(-5/6) 2 + 5(-5/6) + 2  y = 25/12 - 25/6 + 2  y = -1/12  Vertex is (-5/6, -1/12)  y = 3x 2 + 5x + 2  DOO: up, therefore a minimum  x = -b/2a x = -5/2(3) = -5/6  y = 3(-5/6) 2 + 5(-5/6) + 2  y = 25/12 - 25/6 + 2  y = -1/12  Vertex is (-5/6, -1/12)

59. #58 Solve Systems of Equations  x + y = 2  x – 3y =6  -x-y=-2 x+(-1)=2  x-3y=6 x=3  -4y=4  y=-1  x + y = 2  x – 3y =6  -x-y=-2 x+(-1)=2  x-3y=6 x=3  -4y=4  y=-1

60. #59  (3x-5y=6)*7 21x-35y=42  (2x+7y=12)*5 10x+35y=60  (3x-5y=6)*7 21x-35y=42  (2x+7y=12)*5 10x+35y=60

61. #60 Solve Systems of Equations  7x+3y=-1  2x-y=9 y=2x-9  7x+3y=-1  2x-y=9 y=2x-9

62. #61 Solve a system 3x+2y=-24 6x-5y=30 -6x-4y=48 6x-5y=30 -9y=78 y=-78/9 3x+2y=-24 6x-5y=30 -6x-4y=48 6x-5y=30 -9y=78 y=-78/9

63. #62  Is (1, 2) a solution to the following system of inequalities?  x>1, y 2x-1  1>1, 2 2(1)-1  Yes. Yes. No.  The answer is NO.  Is (1, 2) a solution to the following system of inequalities?  x>1, y 2x-1  1>1, 2 2(1)-1  Yes. Yes. No.  The answer is NO.

64. END of Review