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Geometric Representation of ax²+bx+c>0

Geometric Representation of ax²+bx+c>0
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2020-03-01 04:47
FunctionInequality
This example demonstrates the geometric representation of the quadratic inequality ax²+bx+c>0. Key mathematical concepts include quadratic functions, parabolas, and inequalities. Using GeoGebra commands such as Point, Vector, Intersect, etc., the dynamic display of the quadratic function graph and its intersection with the x-axis is shown. Suitable for high school mathematics education and learning about quadratic functions and their properties.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Vector(A, B) → u
  4. Point(yAxis) → C
  5. Point(yAxis) → D
  6. Vector(C, D) → v
  7. OrthogonalLine(D, yAxis) → d
  8. OrthogonalLine(A, xAxis) → e
  9. OrthogonalLine(B, xAxis) → g
  10. Intersect(d, e) → E
  11. Intersect(d, g) → F
  12. Intersect(f, g) → G
  13. Intersect(f, e) → H
  14. Segment(A, B) → h
  15. Point(h) → I
  16. If(f(x(I)) < y(D) ∧ f(x(I)) > 0, f(x(I)), 100) → yhaut
  17. Intersect(yAxis, g) → K
  18. Intersect(h, xAxis) → L
  19. Line(C, xAxis) → i
  20. Intersect(e, i) → M
  21. Intersect(g, i) → N
  22. Intersect(p, xAxis) → O, P
  23. If(f(x(I)) < y(D) ∧ f(x(I)) > y(C) ∧ f(x(I)) ≤ 0, f(x(I)), 100) → ybas
  24. Locus(Q, I) → lieu2
  25. Locus(R, I) → lieu1
  26. Polygon(M, N, F, E) → Poly2, m, n, f_1, e_1
  27. Intersect(xAxis, yAxis) → S
  28. If(Delta ≥ 0 ∧ x(P) < x(B), "" + (x(P))) → Texte1
  29. If(Delta ≥ 0 ∧ x(O) > x(A), "" + (x(O))) → Texte2
  30. If(b > 0, "+", If[b < 0, "-", "0"]) → zz
  31. If(b > 0, "+", If[b < 0, "-", "0"]) → zb
  32. If(b < 0, "-") → sib1
  33. If(b > 0, "+", If[b < 0, "-", "+"]) → sib2
  34. If(c > 0, "+", If[c < 0, "-", "+"]) → zc
  35. If(a > 0 ∧ Delta > 0, "]-∞;" + (x(O)) + "[ U ]" + (x(P)) + "; +∞[") → Texte14
  36. If(a > 0 ∧ Delta > 0, "]" + (x(O)) + ";" + (x(P)) + "[") → Texte27
  37. If(a > 0 ∧ Delta < 0, "pas de solution") → Texte26
  38. If(Delta < 0 ∧ a > 0, "]-∞; +∞[") → Texte15
  39. If(a > 0 ∧ Delta ≟ 0, "Pas de solution") → Texte30
  40. If(Delta ≟ 0, "]-∞;" + ((-(b / ((2 * a))))) + "[ U ]" + ((-(b / ((2 * a))))) + ";+∞[") → Texte23
  41. Polygon(V, W, Z, A_1) → poly1, v_1, w, z, a_1
  42. Segment(B_1, C_1) → l
  43. If(a ≟ 0 ∧ b < 0 ∧ (j ≟ true ∨ k ≟ true), "]-∞; " + ((-(c / b))) + "[") → Texte32
  44. If(a ≟ 0 ∧ b > 0 ∧ (j ≟ true ∨ k ≟ true), "]" + ((-(c / b))) + "; +∞[") → Texte22
  45. If(a ≟ 0 ∧ b > 0 ∧ (j ≟ true ∨ k ≟ true), "]-∞; " + ((-(c / b))) + "[") → Texte31
  46. If(a < 0 ∧ Delta > 0, "]-∞;" + (x(O)) + "[ U ]" + (x(P)) + "; +∞[") → Texte17
  47. If(Delta < 0 ∧ a < 0, "]-∞; +∞[") → Texte18
  48. If(a ≟ 0 ∧ b < 0 ∧ (j ≟ true ∨ k ≟ true), "]" + ((-(c / b))) + "; +∞[") → Texte33
  49. If(a < 0 ∧ Delta > 0, "]" + (x(O)) + ";" + (x(P)) + "[") → Texte28
  50. If(a < 0 ∧ Delta < 0, "Pas de solution") → Texte25
  51. If(a ≟ 0 ∧ b ≟ 0 ∧ c ≟ 0 ∧ (j ≟ true ∨ k ≟ true), "Pas de solution") → Texte38
  52. If(a ≟ 0 ∧ b ≟ 0 ∧ c > 0 ∧ (j ≟ true ∨ k ≟ true), "Pas de solution") → Texte37
  53. If(a ≟ 0 ∧ b ≟ 0 ∧ c < 0 ∧ (j ≟ true ∨ k ≟ true), "]-∞; +∞[") → Texte34
  54. If(a ≟ 0 ∧ b ≟ 0 ∧ c < 0 ∧ (j ≟ true ∨ k ≟ true), "Pas de solution") → Texte35
  55. If(a ≟ 0 ∧ b ≟ 0 ∧ c > 0 ∧ (j ≟ true ∨ k ≟ true), "]-∞; +∞[") → Texte36
  56. Point(f_1) → D_1
  57. Point(f_1) → E_1
  58. Polygon(E_1, N, M, D_1) → poly2, e_2, n_1, m_1, d_1

AI Drawing Prompt Reference

Draw the graph of a quadratic function, showing its intersections with the x-axis and highlighting the regions where the function is greater than zero. Start by drawing the x-axis and y-axis, then mark two points on the x-axis and connect them to form a line segment. Do the same on the y-axis. Draw perpendicular lines from these points to the axes, find their intersections, and use these points to create a polygon. Color the area where the function is positive.
The prompt above was compiled from the original drawing commands and is for reference only.