2. 1 Algebraic Graphs MENU Gradient / Intercept Method Value of ‘c’ Value of ‘ m ‘ basic Giving the Equation of the Line Questions & Answers Giving the Equation of the Line Value of ‘ m ‘ detailed Drawing the Line Questions / Answers Main MENU Do’s and Don’ts for Axes Plotting Curved Graphs Curved Graphs Questions & Answers Transforming Curves y = f ( x ) + a y = f ( a x ) y = - f ( x ) y = a f ( x ) y = f ( x + a ) Coordinates on Line 1 Coordinates on Line 2 Starters Graphical Solutions to Equations

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4. 3 y = mx + c Y = 2x + 1 Y = -3x + 4 Y = 2x - 3 Y = 1x + 5 2 c summary m summary Menu

5. 4 Y = x + 1 Y =x + 2 Y = x + 3 Y = x + 4 Y = x - 1 Y = x - 2 Y = x - 3 Y = x - 4 Y = x The value of C y = mx + c Menu

6. 5 Y = 2x + 1 Y = 2x + 2 Y = 2x + 3 Y = 2x + 4 Y =2 x Y = 2x -1 Y = 2x - 2 Y = 2x - 3 Y = 2x - 4 The value of C y = mx + c Menu

7. 6 The value of C tells us where the line crosses the y – axis. y = mx + c Ex 1. y = 2x + 1 cuts y axis at ( 0, 1 ) Ex 2. y = 2x + 3 cuts y axis at ( 0, 3 ) Ex 3. y = 2x – 4 cuts y axis at ( 0, -4 ) Slide 3 Menu

8. 7 Y = 3x - 2 Y = 1/2x + 1 Y = -2x + 3 Y = -x Menu

9. 8 Y = -2x -3 Y = 1/3x + 3 Y = -x + 1 Y = 2x Y = 3x - 1 Menu

10. 9 Y = x - 2 Y = 2x - 2 Y = 3x - 2 Y = 4x - 2 Y = 5x - 2 Y =1/2 x - 2 Y = 1/3 x - 2 Y =1/4 x -2 Y = 1/5 x - 2 Y = 1/6 x - 2 Y = -20x Y = -1/6x - 2 Y = -1/5x - 2 Y = -1/4x - 2 Y = -1/3x - 2 Y = -1/2x - 2 Y = -x - 2 Y = -2x - 2 Y = -3x - 2 Y= -4x - 2 Y = -5x - 2 The value of m y = mx + c Menu

11. 10 y = mx + c The value of m controls the ‘steepness’ (Gradient) of the line. The ‘bigger’ the number then the steeper the line. x y y x Positive Gradients For example : Y = 2x + 1 Y = 3x -3 Y = 5 + 4x Negative Gradients For example : Y = - 2x + 3 Y = -3x + 2 Y = 4 – 2x Positive and Negative Gradients. Slide 3 Menu

12. 11 y = 1/3 x + c y = 2x + c y = c - 2x y = - x + c Menu

13. 12 Y = 4x + 1 4 1 A closer look at the gradient m Menu

14. 13 Y = 3x + 1 3 1 A closer look at the gradient m Menu

15. 14 Y = 2x + 1 2 1 A closer look at the gradient m Menu

16. 15 Y = 1x + 1 1 1 A closer look at the gradient m Menu

17. 16 Y = ½ x + 1 1 2 A closer look at the gradient m Menu

18. 17 Y = 1/3x + 1 1 3 A closer look at the gradient m Menu

19. 18 Y = ¼ x + 1 1 4 A closer look at the gradient m Menu

20. 19 Y = - 1/4x + 1 4 A closer look at the gradient m Menu

21. 20 Y = -1/3x + 1 3 A closer look at the gradient m Menu

22. 21 Y = -1/2x + 1 2 A closer look at the gradient m Menu

23. 22 Y = -1x + 1 1 A closer look at the gradient m Menu

24. 23 Y = -2x + 1 1 -2 A closer look at the gradient m Menu

25. 24 Y = -3x + 1 1 -3 A closer look at the gradient m Menu

26. 25 Y = 2x + 1 x 1 2 1 x Drawing the line. Menu

27. 26 Y = 3x - 2 x 3 11 x Drawing the line. Menu

28. 27 Y = x 1 2 + 1 x 1 2 x Drawing the line. Menu

29. 28 Y = x - 3 x 1 1 x Drawing the line. Menu

30. 29 Y = x + 2-2 1 x 1 x Drawing the line. Menu

31. 30 Draw the following graphs using The Gradient / Intercept Method 1) y = 2x + 1 2) y = 3x – 2 3) y = 2x – 1 4) y = x + 4 5) y = x - 3 6) y = 3x + 1 7) y = - 2x + 3 8) y = 3 – x 9) y = - 3x - 1 10) y = 1/2x + 3 11) y = 1/3x – 2 12) y = 1/4x 13) Y = 2/3x + 2 14) y = 2 - 1/2x 15) y = - x Ans Menu

32. 31 1) y = 2x + 1 Answers Questions Menu

33. 32 2) y = 3x - 2 Answers Questions Menu

34. 33 3) y = 2x - 1 Answers Questions Menu

35. 34 4) y = x + 4 Answers Questions Menu

36. 35 5) y = x - 3 Answers Questions Menu

37. 36 6) y = 3x + 1 Answers Questions Menu

38. 37 7) y = -2x + 3 Answers Questions Menu

39. 38 8) y = 3 - x Answers Questions Menu

40. 39 9) y = -3x - 1 Answers Questions Menu

41. 40 10) y = ½ x + 3 Answers Questions Menu

42. 41 11) y = 1/3 x - 2 Answers Questions Menu

43. 42 12) y = 1/4 x Answers Questions Menu

44. 43 13) Y = 2/3 x + 2 Answers Questions Menu

45. 44 14) y = 2 - 1/2 x Answers Questions Menu

46. 45 15) y = - x Answers Questions Menu

47. 46 + 1 2 1 Y = x Giving the Equation of the Line Menu

48. 47 + 3 1 2 Y = x Giving the Equation of the Line Menu

49. 48 + 3 -2 1 Y = x Giving the Equation of the Line Menu

50. 49 - 1 2 Y = x Giving the Equation of the Line Menu

51. 50 1) 2) 3) 4) 5) 6) y = 2x + 1 y = 2x - 1 Y = - 2x + 3 Y = 1/2x - 2 Y = - x + 1 Y = -1/2x - 1 Write down the equations of the following lines: Menu

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53. 52 x y y x x x y y 1) 2) 3) 4) 1 2 3-3 -2 -1 0 -3 -2 -1 1 2 3 -2 -1 1 2 3 -2 -1 1 2 0 0 0 321321 -2 -3 321321 -2 -3 43214321 -2 321321 -2 Which axes are O.K ? The x and y axes are the wrong way around ! The scales should be written to the left of the y axis and below the x axis ! No problems ! The axes need to lie over the main grid lines of the graph paper ! Menu

54. 53 5) 6) 7) 8) yy yy x x xx 1 2 3-4 -3 -2 -1 6 4 2 -2 -4 0 1 2 3-2 -1 0 1 2 3 -2 1 2 3-6 -4 -2 642642 -2 0 1 2-2 -1 0 321321 -2 No problems ! The 0 line should be the y axis ! You can’t suddenly change the scales as you go across or up ! You must not change the scales as you go across or up ! Menu

55. 54 x y 1 2 3 4-5 -4 -3 -2 -1 0 120 130 140 150 How can I get around this problem ? I could collapse the axis ! Menu

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57. 56 Graphing the area of a square Menu

58. 57 W Area 1 1 1 1 x 2 2 2 4 x 3 3 3 9 x 4 4 4 16 x 5 5 5 25 x 6 6 6 36 x 7 7 7 49 x 0 0 x A = W 2 W W We must think about negative widths ! It doesn’t make practical sense but it does make mathematical sense ! Menu

59. 58 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 x x x x x x x x + x + = + - x - = + + x - = - - x + = - x-7-6-5-4-3-201234567 y014916253649 A = W 2 y = x 2 1 x x x x x x x 4916253649 y = x 2 This shape is called a PARABOLA Menu

60. 59 x y 1 2 3 4 5-5 -4 -3 -2 -1 50 40 30 20 10 0 + x + = + - x - = + + x - = - - x + = - 2x x Area = 2x 2 y = 2x 2 The rectangle is twice as long as it is wide ! x-5-4-3-2012345 y 1 2 2 x 4 2 x 8 6 3 18 x 8 4 32 x 10 5 50 x 0 x 2 x 8 x 18 x 32 x 50 x y = 2x 2 Menu

61. 60 x y 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 + x + = + - x - = + + x - = - - x + = - y = x 2 + 9 x-6-5-4-3-20123456 y x x 3 3 1 1 3 3 10 x 2 2 13 x 3 3 18 x 4 4 25 x 5 5 34 x 45 x 9 x 10 x 13 x 18 x 25 x 34 x 45 x y = x 2 + 9 6 6 Menu

62. 61 First of all make a table of the x and the corresponding y values and then draw the graphs. 1) y = x 2 { - 7 < x < 7 } 3) y = 3x 2 { - 4 < x < 4 } 5) y = x 2 + 5 { - 6 < x < 6 } 7) y = x 2 + 2x {- 5 < x < 5 } 2) y = 2x 2 { - 4 < x < 4 } 4) y = ½x 2 { - 6 < x < 6 } 6) y = x 2 - 20 { - 5 < x < 7 } 8) y = x 2 - 3x {- 5 < x < 5 } 9) y = x 2 + 3x + 10 {- 5 < x < 5 } 11) y = x 3 {- 3 < x < 3 } 13) y = x 3 + x 2 {- 3 < x < 3 } 10) y = x 2 - 4x - 10 {- 5 < x < 7 } 12) y = x 3 + 2x {- 3 < x < 3 } + x + = + - x - = + + x - = - - x + = - Ans Menu

63. 62 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 x x x x x x x x x- 7- 6- 5- 4- 3- 2- 101234567 y014916253649 1 x x x x x x x 4916253649 y = x 2 Back to questions Menu

64. 63 x y 1 2 3 4 5-5 -4 -3 -2 -1 50 40 30 20 10 0 x- 5- 4- 3- 2- 1012345 y50321882028 3250 x x x x x x x x x x x y = 2x 2 Menu Back to questions

65. 64 x y 1 2 3 4 5-5 -4 -3 -2 -1 50 40 30 20 10 0 x- 4- 3- 2- 101234 y482712303 2748 x x x x x x x x x y = 3x 2 Menu Back to questions

66. 65 x y 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 18 16 14 12 10 8 6 4 2 -2 0 x- 6- 5- 4- 3- 2- 10123456 y1812.584.520.50 24.5812.518 x x x x x x x x x x x x x y = ½x 2 Menu Back to questions

67. 66 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 x- 6- 5- 4- 3- 2- 10123456 y4130211496569 213041 x x x x x x x x x x x x x y = x 2 + 5 Menu Back to questions

68. 67 y x 1 2 3 4 5 6 7-5 -4 -3 -2 -1 0 30 25 20 15 10 5 -5 -10 -15 -20 x- 5- 4- 3- 2- 101234567 y 5- 4- 11- 16- 19- 20- 19- 16- 11- 451629 x x x x x x x x x x x x x y = x 2 - 20 Menu Back to questions

69. 68 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 y = x 2 + 2x x- 5- 4- 3- 2- 1012345 y15830- 1038152435 x x x x x x x x x x x Menu Back to questions

70. 69 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 y = x 2 – 3x x- 5- 4- 3- 2- 1012345 y4028181040- 2 0410 x x x x x x x x x x x Menu Back to questions

71. 70 x y 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 50 40 30 20 10 0 y = x 2 + 3x + 10 x- 5- 4- 3- 2- 1012345 y20141088 1420283850 x x x xx x x x x x x Menu Back to questions

72. 71 y = x 2 - 4x - 10 y x 1 2 3 4 5 6 7 -5 -4 -3 -2 -1 35 30 25 20 15 10 5 -5 -10 -15 -20 0 x- 5- 4- 3- 2- 101234567 y3522112- 5- 10- 13- 14- 13- 10- 5211 x x x x x x x x x x x x x Menu Back to questions

73. 72 y x 0 30 25 20 15 10 5 -5 -10 -15 -20 y = x 3 1 2 3 -3 -2 -1 x- 3- 2- 10123 y- 27- 8- 101827 -25 -30 x x x x x x x Menu Back to questions

74. 73 y x 1 2 3-3 -2 -1 y = x 3 + 2x x- 3- 2- 10123 y-33-12-3031233 0 30 20 10 -10 -20 -30 x x x x x x x Menu Back to questions

75. 74 y = x 3 + x 2 x- 3- 2- 10123 y- 18- 40021236 y x 1 2 3 -3 -2 -1 40 30 20 10 -10 -20 x x xx x x x 0 Menu Back to questions

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77. 76 The notation f(x) means function of x. A function of x is an expression which almost always varies and which will depend upon a given value of x. Examples : f(x) = 2x + 1 f(x) = x 2 f(x) = sin x f(4) = 9 f(4) = 16 f(4) = 0.0698 f(20) = 41 f(20) = 400 f(20) = 0.342 Menu

78. 77 We are going to look at the graphical effects on functions when changes are made to them. For example : f(x) = 2x + 1 f(x) = x 2 f(x) = sin x f(x) + a ( When a = 2 ) f(x) + 2 = 2x + 1 + 2 f(x) + 2 = x 2 + 2 f(x) = sin x + 2 = 2x + 3 f(x) = 2x + 1f(x) = x 2 f(x) = sin x af(x) ( When a = 2 ) 2f(x) = 2(2x + 1) = 4x + 2 2f(x) = 2x 2 2f(x) = 2 sin x Menu

79. 78 We are going to look at the graphical effects on functions when changes are made to them. For example : f(x) = 2x + 1 f(x) = x 2 f(x) = sin x f(ax) ( When a = 2 ) f(x) = 2x + 1 f(2x) = 2×2x + 1 = 4x + 1 f(x) = x 2 f(2x) = (2x) 2 = 4x 2 f(x) = sin x f(2x) = sin 2x f(x + a) ( When a = 2 ) f(x + 2) = 2(x + 2) + 1 = 2x + 5 f(x + 2) = (x + 2) 2 = x 2 + 4x + 4 f(x + 2) = sin (x + 2) Menu

80. 79 We are going to look at the graphical effects on functions when changes are made to them. For example : f(x) = 2x + 1 f(x) = x 2 f(x) = sin x - f(x) f(x) = 2x + 1 - f(x) = - (2x + 1) = -2x - 1 f(x) = x 2 - f(x) = - x 2 f(x) = sin x - f(x) = - sin x Menu

81. 80 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 2 2 f(x) + a y = 2x + 1 y = 2x + 3 y = 2x + 1 y = 2x + 1 + 2 y = 2x + 3 y = x 2 y = x 2 + 2 y = x 2 y = sin x y = sin x + 2 y = x 2 + 2 y = sin x + 2 In what same way have all three graphs been transformed ? 2 2 2 2 2 2 2 2 2 2 2 2 2 They have all been translated 2 units parallel to the y axis. translates ‘a’ units parallel to the y axis. Menu

82. 81 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 y = f(x) x A ( 3, - 1 ) What will the new coordinates of point A be on the graphs y = f(x) + 3 and y = f(x) – 2 ? y = f(x) A ( 1, 2 ) x y = f(x) A ( - 90, - 1 ) x y = f(x) + 3 A ( 3, 2 ) x A ( 3, - 3 ) y = f(x) – 2 x y = f(x) + 3 A ( 1, 5 ) x y = f(x) – 2 A ( 1, 0 ) x y = f(x) + 3 A ( - 90, 2 ) x x y = f(x) – 2 A ( - 90, - 3 ) Menu

83. 82 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 af(x) y = 2x + 1 y = 4x + 2 y = 2x + 1 y = 2(2x + 1) y = 4x + 2 y = x 2 y = 2x 2 y = x 2 y = sin x y = 2 sin x y = 2x 2 y = 2 sin x In what same way have all three graphs have transformed ? They have all had their distances from the x axis doubled. stretches the graph by a scale factor of ‘a’ units parallel to the y axis. 3 6 1 2 1 2 1 2 3 6 1 2 1 2 0.5 1 1 2 Menu

84. 83 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 What will the new coordinates of point A be on the graphs y = 3f(x) ? y = f(x) x A (- 1, - 1) y = f(x) A (2, 0.5) x A (- 180, 0.5) x y = 3f(x) x A (- 1, - 3) y = 3f(x) A (2, 1.5) x. y = 3f(x) A (- 180, 1.5) x y = f(x) Menu

85. 84 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 f(ax) y = 2x + 1 y = 2×2x + 1 y = 4x + 1 y = x 2 y = (2x) 2 y = 4x 2 y = x 2 y = sin x y = sin 2x In what same way have all three graphs have transformed ? They have all been ‘crushed’ in towards the y axis. They are half as ‘wide’. stretches the graph by a scale factor of ‘1/a’ units parallel to the x axis y = 4x + 1 y = 4x 2 2 1 1 0.5 y = sin 2x 0.5 0.25 2 1 2.4 1.2 90 45 270 135450225 Menu

86. 85 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 What will the new coordinates of point A be on the graphs y = f(3x) ? y = f(x) x A (3, - 2 ) x A ( 1, - 2 ) y = f(3x) y = f(x) x A (2, 2 ) x y = f(3x) A ( 2/3, 2 ) y = f(x) x A (180, 1.5 ) A (60, 1.5) x Menu

87. 86 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 2 4 6 -2 -4 -6 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 f(x + a) y = 2x + 1 y = 2x + 5 y = 2x + 1 y = 2(x + 2) + 1 y = 2x + 5 y = x 2 y = (x + 2) 2 y = (x + 2)(x + 2) y = x 2 + 4x + 4 y = x 2 y = sin x y = sin (x + 90) y = 4x 2 + 4x + 4 y = sin (x + 90) y = sin (x + 2) would be too small to see ! In what same way have all three graphs been transformed ? 2 2 2 2 2 2 2 2 2 2 2 90 They have all slid left by ‘a’ units. They slid in the opposite direction to what you might think ! translates the graph ‘a’ units parallel to the x axis. If ‘a’ is positive it translates to the left and if ‘a’ is negative it translates to the right. Menu

88. 87 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 What will the new coordinates of point A be on the graphs y = f(x + 3) and y = f(x – 2) ? What will the new coordinates of point A be on the graphs y = f(x + 180) and y = f(x – 90) ? y = f(x) A (1, 0 ) x y = f(x + 3) A (-2, 0) x A (3, 0) y = f(x – 2 ) x y = f(x) A (1.2, 2 ) x A (-1.8, 2) y = f(x + 3) xx A (3.2, 2) y = f(x – 2 ) y = f(x) x A (90, 1) x A (-90, 1)y = f(x + 180) x y = f(x – 90 ) A (180, 1) Menu

89. 88 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 - f(x) reflects the graph in the x axis. y = 2x + 1 y = - (2x + 1) y = - 2x – 1 y = x 2 y = - x 2 y = sin x y = - sin x In what same way have all three graphs have transformed ? y = 2x + 1 y = - 2x - 1 y = - x 2 y = x 2 y = - sin xy = sin x 3 3 1 1 2 2 3 3 1 1 4 4 2 2 6 6 1 1 1 1 0.5 They have all been reflected in the x axis. Menu

90. 89 yy y xx x 1 2 3-3 -2 -1 01 2 3-3 -2 -1 0 1 2 3 -2 -3 2 4 6 -2 -4 -6 90 180 270 360 450 540 630 720-720 -630 -540 -450 -360 -270 -180 -90 0 1 2 -2 What will the new coordinates of point A be on the graphs y = - f(x) ? y = f(x) x A (1, 0 ) y = - f(x) x A (-2, 1.5 ) x A (-2, -1.5) y = f(x) y = - f(x) A (360, 1.8 ) x x A (360, -1.8) y = f(x) y = - f(x) Menu

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92. 91 Match the coordinates with the graphs they came from : Y = 2x + 1 Y = 4x + 3 Y = 3x Y = 2x - 4 Y = 6x - 1 Y = 3x + 2 ( 5, 23 ) ( 4, 4 ) ( 3, 11 ) ( 3, 7 ) ( 2, 6 ) ( 1, 5 ) Menu

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94. 93 y = 2x - 6 ( 3, 0 ) Does not lie on the lineLies on the line Menu

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97. 96 y = 3x + 2 ( 4, 11 ) Does not lie on the lineLies on the line Menu

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100. 99 y = 4x - 5 ( 2, 3 ) Does not lie on the lineLies on the line Menu

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103. 102 y = 4x - 2 ( -2, 6 ) Does not lie on the lineLies on the line Menu

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106. 105 y = 2 + 3x ( 2, 8 ) Does not lie on the lineLies on the line Menu

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109. 108 y = 2 - 3x ( -2, -6 ) Does not lie on the lineLies on the line Menu

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113. 112 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 Menu Solve algebraically : x + 1 = 3 - 1 x = 2 You could solve this using graphs ! x + 1 = 3 Draw the graph y = x + 1 y = x + 1 y = x + 1 When does the graph have a ‘ y ’ value of 3 ? The y = 3 line ! x y = 3 x = 2

114. 113 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 We can use an accurately drawn graph to solve a great variety of equations, many of which are either difficult or impossible to find exact solutions for when using other methods. y = x 2 + 2x - 3 Given x 2 + 2x – 3 y = solve : x 2 + 2x – 3 = 2 y = 2 xx Approximate solutions : x = - 3.4 & 1.5 Example 1 Menu

115. 114 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 We can use an accurately drawn graph to solve a great variety of equations, many of which are either difficult or impossible to find exact solutions for when using other methods. y = x 2 + 2x - 3 Given x 2 + 2x – 3 y = Example 2 solve : x 2 + 2x – 2 = 0 But this is not the same as our graph ! We will need the line y = - 1 y = - 1 xx Approximate solutions : x = - 2.7 & 0.7 Note : You adapt the equation that you are solving to be the same as the graph which has already been drawn ! Menu

116. 115 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 We can use an accurately drawn graph to solve a great variety of equations, many of which are either difficult or impossible to find exact solutions for when using other methods. y = x 2 + 2x - 3 Given x 2 + 2x – 3 y = Example 3 solve : x 2 + x – 4 = 0 But this is not the same as our graph ! Note : You adapt the equation that you are solving to be the same as the graph which has already been drawn ! + x + 1 We will need the line y = x + 1 x x y = x + 1 Approximate solutions : x = - 2.5 & 1.6 Menu

117. 116 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 We can use an accurately drawn graph to solve a great variety of equations, many of which are either difficult or impossible to find exact solutions for when using other methods. y = x 2 + 2x - 3 Given x 2 + 2x – 3 y = Example 4 solve : x 2 + 4x = x + 3 But this is not the same as our graph ! Note : You adapt the equation that you are solving to be the same as the graph which has already been drawn ! - 2x - 3 We need the line y = - x y = - x x x Approximate solutions : x = - 3.8 & 0.8 Menu

118. 117 x y - 4 - 3 - 2 - 1 0 1 2 3 4 1 2 3 4 5 - 1 - 2 - 3 - 4 Menu

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122. 121 End of Algebraic Graphs Presentation. Return to previous slide.