Translation of a Parabola

Overview

An animation that visualizes the horizontal translation of the parabola $y = x^{2}$ two units to the right, resulting in the function $y = (x-2)^{2}$. The key visual is the replacement of the variable $x$ by $x-2$ and the corresponding smooth physical movement of the graph across the screen.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in at the top. The original graph $y = x^{2}$ is drawn in the left area, axes appear, and a point $(0,0)$ is highlighted.
2	Substitution	~4s	The expression “replace $x$ with $x-2$” appears briefly in the bottom area. An arrow animates from the variable $x$ in the formula to a new placeholder $x-2$. Simultaneously, the entire parabola in the left area translates smoothly rightward, moving continuously across the screen for the full duration of this phase.
3	Translation Complete	~3s	The translated graph $y = (x-2)^{2}$ is now fully positioned in the right area, overlapping the original for comparison. The original graph fades to a lighter shade. The bottom area now shows the final formula $y = (x-2)^{2}$.
4	Outro	~2s	A brief caption “Horizontal shift of 2 units to the right” appears at the bottom, then all elements fade out.

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Layout

┌───────────────────────────────────────┐
│                TOP AREA                │
├───────────────────────┬───────────────┤
│                       │               │
│        LEFT AREA      │   RIGHT AREA  │
│  (Original graph)     │ (Shifted graph)│
│                       │               │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title: “Horizontal Translation of $x^{2}$”	Fades in at start of Intro, stays visible until Outro
Left	Axes and the graph of $y = x^{2}$ (solid blue) with a highlighted origin point	Primary focus during Intro and Substitution; graph moves smoothly rightward during Substitution
Right	Axes (same scale) and the graph of $y = (x-2)^{2}$ (solid red)	Appears during Translation Complete, original graph remains faint for comparison
Bottom	Formula manipulation text (e.g., “replace $x$ with $x-2$”) and final caption	Appears in Substitution and Outro phases

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Notes

• Use the same coordinate limits for both left and right axes (e.g., $x \in [-4,4]$, $y \in [0,16]$) to make the shift evident.
• The original parabola should fade to a light gray (≈30 % opacity) once the translated graph is fully displayed.
• Arrow animation for the substitution should be smooth and point from the “x” in the formula to the newly created “x‑2” term.
• During the Substitution phase, animate the parabola with a continuous translation (e.g., graph.animate.shift(RIGHT*2)) so the movement is physically observable rather than an instantaneous jump.
• Keep total runtime under 12 seconds to maintain brevity while fully conveying the translation concept.
• The entire animation fits within a single Manim Scene class.