Parabola Exploration

Overview

A brief visual exploration of the geometric and algebraic properties of a parabola, illustrating its definition as the set of points equidistant from a focus and a directrix, its vertex form, and the effect of changing parameters. The key takeaway is how the parameters $a$, $h$, and $k$ shape the parabola.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in, then a coordinate plane appears with axes labeled.
2	Definition	~6s	Show focus point and directrix line; animate a point moving on the curve, highlighting equal distances to focus and directrix.
3	Vertex Form	~8s	Introduce the equation $y = a(x-h)^2 + k$; animate sliders for $a$, $h$, $k$ that smoothly transform the parabola while updating a small overlay of the current equation.
4	Standard Form Conversion	~5s	Transition to the expanded form $y = ax^2 + bx + c$; illustrate how $b$ and $c$ relate to $h$ and $k$ by morphing the equation text.
5	Applications (Optional)	~4s	Briefly show a projectile motion curve as a real‑world example, overlaying a small arrow indicating direction of motion.
6	Outro	~3s	Fade out to a summary caption: "Parabolas: symmetry, focus‑directrix definition, and parameter control."

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│                       │               │
│        LEFT AREA      │   RIGHT AREA  │
│   (Main coordinate   │   (Equation   │
│    plane & curve)    │   overlay)   │
│                       │               │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
│   Caption / brief description text    │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Parabola Exploration"	Fade‑in at start of Intro
Left	Coordinate axes, focus point, directrix, and the animated parabola curve	Primary visual focus throughout
Right	Small overlay showing the current equation (either vertex or standard form) with highlighted parameters	Updates each phase, fades in/out as needed
Bottom	Caption or brief explanatory text for each phase (e.g., "Focus‑Directrix Definition", "Changing $a$ stretches the curve")	Small font, appears with each phase transition

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Notes

• Keep total runtime under 30 seconds; durations above are approximate and can be trimmed if needed.
• Use smooth easing for parameter sliders to clearly demonstrate continuous transformation.
• No large blocks of explanatory text; rely on visual cues and minimal captions.
• The scene must be implemented as a single Manim Scene class.
• All text elements should appear only when they add essential information; otherwise, the animation is purely visual.