• Create and sketch graphs of curves given their parametric equations
• Convert parametric equations into a regular y = f(x) form by eliminating the parameter
• Recognize and describe the curve called a cycloid

Designing Motion Paths

You’re working with a local community center to design an interactive art installation that will be accessible to families from all backgrounds. The installation features moving lights that trace beautiful paths in a darkened room, creating patterns that visitors can observe and enjoy together.

The installation has several components:

1. Linear Light Paths: Some lights move along straight lines
2. Circular and Elliptical Orbits: Other lights follow curved paths
3. Special Pattern Wheels: A mechanical component creates cycloid patterns using rolling wheels

Your job is to analyze the mathematical equations that control these light movements and ensure the installation works as designed.