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Equation of a Line Intersecting a Parabola

Equation of a Line Intersecting a Parabola
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2019-01-22 01:49
FunctionCoordinate GeometryDerivativeLimitGraphing
This example demonstrates the intersection of a line with a parabola, involving core math concepts like coordinate geometry, function graphing, derivatives, and limits. It is constructed using GeoGebra commands/objects such as LaTeX, Textfield, Intersect, UnicodeToLetter, and Polygon, making it suitable for high school and above mathematics learners and educators for teaching demonstrations and problem-solving.

GeoGebra Commands & Parameters

  1. LaTeX("\textbf{Instructions: }The graph displays the quadratic relation \(y=x(x-4)\) and a line with a slope of \(-2\).\\Changing the slider changes the \(y\)-intercept of the line. Use the graph to answer the questions below.") → INSTRUCTIONS
  2. LaTeX("\textbf{Question: }What is the equation of a line that intersects a given parabola a set number of times?") → QUESTION
  3. LaTeX((((("\text{" * QUESTION) * "\\") * INSTRUCTIONS) * "}")) → TEXT
  4. Textfield(userAnswerA) → ansBoxA
  5. Textfield(userAnswerC) → ansBoxC
  6. Textfield(userAnswerB) → ansBoxB
  7. Intersect(quadratic, linear) → A, B
  8. LaTeX("\text{a) What is the equation of a line with slope \(-2\)\\that intersects \(y=x(x-4)\) twice?}") → question1
  9. LaTeX("\text{b) What is the equation of a line with slope \(-2\)\\that does not intersect \(y=x(x-4)\) at all?}") → question2
  10. LaTeX("\text{c) What is the equation of the line with slope \(-2\)\\that intersects \(y=x(x-4)\) exactly once?}") → question3
  11. UnicodeToLetter(10003) → cmark
  12. UnicodeToLetter(10007) → xmark
  13. LaTeX("\(y=\)") → eqLabelC
  14. LaTeX("\(y=\)") → eqLabelB
  15. LaTeX("\(y=\)") → eqLabelA
  16. Polygon({(-6.75, -10.75), (-6.75, 10.75), (12.75, 10.75), (12.75, -10.75)}) → viewBox
  17. LaTeX(("y\text{-intercept}\!=\!" * b)) → interceptLabel

AI Drawing Prompt Reference

Draw a coordinate plane with an upward-opening parabola given by y=x(x-4) and a line with slope -2 whose y-intercept can be adjusted via a slider. Mark the two intersection points of the line and parabola when they cross. Above the graph, display text with three questions: find the equation of a line with slope -2 that intersects the parabola twice, does not intersect it, and intersects exactly once. Below the graph, place three input boxes labeled with 'y=' for answers, and show the current y-intercept value of the line on the side.
The prompt above was compiled from the original drawing commands and is for reference only.