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What Makes a Function Even? A Dynamic Visual Definition

What Makes a Function Even? A Dynamic Visual Definition
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2016-11-24 11:36
FunctionTransformationGeometric ConstructionSlider AnimationGraphing
① This interactive GeoGebra example dynamically illustrates the defining property of an even function: its graph is symmetric about the y-axis, and the function can be modified by dragging points to explore how symmetry holds. ② Core mathematical concepts include the formal definition of even functions (f(-x)=f(x)), symmetry, function transformation, and graphical analysis. ③ Construction highlights: the applet uses commands such as Point, PointIn, and FitPoly to create draggable points and fit a polynomial function; Mirror generates the reflection across the y-axis; Intersect, Dilate, Segment, Rotate, Vector, and Translate build auxiliary geometric elements and dynamic annotations; RotateText rotates labels; buttons (button1, button2) control interactivity; and text objects (text1-text4) display the definition and explanations. ④ This resource is suitable for high school or introductory college mathematics, ideal for teaching function symmetry, exploring even-function properties, and serving as a hands-on demonstration for teachers or self-study for students.

GeoGebra Commands & Parameters

  1. Point(x = 0) → A
  2. PointIn(a) → B
  3. PointIn(b) → C
  4. PointIn(c) → D
  5. PointIn(d) → E
  6. FitPoly(list, 4) → f
  7. Mirror(g, x = 0) → g_1
  8. Point(g) → F
  9. Intersect(h, y = 0) → G
  10. Mirror(G, x = 0) → G'
  11. Mirror(F, x = 0) → F'
  12. Dilate(H, If[0 < e ≤ 1, 1 - e, 0], G) → H'
  13. Segment(H, H') → i
  14. Dilate(G, If[1 < e ≤ 2, 2 - e, 0], F) → G'_1
  15. Segment(G, G'_1) → j
  16. Rotate(H, If[2 < e ≤ 3, ((-(e - 2)) * pi), (-pi)], H) → H'_1
  17. Rotate(H', If[2 < e ≤ 3, ((-(e - 2)) * pi), (-pi)], H) → H''
  18. Segment(H'_1, H'') → i'
  19. Dilate(G', If[3 < e ≤ 4, 4 - e, 0], F') → G''
  20. Segment(G', G'') → k
  21. Dilate(F, If[4 < e ≤ 5, 5 - e, 0], F') → F'_1
  22. Vector(F, F'_1) → u
  23. Translate(G, u) → G'_2
  24. Translate(G'_1, u) → I
  25. Segment(G'_2, I) → j'
  26. RotateText(text1, pi / 2) → text2
  27. RotateText(text3, pi / 2) → text4
  28. RotateText(text3, pi / 2) → text5

AI Drawing Prompt Reference

Draw a graph of an even function symmetric about the y-axis, generated by fitting a polynomial through four draggable points. Place a point on the curve and its reflection across the y-axis, connecting them with a segment to show equal distances from the axis. Add a slider that controls a sequence of dynamic transformations: first shrink the segment to a point, then rotate one point around the other, then translate the reflected point horizontally, and finally translate the whole figure, demonstrating that symmetry persists. Include rotated text labels stating the definition of an even function.
The prompt above was compiled from the original drawing commands and is for reference only.