Pencil Sketch of Linear Function with Slope and Intercept

Scene Overview

• Scene Class: Single Manim Scene (e.g., PencilGraphScene).
• Purpose: Visually illustrate the linear function $p(x)=2x-1$ being drawn as if by a hand‑held pencil, emphasizing the slope and intercept.
• Duration: Approximately 12 seconds total.

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1. Animation Description & Flow

1. Setup (0 s – 2 s)
   • Fade in a Cartesian coordinate plane with axes ranging from $x=-3$ to $x=3$ and $y=-4$ to $y=4$.
   • Axes are drawn in a light gray color; tick marks and numeric labels appear instantly.
2. Pencil Introduction (2 s – 3 s)
   • A thin, semi‑transparent pencil tip (a small brown ellipse with a subtle shadow) slides in from the left edge of the screen to the origin $(0,0)$. This motion lasts 1 second.
3. Drawing the Line (3 s – 9 s)
   • The pencil tip follows the graph of $p(x)=2x-1$ from $x=-2$ (point $(-2,-5)$ clipped to the visible frame) to $x=2$ (point $(2,3)$).
   • As the tip moves, a dark gray line is progressively revealed behind it, mimicking a pencil stroke.
   • The line width is set to mimic a real pencil (≈3 pt) with a slight texture effect (optional shading).
4. Highlight Key Points (9 s – 11 s)
   • Small, semi‑transparent circles appear at the intercepts:
     • Y‑intercept at $(0,-1)$ – a blue dot that fades in over 0.5 s.
     • X‑intercept at $(0.5,0)$ – a red dot that fades in over 0.5 s.
   • Tiny, hand‑drawn style arrows (thin lines with arrowheads) label the slope "2" and the intercept "-1" near the line, each appearing with a quick sketch animation (0.5 s each).
5. Conclusion (11 s – 12 s)
   • The pencil tip lifts off the screen, moving upward and fading out.
   • The completed graph remains on screen for a brief pause, then all elements fade out together.

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2. Mathematical Elements & Formulas

• Function: $$p(x) = 2x - 1$$
• Key Points:
  • Y‑intercept: $(0, -1)$
  • X‑intercept: $(\tfrac{1}{2}, 0)$
• Slope Indicator: The label "2" placed parallel to the line segment between $(-1, -3)$ and $(1, 1)$.

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3. Visual Elements

Element	Description	Color / Style
Axes	Light gray, thin lines; tick marks visible	#CCCCCC
Pencil tip	Small brown ellipse with subtle shadow, semi‑transparent	#8B4513
Drawn line	Dark gray, textured to resemble graphite	#333333
Y‑intercept dot	Blue, radius ≈0.08 units, semi‑transparent	#1E90FF
X‑intercept dot	Red, radius ≈0.08 units, semi‑transparent	#FF4500
Slope label "2"	Hand‑sketched style number, black	#000000
Intercept label "-1"	Hand‑sketched style number, black	#000000

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4. Timing & Transitions

• Fade‑in / Fade‑out: 0.5 s each for axes and final exit.
• Pencil movement: Linear easing for the tip travel; line reveal follows the tip exactly.
• Dot appearances: Opacity transition from 0 % to 100 % over 0.5 s.
• Arrow/label sketches: Quick “draw‑on‑air” effect (stroke appears from start to end) over 0.5 s.
• Overall pacing: Keep a steady rhythm; no pause longer than 0.5 s between steps.

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5. Camera Angles & Perspective

• Static camera: Fixed orthographic view centered on the origin; no zoom or pan.
• Optional subtle zoom: At 3 s, a slight (≈5 %) zoom‑in to emphasize the pencil tip, then return to original framing by 9 s.

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6. Additional Details

• Background: Plain white canvas to emulate paper.
• Texture: Optional faint paper grain overlay (very low opacity) to enhance the “pencil on paper” feel.
• No text blocks: All information conveyed visually; only the small hand‑drawn numeric labels are used.
• Scene length: 12 seconds, well within the 30‑second limit.