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Graphs of Quadratic Equations: Exploring Parabola Vertices and Intercepts

Graphs of Quadratic Equations: Exploring Parabola Vertices and Intercepts
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2015-09-29 09:16
FunctionCoordinate GeometryDerivativeOptimizationGraphing
This GeoGebra example demonstrates plotting quadratic equations y=ax²+bx+c (parabolas). Users can adjust parameters a, b, and c via three input boxes to observe real-time changes in the parabola's shape and position. Key mathematical concepts covered include: (1) Characteristics of quadratic function graphs—opening direction determined by the sign of a, vertex formula at (-b/2a, c-b²/4a); (2) Finding intercepts with coordinate axes (set y=0 for x-intercepts, set x=0 for y-intercept); (3) Using the Extremum command to automatically locate the parabola's vertex. Construction details: Textfield commands create parameter input boxes, Extremum finds vertex coordinates, Intersect plots axis intersections, and Point marks key locations. Suitable for middle to high school mathematics teaching, ideal for exploring quadratic function properties and analyzing how parameters affect graph transformations.

GeoGebra Commands & Parameters

  1. Textfield(a) → inputBox1
  2. Textfield(b) → inputBox2
  3. Textfield(c) → inputBox3
  4. Extremum(a * x^(2) + b * x + c) → A
  5. Intersect(d, xAxis, 1) → B
  6. Intersect(d, xAxis, 2) → C
  7. Intersect(d, yAxis, 1) → D
  8. Line(A, yAxis) → e
  9. Intersect(e, xAxis) → E
  10. Point(d) → F

AI Drawing Prompt Reference

Draw a graph of a quadratic function y=ax²+bx+c, where a, b, and c are adjustable parameters. Include three input boxes for these parameters. The parabola changes shape and position in real time as parameters vary, with the opening direction determined by the sign of a. Mark the vertex of the parabola and display its coordinates. Also mark the two x-intercepts (if they exist) and the y-intercept. Draw a vertical line from the vertex to the y-axis, and mark its intersection with the x-axis. Place an arbitrary point on the parabola and label it. The graph should clearly show the vertex, intercepts, and how parameters affect the curve.
The prompt above was compiled from the original drawing commands and is for reference only.