Quadratic Functions (Applied)
🎯 Learning Goals
- Solve real-world problems using quadratic functions
- Understand the relationship between moving objects and parabolas
💡 Why Learn This?
When you throw a basketball, drop an object, or design a bridge, the trajectory follows a quadratic curve. Knowing how to calculate the peak height or the landing time is essential in physics and engineering.
Motion and Trajectory
The path of a projectile (like a thrown ball) can be modeled by a quadratic function y = -ax² + bx + c. The negative 'a' means the parabola opens downwards, perfectly matching gravity pulling the object back to earth.
The Thrown Ball
- ・ If height y = -5x² + 20x (where x is time in seconds).
- ・ The peak height occurs when x = 2 seconds, reaching a height of 20 meters.
⚠️ Common Pitfalls
A common mistake is forgetting that 'x' usually represents time in physics problems, not horizontal distance. When finding when the object hits the ground, you are looking for the x-intercepts (where y = 0).
Projectile Simulator
Adjust the initial velocity to see how high and how far the ball travels.
y = -0.5x² + 5x
📝 Summary & Recap
- Quadratic functions perfectly model gravity and projectile motion.
- The vertex of the parabola tells you the maximum height and when it happens. The x-intercepts tell you when the object starts and lands.
Quick Drill
Test your understanding of applied quadratic functions!
🔍 Deep Dive (Optional)
The Gateway Arch in St. Louis looks like a parabola, but it's actually an 'inverted catenary' curve! While parabolas model gravity on moving objects, catenary curves model the shape of a hanging chain. They look similar but have different mathematical formulas.