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

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

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

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

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

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

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

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

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

12 Y = 4x A closer look at the gradient m Menu

13 Y = 3x A closer look at the gradient m Menu

14 Y = 2x A closer look at the gradient m Menu

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

16 Y = ½ x A closer look at the gradient m Menu

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

18 Y = ¼ x A closer look at the gradient m Menu

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

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

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

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

23 Y = -2x A closer look at the gradient m Menu

24 Y = -3x A closer look at the gradient m Menu

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

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

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

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

29 Y = x x 1 x Drawing the line. Menu

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 ) y = 1/2x ) y = 1/3x – 2 12) y = 1/4x 13) Y = 2/3x ) y = 2 - 1/2x 15) y = - x Ans Menu

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

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

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

34 4) y = x + 4 Answers Questions Menu

35 5) y = x - 3 Answers Questions Menu

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

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

38 8) y = 3 - x Answers Questions Menu

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

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

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

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

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

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

45 15) y = - x Answers Questions Menu

Y = x Giving the Equation of the Line Menu

Y = x Giving the Equation of the Line Menu

Y = x Giving the Equation of the Line Menu

Y = x Giving the Equation of the Line Menu

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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52 x y y x x x y y 1) 2) 3) 4) 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

53 5) 6) 7) 8) yy yy x x xx 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

54 x y How can I get around this problem ? I could collapse the axis ! Menu

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56 Graphing the area of a square Menu

57 W Area x x x x x x 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

58 x y x x x x x x x x + x + = + - x - = + + x - = - - x + = - x y A = W 2 y = x 2 1 x x x x x x x y = x 2 This shape is called a PARABOLA Menu

59 x y x + = + - x - = + + x - = - - x + = - 2x x Area = 2x 2 y = 2x 2 The rectangle is twice as long as it is wide ! x y x 4 2 x x x x 0 x 2 x 8 x 18 x 32 x 50 x y = 2x 2 Menu

60 x y x + = + - x - = + + x - = - - x + = - y = x x y x x x x x x x 45 x 9 x 10 x 13 x 18 x 25 x 34 x 45 x y = x Menu

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 { - 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 { - 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

62 x y x x x x x x x x x y x x x x x x x y = x 2 Back to questions Menu

63 x y x y x x x x x x x x x x x y = 2x 2 Menu Back to questions

64 x y x y x x x x x x x x x y = 3x 2 Menu Back to questions

65 x y x y x x x x x x x x x x x x x y = ½x 2 Menu Back to questions

66 x y x y x x x x x x x x x x x x x y = x Menu Back to questions

67 y x x y x x x x x x x x x x x x x y = x Menu Back to questions

68 x y y = x 2 + 2x x y x x x x x x x x x x x Menu Back to questions

69 x y y = x 2 – 3x x y x x x x x x x x x x x Menu Back to questions

70 x y y = x 2 + 3x + 10 x y x x x xx x x x x x x Menu Back to questions

71 y = x 2 - 4x - 10 y x x y x x x x x x x x x x x x x Menu Back to questions

72 y x y = x x y x x x x x x x Menu Back to questions

73 y x y = x 3 + 2x x y x x x x x x x Menu Back to questions

74 y = x 3 + x 2 x y y x x x xx x x x 0 Menu Back to questions

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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) = f(20) = 41 f(20) = 400 f(20) = Menu

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 f(x) + 2 = x 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

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

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 f(x) = - (2x + 1) = -2x - 1 f(x) = x 2 - f(x) = - x 2 f(x) = sin x - f(x) = - sin x Menu

80 yy y xx x f(x) + a y = 2x + 1 y = 2x + 3 y = 2x + 1 y = 2x y = 2x + 3 y = x 2 y = x y = x 2 y = sin x y = sin x + 2 y = x y = sin x + 2 In what same way have all three graphs been transformed ? They have all been translated 2 units parallel to the y axis. translates ‘a’ units parallel to the y axis. Menu

81 yy y xx x 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

82 yy y xx x 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 Menu

83 yy y xx x 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

84 yy y xx x 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 y = sin 2x Menu

85 yy y xx x 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

86 yy y xx x 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 ? 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

87 yy y xx x 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

88 yy y xx x 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 They have all been reflected in the x axis. Menu

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

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

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

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

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

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

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112 x y Menu Solve algebraically : x + 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

113 x y 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 = & 1.5 Example 1 Menu

114 x y 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 = & 0.7 Note : You adapt the equation that you are solving to be the same as the graph which has already been drawn ! Menu

115 x y 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 = & 1.6 Menu

116 x y 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 = & 0.8 Menu

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