Look around. Sunflowers spiral logarithmically. River cross-sections are parabolas. Hanging chains form catenaries. Functions aren't abstract — they're how the world looks.
Drag the sliders
y = a·pow(x, 2)y = A·sin(x)meet at x ≈ 2.09
Same curve, many disguises
A water arc, a satellite dish cross-section and a thrown ball all trace parabolas — change a and you stretch or flip every one of them. Ripples, AC current and a plucked string all ride sine waves — change A and you make them louder or quieter.
Your turn
A fountain's water leaves the nozzle, rises, and falls back. Which family of curve is the jet of water?
A hanging chain looks like a parabola but isn't — it's a catenary (cosh x). Galileo guessed parabola; it took 70 more years to prove him wrong.
Spot these in the wild
- Parabola — water from a hose, a thrown ball's path
- Sine wave — ripples on water, vibrating string
- Exponential — bacterial growth, compound interest
- Logarithmic spiral — nautilus shell, hurricane
- Catenary — hanging chain, suspension bridge cable