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二维一次不等式的图形表示与解集判定

二维一次不等式的图形表示与解集判定
几何画板及绘图内容加载中，请稍等……
2023-09-25 20:29
解析几何不等式几何作图函数图像
该范例演示如何在二维坐标平面中绘制一次不等式（含边界是否取等）的图形，并用测试点判断不等式的解集区域。核心数学概念包括：线性不等式、半平面/区域表示、边界直线（严格不等与非严格不等对应实线或虚线）、以及代数/图形法解二元一次不等式。作图要点来自构造摘要中的 GeoGebra 元素与命令：用 Inequality 生成不等式模型，并借助 Line 构造边界直线；用 IsInRegion 对测试点 TestPointInReg 判断其是否落在不等式解区域内；使用 Segment、Polygon、IntersectPath 等进行图形元素组织与区域描绘；通过 Split 将对象按边界分割，并用 Zip/IndexOf/AreEqual 处理分支与情形切换；文本与交互反馈使用 Textfield、Text、Translate 生成提示/说明；教学流程与引导显示用 onboardingTextOffset、onboardingStage、showhint1 以及多对象状态对象（如 selectedSide1/selectedSide2、side1/side2、InSide1/InSide2）来实现随机测试与反馈。本资源适合 9-12 年级的代数学习，也适合作为课堂探究、分层练习或图形法/代数法结合教学的交互材料。

GeoGebra 命令及参数

  1. Sequence(Random[0, 99] / 100, A, 1, 200) → allRandomSeeds, {0.39, 0.65, 0.37, 0.82, 0.69, 0.31, 0.54, 0.23, 0.13, 0.95, 0.22, 0.94, 0.8, 0.59, 0.16, 0.17, 0.95, 0.96, 0.11, 0.03, 0.62, 0.42, 0.72, 0.7, 0.15, 0.93, 0.34, 0.58, 0.85, 0.92, 0.12, 0.31, 0.26, 0.11, 0.82, 0.73, 0.76, 0.16, 0.38, 0.19, 0.51, 0.84, 0.98, 0.97, 0.53, 0.87, 0.02, 0.79, 0.12, 0.14, 0.47, 0.23, 0.86, 0, 0.39, 0.54, 0.35, 0.89, 0.49, 0.76, 0.43, 0.47, 0.06, 0.55, 0.71, 0.39, 0.5, 0.11, 0.65, 0.68, 0.81, 0.12, 0.2, 0.91, 0.49, 0.23, 0.69, 0.72, 0.17, 0.9, 0.89, 0.12, 0, 0.92, 0.53, 0.44, 0.56, 0.41, 0.93, 0.54, 0.43, 0.12, 0.5, 0.83, 0.62, 0.31, 0.01, 0.49, 0.5, 0.59, 0.56, 0.62, 0.36, 0.02, 0.07, 0.5, 0.93, 0.49, 0.17, 0.51, 0.56, 0.95, 0.22, 0.63, 0.71, 0.45, 0.93, 0.44, 0.57, 0.87, 0.31, 0.79, 0.78, 0.53, 0.75, 0.46, 0.33, 0.47, 0.05, 0.41, 0.36, 0.06, 0.57, 0.76, 0.66, 0.21, 0.89, 0.96, 0.25, 0.63, 0.2, 0.07, 0.32, 0.27, 0.06, 0.22, 0.9, 0.78, 0.93, 0.38, 0.69, 0.68, 0.14, 0.78, 0.51, 0.84, 0.21, 0.21, 0.39, 0.7, 0.19, 0.48, 0.21, 0.27, 0.65, 0.87, 0.64, 0.12, 0.06, 0.37, 0.93, 0.44, 0.9, 0.09, 0.24, 0.92, 0.31, 0.49, 0.62, 0.32, 0.26, 0.7, 0.74, 0.72, 0.22, 0.63, 0.86, 0.8, 0.76, 0.71, 0.03, 0.98, 0.34, 0.24, 0.9, 0.19, 0.43, 0.78, 0.73, 0.09}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. If(currentRandomSeeds(3) ≥ 0.5, 1, 2) → Case
  4. Line((If[randomNum1 ≟ 0, 1, randomNum1], 0), (0, If[randomNum2 ≟ 0, 1, randomNum2])) → lineCase1
  5. If(Case ≟ 1, AreCollinear[TestPoint, Intersect[lineCase1, xAxis], Intersect[lineCase1, yAxis]], AreCollinear[TestPoint, (0, randomNum2), (1, randomNum1 + randomNum2)]) → TestPointInLine
  6. If(Mod[Inequality, 2] ≟ 1, y < (randomNum1 * x) + randomNum2, y > (randomNum1 * x) + randomNum2) → ineq2
  7. Element({"≤", "≥", "<", ">"}, Inequality) → InequalityText
  8. If(invert, If[Mod[Inequality, 2] ≟ 1, y > (Slope[lineCase1] * x) + If[randomNum2 ≟ 0, 1, randomNum2], y < (Slope[lineCase1] * x) + If[randomNum2 ≟ 0, 1, randomNum2]], If[Mod[Inequality, 2] ≟ 1, y < (Slope[lineCase1] * x) + If[randomNum2 ≟ 0, 1, randomNum2], y > (Slope[lineCase1] * x) + If[randomNum2 ≟ 0, 1, randomNum2]]) → ineq1
  9. IsInRegion(TestPoint, If[Case ≟ 1, ineq1, ineq2]) → TestPointInReg
  10. Split(TextCase1, {"≥", "≤", ">", "<"}) → Text1Split
  11. Zip(AreCollinear[PP, (0, randomNum2), (1, randomNum1 + randomNum2)], PP, TestPoints) → InLineCase2
  12. Zip(AreCollinear[PP, Intersect[lineCase1, xAxis], Intersect[lineCase1, yAxis]], PP, TestPoints) → InLineCase1
  13. IndexOf(false, If[Case ≟ 1, InLineCase1, InLineCase2]) → pointNum
  14. Element({"Drag to draw a line", "Click to change the line style", "Click to select a region"}, onboardingStage) → onboardingText
  15. Line(P1, P2) → studLine
  16. If(Defined[Slope[studLine]], y < (Slope[studLine] * x) + studLine(0) ∧ (-GVsize) ≤ x ≤ GVsize ∧ (-GVsize) ≤ y ≤ GVsize, x(P1) < x < GVsize ∧ (-GVsize) ≤ y ≤ GVsize) → side2
  17. Zip(IsInRegion[PP, side2], PP, TestPoints) → InSide2
  18. If(Defined[Slope[studLine]], y > (Slope[studLine] * x) + studLine(0) ∧ (-GVsize) ≤ x ≤ GVsize ∧ (-GVsize) ≤ y ≤ GVsize, (-GVsize) < x < x(P1) ∧ (-GVsize) ≤ y ≤ GVsize) → side1
  19. Zip(IsInRegion[PP, side1], PP, TestPoints) → InSide1
  20. AreEqual(studLine, If[Case ≟ 1, lineCase1, lineCase2]) → correct1
  21. Zip(IsInRegion[PP, ineq2], PP, TestPoints) → InIneq2
  22. Zip(IsInRegion[PP, ineq1], PP, TestPoints) → InIneq1
  23. Textfield(input1) → InputBox1
  24. Textfield(input2) → InputBox2
  25. Corner(5) → layoutAppSize
  26. PointToPixels(Corner[onboardingDisplayText, 2], Corner[1], Corner[3], Corner[5]) → layoutOnboarding
  27. Polygon({((-GVsize), (-GVsize)), (GVsize, (-GVsize)), (GVsize, GVsize), ((-GVsize), GVsize)}) → q1
  28. Text("x", (GVsize + 0.5, 0.1), true, true, 0, 0) → text1
  29. Text("y", (0, GVsize + 0.5), true, true, 0, 0) → text2
  30. Translate(hGL, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → hGL'
  31. Translate(vGL, Vector[(Vector[Corner[1], Corner[2]] / 800 * 16)]) → vGL'
  32. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  33. circlePixels(P1, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject
  34. circlePixels(P2, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject2
  35. IntersectPath(studLine, q1) → f
  36. expandSegment(Vertex[f, 1], Vertex[f, 2], (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject3
  37. Split(Text1Replace, {"≥", "≤", ">", "<"}) → l1
  38. Split(TextCase2, {"≥", "≤", ">", "<"}) → Text2Split
  39. ReplaceAll(ReplaceAll[Text1Split(1), "x", (("(" * x(TestPoint)) * ")")], "y", (("(" * y(TestPoint)) * ")")) → text3
  40. Segment(((-GVsize) + a, 0), (GVsize - a, 0)) → fakeXAxis
  41. Segment((0, (-GVsize) + a), (0, GVsize - a)) → fakeYAxis
  42. circlePixels(TestPoint, 9 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingSol

AI 绘图提示语参考

在二维坐标系中绘制线性不等式，展示边界线和解区域。首先生成随机点和直线，根据不等式类型确定边界线是实线还是虚线。通过测试点判断点是否在解区域内，并用不同颜色或样式区分。最后，添加坐标轴、标签和说明文字，解释不等式的解法和解区域的意义。
以上提示语为根据原始绘图命令整理而来，仅供参考。