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Understanding Cubic Graphs

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Understanding Cubic Graphs. 5. Exercise 2A – Multiple Transformation. 1. Introduction to cubic curves. 2. Exercise 1A – Equations and curves. 6. Exercise 2C – Finding Mult. Trans. Equations. 3. Notes of transformations. 4. Exercise 1B – Single Transformations.

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Understanding Cubic Graphs
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  1. Understanding Cubic Graphs 5. Exercise 2A – Multiple Transformation 1. Introduction to cubic curves 2. Exercise 1A – Equations and curves 6. Exercise 2C – Finding Mult. Trans. Equations 3. Notes of transformations 4. Exercise 1B – Single Transformations

  2. Predicting Ths Shape Of A Cubic Graph From Its Equations It is very difficult to to tell the shape of a cubic graph when the equation is in its General Formof y = ax3 + bx2 + cx + d So it is very hard to predict what the following graphs will look like: y = 8x3 – 36x2 + 54x – 27 y = 2x3 + x2 + 8x + 4 y = 3x3 + 10x2 + 9x + 2

  3. But if we factorise the equations….. We notice they fall into one three different categories

  4. y = (2x + 3)3 Stationary Point of Inflection A point of inflection is where the curvature of graph changes (where the graph changes from part happy fact to part sad face or visa versa).

  5. y = (2x + 1)(x2 + 4) Non Stationary Point of Inflection

  6. y = (3x + 1)(x + 2)(x + 1) Local Maximum Local Minimum Non stationary point of inflection

  7. Other Examples

  8. Graphs of cubics with a negative x3 co-efficients Graphs with negative x3 co-efficients all factorise with a negative sign or multiple out the front. All these graphs eventually drop from left to right.

  9. Exercise 1A

  10. Exercise 1A

  11. Exercise 1A

  12. -2 -4 1 Exercise 1A 2a) y = – (x + 4) (x + 2) (x – 1)

  13. -1 2 Exercise 1A 2b) y = (x + 1) . x . (x – 2) = x (x + 1) (x – 2)

  14. -1 2 -1 Exercise 1A 2c) y = (x + 1) (x – 2) 2

  15. 2 Exercise 1A 2d) y = – x 2 (x – 2)

  16. -2 Exercise 1A 2e) y = (x + 2) 2 . x = x (x + 2) 2

  17. 1 -3 -1 Exercise 1A 2f) y = (x + 3) (x + 1) (x – 1)

  18. Exercise 1A 3a) y = a (x + 2) (x + 1) (x – 1) sub in (0, 8) 8 = a . 2 . 1 . –1 8 = –2a –4 = a soy = –4(x + 2) (x + 1) (x – 1)

  19. Exercise 1A 3b) y = a (x + 2) (x – 1) 2 sub in (0, 6) 6 = a . 2 . (1)2 6 = 2a 3 = a soy = 3(x + 2) (x – 1)2

  20. Exercise 1A 3b) y = a (x + 2) (x – 1) 2 sub in (-1, 12) 12 = a . (-1 + 2) . (-1 – 1)2 12 = a . 1 . (-2)2 12 = 4a 3 = a soy = 3(x + 2) (x – 1)2

  21. Exercise 1A 4a) A cubic regression on the known points on the graphic calculator gives: y = –4x3 – 8x2 + 4x + 8 which factorises on the calculator to y = –4(x + 2) (x + 1) (x – 1) which is the same result as attained by algebra

  22. Exercise 1A 4b) Work out the same as q4a 5. Check your answer up the back of the book 6. Check your answer up the back of the book

  23. Exercise 1A 7a) y = x3 + 3x2 + 2x + 6 y intercept x intercepts local maxima local minima point of inflection

  24. Exercise 1A 7b) y = 2(x+ 3) (x+ 1) (x– 1) y intercept x intercepts local maxima local minima point of inflection

  25. Exercise 1A 8. Only do Ex 7H Q1,2,3,4,5a and check your answers up the back of the book

  26. Three Types of Transformations Translations = sliding to the left, right, up or down Dilations =stretching or compressing from the x or y axis Reflections = folding about the x or y axis. Each of these transformations has a hand action that I will be using.

  27. What Mathematicians Have Found! For operations outside the cube General rule: The operation you see operates on the y co-ordinates. Specific rules:  the y values by the number involved (= dilation by factor from the x axis) + – + - the number to the y values (=translation in positive y or negative y direction ) neg = reflection (folding) of y values about the x axis

  28. y = x3 + 5 Add 5 to every y value = translate every y value 5 units in the positive y direction When transforming we usually start with the point of inflection

  29. y = 2x3 Multiply every y value by 2 = dilate every y value by a factor of 2 from the x axis When transforming we usually start with the point of inflection

  30. y = – x3 Reflect every y value about the x axis = fold every y value over the x axis When transforming we usually start with the point of inflection

  31. y = (x + 3)3 Take 3 to every x value = translate every x value 3 units in the negative x direction When transforming we usually start with the point of inflection

  32. y = (2x)3 Halve every x value = dilate every x value by a factor of ½ from the y axis When transforming we usually start with the point of inflection

  33. y = (– x)3 Reflect every x value about the y axis = fold every x value over the y axis

  34. Exercise 1B

  35. Exercise 1B1a) y = 3x3 Multiply every y value by 3 = dilate every y value by a factor of 3 from the x axis

  36. Exercise 1B1b) Halve every y value = dilate every y value by a factor of ½ from the x axis

  37. Multiply every x value by 2 = dilate every y value by a factor of 2 from the y axis Exercise 1B1c)

  38. Exercise 1B1d) y = (x – 1)3 Add 2 to every x value = translate every x value 1 unit in the positive x direction

  39. Exercise 1B1e) y = x3 – 10 Take 10 from every y value = translate every y value 10 units in the negative y direction

  40. Exercise 1B2a) Dilate y = x3 by a factor of 3 from the y axis New Equation:

  41. Exercise 1B2b) Translate y = x3 by 3 in the positive y direction New Equation: y = x3 + 3

  42. Exercise 1B2c) Dilate y = x3 by a factor of 3 from the x axis New Equation: y = 3x2

  43. Exercise 1B2d) Translate y = x3 by 2 in the negative x direction New Equation: y = (x + 2)3

  44. Exercise 1B2e) Reflect about the y axis New Equation: y = (-x)3

  45. Exercise 1B2f) Reflect about the x axis New Equation: y = -x3

  46. Exercise 2A

  47. x + 1 y + 5 x – 1 y 2 As a flowcharty = x3 As a flowcharty = x3 Exercise 2A1a)y = (x – 1)3 + 5 In words Trans 1 = translation 1 unit in the positive x direction Trans 2 = translation 5 units in the positive y direction 1b)y = 2(x + 1)3 In words Trans 1 = translation 1 unit in the negative x direction Trans 2 = dilation by a factor of 2 from x axis

  48. – y y – 10 As a flowcharty = x3 Exercise 2A1c)y = –(x)3 – 10 In words Trans 1 = reflection in x axis Trans 2 = translation 10 units in the negative y direction

  49. description x + 1 y + 5 Exercise 2A2a)y = (x – 1)3 + 51: translation 1 unit in the positive x direction 2: translation 5 units in the positive y direction flowcharty = x3

  50. description y 2 x– 1 Exercise 2A2b)y = 2(x + 1)31: dilation by a factor of 2 from x axis 2: translation 1 unit in the negative x direction flowcharty = x3