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Quadratic Roots and Parabola Graph: Interactive Nature of Solutions

Quadratic Roots and Parabola Graph: Interactive Nature of Solutions
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2021-01-26 00:25
FunctionGeometric ConstructionSlider AnimationGraphing
This example demonstrates how the nature of solutions of a quadratic equation ax^2+bx+c=0 corresponds to the graph of a parabola. It visually links the discriminant Δ=b^2−4ac to root types: Δ>0 (two distinct real roots), Δ=0 (one repeated root), and Δ<0 (no real roots). The key math concepts include quadratic equations, discriminant-based classification, the relation between x-intercepts and real roots, and the geometric meaning of a quadratic function’s opening direction and intercept structure. Construction-wise, the design relies on GeoGebra tools such as Textfield and InputBox objects to enter parameters (a, b, c and related values), uses Sum and If to compute/compare expressions involving Δ and select the correct case, and outputs the results via TableText. Named text objects (e.g., text1) and table entries (multiple labeled cells) support synchronized explanations alongside the plotted diagram. Suitable for middle and high school students, classroom interactive lessons, and exam preparation focused on translating algebraic conditions into graph features.

GeoGebra Commands & Parameters

  1. Textfield(a) → InputBox1
  2. Textfield(b) → InputBox2
  3. Textfield(c) → InputBox3
  4. Textfield(m) → InputBox4
  5. Textfield(n) → InputBox5
  6. Sum(B4:B6) → B7
  7. Sum(C4:C6) → C7
  8. Sum(D4:D6) → D7
  9. Sum(E4:E6) → E7
  10. Sum(F4:F6) → F7
  11. Sum(G4:G6) → G7
  12. Sum(H4:H6) → H7
  13. Sum(I4:I6) → I7
  14. Sum(J4:J6) → J7
  15. If(w > A10, (("" * A3) * "")) → A17_{1}
  16. If(w > A10, (("" * A3) * "")) → A17
  17. If(w > B10, (("" * B3) * "")) → B17
  18. If(w > C10, (("" * C3) * "")) → C17
  19. If(w > D10, (("" * D3) * "")) → D17
  20. If(w > E10, (("" * E3) * "")) → E17
  21. If(w > F10, (("" * F3) * "")) → F17
  22. If(w > G10, (("" * G3) * "")) → G17
  23. If(w > H10, (("" * H3) * "")) → H17
  24. If(w > I10, (("" * I3) * "")) → I17
  25. If(w > A11, (("" * A4) * "")) → A18
  26. If(w > A12, (("" * A5) * "")) → A19
  27. If(w > A13, (("" * A6) * "")) → A20
  28. If(w > A14, (("" * A7) * "")) → A21
  29. If(w > B11, (("" * B4) * "")) → B18
  30. If(w > C11, (("" * C4) * "")) → C18
  31. If(w > D11, (("" * D4) * "")) → D18
  32. If(w > E11, (("" * E4) * "")) → E18
  33. If(w > F11, (("" * F4) * "")) → F18
  34. If(w > G11, (("" * G4) * "")) → G18
  35. If(w > H11, (("" * H4) * "")) → H18
  36. If(w > I11, (("" * I4) * "")) → I18
  37. If(w > B12, (("" * B5) * "")) → B19
  38. If(w > C12, (("" * C5) * "")) → C19
  39. If(w > D12, (("" * D5) * "")) → D19
  40. If(w > E12, (("" * E5) * "")) → E19
  41. If(w > F12, (("" * F5) * "")) → F19
  42. If(w > G12, (("" * G5) * "")) → G19
  43. If(w > H12, (("" * H5) * "")) → H19
  44. If(w > I12, (("" * I5) * "")) → I19
  45. If(w > B13, (("" * B6) * "")) → B20
  46. If(w > C13, (("" * C6) * "")) → C20
  47. If(w > D13, (("" * D6) * "")) → D20
  48. If(w > E13, (("" * E6) * "")) → E20
  49. If(w > F13, (("" * F6) * "")) → F20
  50. If(w > G13, (("" * G6) * "")) → G20
  51. If(w > H13, (("" * H6) * "")) → H20
  52. If(w > I13, (("" * I6) * "")) → I20
  53. If(w > B14, (("" * B7) * "")) → B21
  54. If(w > C14, (("" * C7) * "")) → C21
  55. If(w > D14, (("" * D7) * "")) → D21
  56. If(w > E14, (("" * E7) * "")) → E21
  57. If(w > F14, (("" * F7) * "")) → F21
  58. If(w > G14, (("" * G7) * "")) → G21
  59. If(w > H14, (("" * H7) * "")) → H21
  60. If(w > I14, (("" * I7) * "")) → I21
  61. If(w > J10, (("" * J3) * "")) → J17
  62. If(w > J11, (("" * J4) * "")) → J18
  63. If(w > J12, (("" * J5) * "")) → J19
  64. If(w > J13, (("" * J6) * "")) → J20
  65. If(w > J14, (("" * J7) * "")) → J21
  66. If(w > A24, (("" * A3) * "")) → A27
  67. If(w > B24, (("" * B3) * "")) → B27
  68. If(w > C24, (("" * C3) * "")) → C27
  69. If(w > D24, (("" * D3) * "")) → D27
  70. If(w > E24, (("" * E3) * "")) → E27
  71. If(w > F24, (("" * F3) * "")) → F27
  72. If(w > G24, (("" * G3) * "")) → G27
  73. If(w > H24, (("" * H3) * "")) → H27
  74. If(w > I24, (("" * I3) * "")) → I27
  75. If(w > J24, (("" * J3) * "")) → J27
  76. If(w > A25, (("" * A7) * "")) → A28
  77. If(w > B25, (("" * B7) * "")) → B28
  78. If(w > C25, (("" * C7) * "")) → C28
  79. If(w > D25, (("" * D7) * "")) → D28
  80. If(w > E25, (("" * E7) * "")) → E28
  81. If(w > F25, (("" * F7) * "")) → F28
  82. If(w > G25, (("" * G7) * "")) → G28
  83. If(w > H25, (("" * H7) * "")) → H28
  84. If(w > I25, (("" * I7) * "")) → I28
  85. If(w > J25, (("" * J7) * "")) → J28
  86. TableText({{A17, B17, C17, D17, E17, F17, G17, H17, I17, J17}, {A18, B18, C18, D18, E18, F18, G18, H18, I18, J18}, {A19, B19, C19, D19, E19, F19, G19, H19, I19, J19}, {A20, B20, C20, D20, E20, F20, G20, H20, I20, J20}, {A21, B21, C21, D21, E21, F21, G21, H21, I21, J21}}, "|_llllllllll") → text2
  87. If(d ≟ 0, "ROOTS ARE REAL AND EQUAL", If[d < 0, "ROOTS ARE UNREAL", If[d > 0, "ROOTS ARE REAL AND DISTINCT"]]) → text6

AI Drawing Prompt Reference

Draw a parabola graph given by y = ax^2 + bx + c. Include a discriminant calculation Δ = b^2 - 4ac. Display text explanations based on Δ: if Δ > 0, show "two distinct real roots"; if Δ = 0, show "one repeated root"; if Δ < 0, show "no real roots". The construction order should be: input parameters a, b, c; compute Δ; then display the appropriate text based on Δ's value. The graph should clearly illustrate the relationship between the parabola and the roots.
The prompt above was compiled from the original drawing commands and is for reference only.