Deriving the Maximum or Minimum of a Quadratic Function via the Discriminant Method

Overview

A concise visual derivation showing how the discriminant of a quadratic function determines whether the parabola opens upward or downward, and how the vertex gives the function’s maximum or minimum value. Viewers will see the algebraic steps linked to geometric features of the parabola.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in, then the general quadratic $f(x)=ax^2+bx+c$ appears centered.
2	Parabola Sketch	~5s	A coordinate grid is drawn; the parabola is plotted for a generic $a>0$ (upward) and then for $a<0$ (downward) to illustrate the two cases.
3	Completing the Square	~8s	The expression is transformed step‑by‑step into vertex form $a\bigl(x+\frac{b}{2a}\bigr)^2 - \frac{\Delta}{4a}$ where $\Delta = b^2-4ac$. Each algebraic manipulation is animated alongside a moving highlight on the graph showing the vertex.
4	Discriminant Insight	~5s	The term $-\frac{\Delta}{4a}$ is highlighted; a brief visual cue explains that the sign of $a$ determines whether the vertex is a minimum (if $a>0$) or maximum (if $a<0$).
5	Vertex as Extrema	~5s	The vertex point is marked; a small vertical line indicates the extremum value $f_{ext}= -\frac{\Delta}{4a}$. The line animates to the axis to show the extremum location.
6	Summary & Outro	~4s	A concise statement appears: "For $f(x)=ax^2+bx+c$, the extremum occurs at $x=-\frac{b}{2a}$ with value $-\frac{\Delta}{4a}$. The sign of $a$ decides max vs. min." Then everything fades out.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA (Title)            │
├──────────────────────┬──────────────────────┤
│                      │                      │
│   LEFT AREA (Graph)  │   RIGHT AREA (Algebra)│
│                      │                      │
├──────────────────────┴──────────────────────┤
│          BOTTOM AREA (Key Formula)          │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Animation title "Maximum/Minimum via Discriminant"	Fades in at start of each major segment
Left	Coordinate grid with the parabola, vertex point, and extremum line	Primary visual focus; updates as algebra progresses
Right	Step‑by‑step algebraic transformation to vertex form, highlighting $\Delta$	Synchronized with left‑area changes
Bottom	Final compact formula $x_{ext} = -\frac{b}{2a},\\; f_{ext}= -\frac{\Delta}{4a}$ and a brief note on max/min condition	Appears in the last phase and remains until fade‑out

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Notes

• Keep total runtime under 30 seconds; the durations above sum to ~30 seconds.
• Use smooth fade and transform transitions to maintain visual continuity.
• No textual narration beyond essential mathematical symbols; rely on visual cues and brief on‑screen statements.
• The scene must be implemented as a single Manim Scene class.
• Emphasize the discriminant $\Delta = b^2-4ac$ as the bridge between algebra and geometry.