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Linear Equation Optimization Example

Linear Equation Optimization Example
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2017-08-06 22:36
FunctionOptimization
This example demonstrates solving optimization problems using linear equations. Key mathematical concepts include linear equations and optimization problems. GeoGebra commands such as Sequence, Zip, Segment, and Slope are used to construct the graphical representation. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Move the slider to solve the problem.}") → text0
  3. LaTeX("\text{Use the buttons below to return to the original}\cr\text{problem or randomly generate a similar practice}\cr\text{problem.}") → RandomizationText
  4. Corner(2, 3) → topright
  5. Corner(2, 1) → bottomleft
  6. Sequence(k, k, 2010, 2017, 0.25) → Years
  7. LaTeX((((("\text{The graph on the right shows how the value of a stock}\cr\text{has changed over a period of time. What is the}\cr\text{average rate of change from " * startYear) * " to ") * endYear) * "?}")) → Problem
  8. LaTeX("\text{Determine the interval to be observed.}") → text1
  9. LaTeX((((((((("\text{The interval to be observed is }\textcolor{#000099}{[" * startYear) * ", ") * endYear) * "]}\text{, or}\cr\text{where }\textcolor{#000099}{") * startYear) * " ≤ x ≤ ") * endYear) * "}\text{.}")) → text2
  10. LaTeX("\text{Identify the initial point of the interval.}") → text3
  11. LaTeX("\text{Identify the endpoint of the interval.}") → text5
  12. LaTeX((((((("\text{The }\textcolor{#000099}{x}\text{-value of the initial point is }\textcolor{#000099}{" * startYear) * "}\text{. The}\cr\text{corresponding }\textcolor{#000099}{y}\text{-value is approximately }\textcolor{#000099}{") * y(StartPoint)) * "}\text{.}\cr\text{The coordinates of the initial point of the interval}\cr\text{are about }\textcolor{#000099}{") * StartPoint) * "}\text{.}")) → text4
  13. Zip((j, k), j, Years, k, SharePrices) → Data
  14. LaTeX((((((("\text{The }\textcolor{#000099}{x}\text{-value of the final point is }\textcolor{#000099}{" * endYear) * "}\text{. The}\cr\text{corresponding }\textcolor{#000099}{y}\text{-value is approximately }\textcolor{#000099}{") * y(EndPoint)) * "}\text{.}\cr\text{The coordinates of the final point of the interval}\cr\text{are about }\textcolor{#000099}{") * EndPoint) * "}\text{.}")) → text6
  15. LaTeX("\text{Substitute the coordinates of the two points into the}\cr\text{slope formula to calculate the average rate of change.}") → text7
  16. Segment(StartPoint, EndPoint) → f
  17. Slope(f) → a
  18. If(round(a, 0) ≟ changeRate, LaTeX[(abs(changeRate) * ".00")], LaTeX[(abs(changeRate) * "0")]) → rateText
  19. If(a < 0, "-", "") → rateSign
  20. TableText({{LaTeX["\textcolor{#000099}{m=\frac{y_2-y_1}{x_2-x_1}}"], LaTeX["\text{Slope formula}"]}}) → text8
  21. LaTeX((((((((("\text{The average rate of change for the price of this stock over the interval}\cr\text{[}" * startYear) * ", ") * endYear) * "\text{] is approximately }") * rateSign) * "\$") * rateText) * "\text{ per year.}")) → text12
  22. TableText({{LaTeX["\textcolor{#000099}{m=\frac{y_2-y_1}{x_2-x_1}}"], LaTeX["\text{Slope formula}"]}, {LaTeX[(((((((("\textcolor{#000099}{m=\frac{(" * y(EndPoint)) * ")-(") * y(StartPoint)) * ")}{(") * x(EndPoint)) * ")-(") * x(StartPoint)) * ")}}")], LaTeX[(("\text{Substitute }\textcolor{#000099}{" * StartPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_1,y_1)}"]}, {LaTeX["\text{}"], LaTeX[(("\text{and }\textcolor{#000099}{" * EndPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_2,y_2)}\text{.}"]}}) → text9
  23. TableText({{LaTeX["\textcolor{#000099}{m=\frac{y_2-y_1}{x_2-x_1}}"], LaTeX["\text{Slope formula}"]}, {LaTeX[(((((((("\textcolor{#000099}{m=\frac{(" * y(EndPoint)) * ")-(") * y(StartPoint)) * ")}{(") * x(EndPoint)) * ")-(") * x(StartPoint)) * ")}}")], LaTeX[(("\text{Substitute }\textcolor{#000099}{" * StartPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_1,y_1)}"]}, {LaTeX["\text{}"], LaTeX[(("\text{and }\textcolor{#000099}{" * EndPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_2,y_2)}\text{.}"]}, {LaTeX[(((("\textcolor{#000099}{m=\frac{" * rise) * "}{") * run) * "}}")], LaTeX["\text{Subtract.}"]}}) → text10
  24. TableText({{LaTeX["\textcolor{#000099}{m=\frac{y_2-y_1}{x_2-x_1}}"], LaTeX["\text{Slope formula}"]}, {LaTeX[(((((((("\textcolor{#000099}{m=\frac{(" * y(EndPoint)) * ")-(") * y(StartPoint)) * ")}{(") * x(EndPoint)) * ")-(") * x(StartPoint)) * ")}}")], LaTeX[(("\text{Substitute }\textcolor{#000099}{" * StartPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_1,y_1)}"]}, {LaTeX["\text{}"], LaTeX[(("\text{and }\textcolor{#000099}{" * EndPoint) * "}")]}, {LaTeX["\text{}"], LaTeX["\text{for }\textcolor{#000099}{(x_2,y_2)}\text{.}"]}, {LaTeX[(((("\textcolor{#000099}{m=\frac{" * rise) * "}{") * run) * "}}")], LaTeX["\text{Subtract.}"]}, {LaTeX[(("\textcolor{#000099}{m=" * changeRate) * "}")], LaTeX["\text{Divide.}"]}}) → text11