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通过三个数据点求解二次方程

通过三个数据点求解二次方程
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2014-07-29 22:08
函数解析几何
所属专题 CCGPS AA Unit 6 Lesson 1
①本范例演示如何通过三个数据点(0,0), (1,2), (2,8)求解一个二次方程；②涉及的核心数学概念包括二次函数、方程组求解、解析几何；③作图要点提炼自构造摘要，使用了LaTeX, If, TableText等GeoGebra命令/对象进行条件判断和文本显示；④适合高中阶段学习二次函数和方程组的学生使用。

GeoGebra 命令及参数

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  3. LaTeX("\text{Use the given data points to write three equations using the standard form of a}\cr\text{quadratic equation.}") → text1
  4. LaTeX(((((A * "\text{, }") * B) * "\text{, }") * C)) → Problem
  5. LaTeX("\text{Substitute the }\textcolor{#000099}{x}\text{- and }\textcolor{#000099}{y}\text{-values from each of the three data points into }\cr\textcolor{#000099}{y=ax^2 +bx+c}\text{ and simplify.}") → text2
  6. If(x(A) ≟ 0, LaTeX[""], If[x(A) ≟ 1, LaTeX["-b"], If[x(A) ≟ -1, LaTeX["+b"], LaTeX[(Simplify[(-x(A))] * "b")]]]) → cb
  7. If(x(A) ≟ 0, LaTeX[""], If[x(A) ≟ 1, LaTeX["-a"], If[x(A) ≟ -1, LaTeX["-a"], LaTeX[(Simplify[(-x(A)^(2))] * "a")]]]) → ca
  8. If(y(A) ≟ 0, LaTeX[""], If[y(A) ≟ 1, LaTeX["1"], If[y(A) ≟ -1, LaTeX["-1"], LaTeX[y(A)]]]) → cy
  9. If(x(A) + y(A) ≟ 0, LaTeX["0"], LaTeX[""]) → cfix
  10. LaTeX("\text{Substitute }\textcolor{#000099}{c}\text{ and }\textcolor{#000099}{b}\text{ from the previous steps into the equation using the third}\cr\text{point and solve for }\textcolor{#000099}{a}.") → text7
  11. LaTeX("\text{Substitute the value of }\textcolor{#000099}{a}\text{ into the equation for }\textcolor{#000099}{b}\text{ to find the value for }\textcolor{#000099}{b}.") → text10
  12. LaTeX((("\textcolor{#000099}{b=" * b) * "}")) → text11
  13. LaTeX("\text{Substitute the values for }\textcolor{#000099}{a}\text{ and }\textcolor{#000099}{b}\text{ into the formula for }\textcolor{#000099}{c}\text{ to find the value for }\textcolor{#000099}{c}.") → text12
  14. LaTeX((("\textcolor{#000099}{c=" * c) * "}")) → text13
  15. LaTeX("\text{Use the values determined for }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{b}\text{, and }\textcolor{#000099}{c}\text{ to write the quadratic equation that is}\cr\text{true for the three data points.}") → text14
  16. LaTeX((((((("\text{Substitute }\textcolor{#000099}{a = " * a) * "}\text{, }\textcolor{#000099}{b = ") * b) * "}\text{, and }\textcolor{#000099}{c = ") * c) * "}\text{ into the standard form of a quadratic}\cr\text{equation and simplify.}")) → text15
  17. LaTeX("\text{Check the resulting equation by substituting the x- and y-values from each of the}\cr\text{three data points to see if a true statement results. All three data points should}\cr\text{produce true statements.}") → text17
  18. LaTeX((("\text{All three data points produce true statements for }\textcolor{#000099}{y = " * f) * "}.")) → text19
  19. TableText({{LaTeX[("\text{For }" * A)], LaTeX[("\text{For }" * B)], LaTeX[("\text{For }" * C)]}, {LaTeX[(("\textcolor{#000099}{y=" * f) * "}")], LaTeX[(("\textcolor{#000099}{y=" * f) * "}")], LaTeX[(("\textcolor{#000099}{y=" * f) * "}")]}, {LaTeX[(((("\textcolor{#000099}{" * y(A)) * "=") * f(x(A))) * "}")], LaTeX[(((("\textcolor{#000099}{" * y(B)) * "=") * f(x(B))) * "}")], LaTeX[(((("\textcolor{#000099}{" * y(C)) * "=") * f(x(C))) * "}")]}}) → text18
  20. LaTeX((("\text{Substitute }\textcolor{#000099}{" * x4) * "}\text{ for }\textcolor{#000099}{x}\text{ in the equation and solve for }\textcolor{#000099}{y}.")) → text22
  21. LaTeX((((("\textcolor{#000099}{y=" * f) * "}\cr\textcolor{#000099}{y=") * f(x4)) * "}")) → text23
  22. LaTeX((((("\text{The fourth point is }\textcolor{#000099}{(" * x4) * ",") * f(x4)) * ")}.")) → text24
  23. TableText({{LaTeX[("\text{For }" * A)], LaTeX[("\text{For }" * B)], LaTeX[("\text{For }" * C)]}, {LaTeX["\textcolor{#000099}{y=ax^2+bx+c}"], LaTeX["\textcolor{#000099}{y=ax^2+bx+c}"], LaTeX["\textcolor{#000099}{y=ax^2+bx+c}"]}, {LaTeX[(((((("\textcolor{#000099}{" * y(A)) * "=a(") * x(A)) * ")^2+b(") * x(A)) * ")+c}")], LaTeX[(((((("\textcolor{#000099}{" * y(B)) * "=a(") * x(B)) * ")^2+b(") * x(B)) * ")+c}")], LaTeX[(((((("\textcolor{#000099}{" * y(C)) * "=a(") * x(C)) * ")^2+b(") * x(C)) * ")+c}")]}, {LaTeX[(((((("\textcolor{#000099}{c=" * y(A)) * "-a(") * x(A)) * ")^2-b(") * x(A)) * ")}")], LaTeX[""], LaTeX[""]}}) → text3
  24. LaTeX("\text{Move the slider to solve the problem.}") → text0
  25. LaTeX("\text{If we designate the three given points from left to right as }\textcolor{#000099}{(x_1,y_1)}\text{, }\textcolor{#000099}{(x_2,y_2)}\text{, }\cr\text{and }\textcolor{#000099}{(x_3,y_3)}\text{, then the general formula for }\textcolor{#000099}{b}\text{ is }\cr\textcolor{#000099}{b=\frac{y_2-y_1-a[(x_2)^2-(x_1)^2]}{x_2-x_1}}.") → text5
  26. LaTeX((((((((((((((((((("\text{Substituting in the given values gives }\textcolor{#000099}{b=\frac{(" * y(B)) * ")-(") * y(A)) * ")-a[(") * x(B)) * ")^2-(") * x(A)) * ")^2]}{(") * x(B)) * ")-(") * x(A)) * ")}=\frac{") * y2my1) * "-a(") * x2mx1sq) * ")}{") * x2mx1) * "}}.")) → text6
  27. LaTeX("\text{The general formula for }\textcolor{#000099}{a}\text{ is }\textcolor{#000099}{a=\frac{(y_3-y_1)(x_2-x_1)-(y_2-y_1)(x_3-x_1)}{[(x_3)^2-(x_1)^2](x_2-x_1)-(x_3-x_1)[(x_2)^2-(x_1)^2]}}.") → text8
  28. TableText({{LaTeX["\textcolor{#000099}{y=ax^2+bx+c}"], LaTeX["\text{Standard form of a quadratic equation}"]}, {LaTeX[(((((("\textcolor{#000099}{y=(" * a) * ")x^2+(") * b) * ")x+(") * c) * ")}")], LaTeX[(((((("\text{Substitute }\textcolor{#000099}{" * a) * "}\text{ for }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{") * b) * "}\text{ for }\textcolor{#000099}{b}\text{, and }\textcolor{#000099}{") * c) * "}\text{ for }\textcolor{#000099}{c}.")]}, {LaTeX[(("\textcolor{#000099}{y=" * f) * "}")], LaTeX["\text{Simplify.}"]}}) → text16
  29. LaTeX((("\text{Use the resulting quadratic equation model for the three given points to predict}\cr\text{the location of a fourth point for which }\textcolor{#000099}{x = " * x4) * "}.")) → text20
  30. LaTeX("\text{Substitute the value for }\textcolor{#000099}{c}\text{ found from the first point into the equation from the second}\cr\text{point and solve for }\textcolor{#000099}{b}.") → text4
  31. LaTeX((((((((((((((((((((((((((((((((((("\text{Substitute values using the given points. }\cr\textcolor{#000099}{a=\frac{[" * y(C)) * "-(") * y(A)) * ")][") * x(B)) * "-(") * x(A)) * ")]-[") * y(B)) * "-(") * y(A)) * ")][") * x(C)) * "-(") * x(A)) * ")]}{[(") * x(C)) * ")^2-(") * x(A)) * ")^2][") * x(B)) * "-(") * x(A)) * ")]-[") * x(C)) * "-(") * x(A)) * ")][(") * x(B)) * ")^2-(") * x(A)) * ")^2]}=") * a) * "}")) → text9
  32. LaTeX((("\text{Since }\textcolor{#000099}{y = " * f) * "}\text{ is true for all three given points, it will be true for all other}\cr\text{points on the graph. Therefore, the equation can be used to predict other points.}")) → text21

AI 绘图提示语参考

画一个包含三个数据点 (0,0), (1,2), (2,8) 的图形，并展示如何通过这些点求解一个二次方程。首先标出这三个点，然后绘制一条通过它们的二次曲线。接着，在旁边用数学公式和步骤展示如何通过代入这些点的坐标到二次方程的标准形式 y=ax^2+bx+c 中，解出系数 a、b、c 的值。最后，将求得的系数代入方程，验证曲线确实通过所有三个点，并展示如何用该方程预测第四个点的位置。
以上提示语为根据原始绘图命令整理而来，仅供参考。