Logo Design Using Mathematical Functions
📌 The video demonstrates how to design the McDonald's 'M' logo using mathematical functions, specifically quadratic functions (parabolas) and constant functions.
📐 The logo is formed by two parabolas intersecting with a horizontal line, which is simulated using constant functions ($y = c$).
⚙️ The process relies on transforming the basic quadratic function ($y = x^2$) through vertical shifts (adding/subtracting constants to $y$), horizontal shifts (replacing $x$ with $x \pm c$), and reflections (multiplying by -1).

Quadratic Function Transformations
📏 Basic transformations include shifting the function vertically by adding a constant to the entire function, or horizontally by manipulating the independent variable $x$.
↔️ Horizontal shifts work in reverse: subtracting 5 units shifts the graph 5 units to the right, and adding 5 units shifts it to the left.
🌐 To create a specific segment of the parabola, the domain must be limited using the `function(f, start, end)` command in GeoGebra, defining where the curve begins and ends.

Logo Construction and Finalization
✅ The McDonald's 'M' was constructed using four parabolic segments and three segments of constant functions for the base line.
🎨 After defining the functions in GeoGebra, the grid lines were removed, and the figure was colored yellow using the fill tool in an external program like Paint to complete the logo visualization.
💡 This practical exercise confirms that mathematics is used to model and parameterize real-world designs like famous logos.

Key Points & Insights
➡️ The core requirement for drawing the 'M' involves manipulating the basic function $y = x^2$ using translations, scaling, and domain restriction.
➡️ Utilize free software like GeoGebra for visualizing and testing these complex function transformations interactively.
➡️ Recognize that famous logos are often created via mathematical modeling and computer generation, rather than being entirely hand-drawn.

📸 Video summarized with SummaryTube.com on Jan 15, 2026, 08:58 UTC