Parabolas – The Shape Behind Every Quadratic
A parabola is the smooth, U-shaped curve produced by graphing any quadratic equation y = ax² + bx + c. Whether the curve opens upward (like a smiling U) or downward (like a frown) depends entirely on the sign of a: positive a opens upward, negative a opens downward. Every parabola is perfectly symmetric – its left and right halves are exact mirror images of each other.
The name “parabola” comes from the ancient Greek mathematician Apollonius of Perga, who studied these curves around 200 BCE as one of several “conic sections” – the shapes formed when a flat plane slices through a cone. Nearly 2,000 years later, Galileo proved that a thrown object's path through the air also traces a parabola, connecting this ancient geometric curiosity to real-world motion.
Recognising a Parabola
An equation with an x² term graphs as a parabola. If a > 0 it opens upward; if a < 0 it opens downward.
The coefficient of x² is −3, which is negative: it opens downward.
Real-Life Application
- Satellite dishes: parabolic dishes focus signals to a single point.
- Projectile motion: a thrown ball or fired rocket follows a parabolic path.
- Suspension bridges: the main support cables often form a parabolic curve.
Key Takeaways
- A parabola is the U-shaped graph of any quadratic equation.
- It opens upward if a > 0, and downward if a < 0.
- The shape was studied as a conic section by Apollonius around 200 BCE.
Practice: Parabolas
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