Parabola Saga (v2)

“Whether ‘down-y’ or an ‘up-y’, I can classify that puppy…
But I sure need a vertex!”

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🧮 What This App Is About

The quadratic function y = ax² + bx + c produces a parabola. Fix a and c, then vary b across a range and graph all the resulting parabolas at once. The pattern that emerges reveals exactly what b controls: the horizontal position of the vertex (via h = −b / (2a)), and the slope of the curve at the y-intercept.

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📈 Phase 2: Single Parabola

Phase 2 adds the Parabola Design card. Use the three sliders to set a, b, and c for y = ax² + bx + c. The curve draws immediately. The vertex (h, k) is computed from h = −b / (2a) and marked with a labeled dot on the canvas. The direction indicator (“Opens UP — minimum”) connects the a sign to Verse 2 of the “If I Only Had a Vertex” parody.

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🎮 Controls

• Grid size — sets the axis range from −N to +N.
• Grid step — spacing between grid lines and tick labels.
• a slider — controls opening direction and width; positive = opens up (minimum), negative = opens down (maximum).
• b slider — shifts the axis of symmetry: h = −b / (2a).
• c slider — vertical shift; (0, c) is always on the parabola (y-intercept).
• Vertex display — shows (h, k) and opening direction; updates live.
• Show vertex label — when checked, the V(h, k) coordinate label appears on the graph next to the vertex dot. The dot is always visible; only the text is toggled.
• Trail — drag to 1–85% and then move any slider slowly to leave ghost parabolas on the canvas. Drag the b slider from end to end to preview the b-coefficient family. Reset clears the trail.
• Press G to toggle the grid  |  E to toggle axis labels.
• ↺ Reset — restores grid to ±10, step 1 and coefficients to a=1, b=0, c=0.
• 💾 Save Image — saves a PNG snapshot of the canvas.

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🧑‍💻 Developer Notes

Developer

klp

Date

09/09/2026 (Phase 2: 09/10/2026)

Commentary

Phase 2 built on SketchWaveJS 2026. Uses SWGrid for the coordinate system and a 600-segment p5.js polyline for the parabola curve. Phase 3 will plot a family of parabolas for a range of b values with fixed a and c.