Quadratic Functions (Basics)
🎯 Learning Goals
- Understand the shape and meaning of the parabola y = ax²
- Learn how the coefficient 'a' affects the width and direction of the graph
💡 Why Learn This?
Quadratic functions describe motions like a thrown ball's trajectory (parabola), car braking distances, and satellite dish shapes. They are fundamental in physics, engineering, and optimizing curves.
The Power of Squared: y = ax²
Unlike linear functions that grow at a steady rate, quadratic functions have a variable 'x' that is squared. This means as 'x' grows, 'y' grows much faster, creating a curved U-shape called a parabola.
Properties of the Parabola
- ・ It is symmetrical around the y-axis.
- ・ The lowest (or highest) point is called the vertex (for y = ax², it is at the origin 0,0).
⚠️ Common Pitfalls
A common mistake is forgetting that squaring a negative number results in a positive number. For example, (-3)² is 9, not -9. This is why the graph curves upwards on both the left and right sides (when a > 0).
Parabola Explorer
Adjust the value of 'a' to see how the parabola y = ax² changes its shape and direction.
y = 1x²
📝 Summary & Recap
- The graph of y = ax² is a parabola passing through the origin.
- If 'a' is positive, it opens upwards. If 'a' is negative, it opens downwards. A larger absolute value of 'a' makes the parabola narrower.
Quick Drill
Test your understanding of parabolas!
🔍 Deep Dive (Optional)
Telescopes and satellite dishes use parabolic mirrors. Because of a mathematical property of the parabola, all incoming parallel rays of light or radio waves bounce off the curve and hit a single 'focal point', creating a strong signal!