Geometric Construction of Volume Functions

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2014-07-29 04:36

In collections CCGPS AA Unit 6 Lesson 5

This example demonstrates how two functions, [math]f(x) = –2(x – 5)^{2} + 3[/math] and [math]g(x) = –4x^{2} + 8x + 3[/math], represent the volumes of two rectangular solids. It covers core mathematical concepts such as functions, volume calculation, and coordinate geometry. Using GeoGebra commands like LaTeX, Corner, If, and TableText, it dynamically shows the function graphs and their corresponding volume changes. Suitable for high school mathematics education, particularly for teaching coordinate geometry and function graphs.

GeoGebra Commands & Parameters

LaTeX(steps) → counter
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
Corner(1) → A
Corner(3) → B
LaTeX("\text{Determine the coordinates of the maximum point of }\textcolor{#000099}{f(x)}.") → text1
LaTeX("\text{To determine the coordinates of the maximum point of a function, first determine}\cr\text{the type of function and compare it to the standard form for that function.}") → text2
LaTeX((("\text{The function }\textcolor{#000099}{f(x) = " * f) * "}\text{ is a quadratic function.}")) → text3
If(ygmax > fc, LaTeX["\textcolor{#006633}{g(x)}\text{ is the function that}\cr\text{represents the greatest maximum volume for the associated rectangular solid}"], If[ygmax < fc, LaTeX["\textcolor{#000099}{f(x)}\text{ is the function that}\cr\text{represents the greatest maximum volume for the associated rectangular solid}"], LaTeX["\textcolor{#000099}{f(x)}\text{ and }\textcolor{#006633}{g(x)}\text{ represent}\cr\text{identical maximum volumes for their associated rectangular solids.}"]]) → winner
LaTeX((((((("\text{The function value of the maximum point of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{" * fc) * "}\text{, and the function value of}\cr\text{the maximum point of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{") * ygmax) * "}\text{. Therefore, }") * winner) * ".")) → text17
LaTeX((("\text{The y-coordinate of the maximum point of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{" * ygmax) * "}.")) → text14
LaTeX((((("\text{The maximum point of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{(" * xgmax) * ", ") * ygmax) * ")}.")) → text15
LaTeX("\text{Determine the coordinates of the maximum point of }\textcolor{#006633}{g(x)}.") → text6
LaTeX((("\text{The x-coordinate of the maximum point of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{" * xgmax) * "}.")) → text11
LaTeX((("\text{Since }\textcolor{#006633}{a=" * ga) * "}\text{ is less than 0, }\textcolor{#006633}{g(x)}\text{ will have a maximum.}")) → text9
LaTeX("\text{To determine the y-coordinate of the maximum, substitute the value of }\textcolor{#006633}{x}\text{ into the function }\textcolor{#006633}{g(x)}.") → text12
TableText({{LaTeX[(("\textcolor{#006633}{g(x)=" * g) * "}")], LaTeX["\text{Given function}"]}, {LaTeX[(((((((((((("\textcolor{#006633}{g(" * xgmax) * ")=") * ga) * "(") * xgmax) * ")^2+(") * gb) * ")(") * xgmax) * ")+(") * gc) * ")}")], LaTeX[(("\text{Substitute }\textcolor{#006633}{" * xgmax) * "}\text{ for }\textcolor{#006633}{x}.")]}, {LaTeX[(((("\textcolor{#006633}{g(" * xgmax) * ")=") * ygmax) * "}")], LaTeX["\text{Simplify.}"]}}) → text13
LaTeX("\text{Determine which function models the greatest maximum volume of its associated rectangular solid.}") → text16
LaTeX("\text{Move the slider to solve the problem.}") → text0
LaTeX("\text{The vertex form for a quadratic function is }\textcolor{#000099}{f(x) = a(x – h)^2 + k}\text{, where the}\cr\text{coordinates of the maximum }\textcolor{#000099}{(a < 0)}\text{ or minimum }\textcolor{#000099}{(a > 0)}\text{ function value are }\textcolor{#000099}{(h, k)}.") → text4
LaTeX((((((((((((("\text{Since }\textcolor{#000099}{f(x) =" * f) * "}\text{ is already written in vertex form, it can be seen that for }\cr\textcolor{#000099}{f(x)}\text{, }\textcolor{#000099}{h=") * fb) * "}\text{, }\textcolor{#000099}{k=") * fc) * "}\text{, and }\textcolor{#000099}{a=") * fa) * "}\text{, which is less than 0; therefore, the maximum point of }\textcolor{#000099}{f(x)}\cr\text{is located at }\textcolor{#000099}{(") * fb) * ", ") * fc) * ")}.")) → text5
LaTeX((((("\textcolor{#000099}{f(x)=" * f) * "}\cr\textcolor{#006633}{g(x)=") * g) * "}")) → Problem
LaTeX((((((((("\text{The function }\textcolor{#006633}{g(x) = " * g) * "}\text{ is also a quadratic function, but it is written}\cr\text{in standard form, }\textcolor{#006633}{f(x)=ax^2 +bx+c}\text{, where }\textcolor{#006633}{a=") * ga) * "}\text{, }\textcolor{#006633}{b=") * gb) * "}\text{, and }\textcolor{#006633}{c=") * gc) * "}.")) → text7
LaTeX("\text{Recall that the x-coordinate of the maximum or minimum of a quadratic in this}\cr\text{form is }\textcolor{#006633}{x=-\frac{b}{2a}}\text{, which is a maximum if }\textcolor{#006633}{a<0}.") → text8
TableText({{LaTeX["\textcolor{#006633}{x=-\frac{b}{2a}}"], LaTeX["\text{Formula for the x-coordinate}"]}, {LaTeX[(((("\textcolor{#006633}{x=-\frac{(" * gb) * ")}{2(") * ga) * ")}}")], LaTeX[(((("\text{Substitute }\textcolor{#006633}{" * gb) * "}\text{ for }\textcolor{#006633}{b}\text{ and }\textcolor{#006633}{") * ga) * "}\text{ for }\textcolor{#006633}{a}.")]}, {LaTeX[(((("\textcolor{#006633}{x=-\frac{" * gb) * "}{") * twoa) * "}}")], LaTeX["\text{Simplify.}"]}, {LaTeX[(("\textcolor{#006633}{x=" * xgmax) * "}")], LaTeX["\text{Simplify.}"]}}) → text10

AI Drawing Prompt Reference

Draw the graphs of two quadratic functions, f(x) = –2(x – 5)² + 3 and g(x) = –4x² + 8x + 3, and find their maximum points. First, draw f(x) and locate its vertex at (5, 3). Then, draw g(x) and calculate its vertex x-coordinate as 2, and substitute to find the y-coordinate as 7. Finally, compare the maximum values of the two functions to determine which one represents the greater volume.

The prompt above was compiled from the original drawing commands and is for reference only.