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Solving a Quadratic Equation Through Three Points

Solving a Quadratic Equation Through Three Points
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2014-07-29 22:08
FunctionCoordinate Geometry
In collections CCGPS AA Unit 6 Lesson 1
This example demonstrates how to find a quadratic equation that fits three data points (0,0), (1,2), and (2,8) by solving a system of equations. It involves core math concepts like quadratic functions, systems of equations, and coordinate geometry. The construction uses GeoGebra commands/objects such as LaTeX, If, TableText, etc. Ideal for high school students learning quadratic functions and systems of equations.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a graph showing how to find a quadratic equation through three points (0,0), (1,2), and (2,8). Plot these three points and draw the quadratic curve that passes through them. Then, display the mathematical steps to solve for the coefficients a, b, and c of the quadratic equation y=ax^2+bx+c by substituting the coordinates of these points. Finally, verify that the curve passes through all three points by substituting the found coefficients back into the equation and show how to predict the location of a fourth point using the equation.
The prompt above was compiled from the original drawing commands and is for reference only.