The Parabola
Explore the most elegant curve in algebra: understand, visualize and experiment
Start exploring โThe Parabola in the Real World
Telescopes & Antennas
Parabolic mirrors focus light (or radio waves) at the focal point, exploiting the reflective property of the parabola.
Projectile Trajectories
In the absence of air resistance, a thrown object follows a parabolic trajectory determined by gravity.
Suspension Bridges
The main cables of suspension bridges, under uniform load, take the shape of a parabola.
Solar Panels
Parabolic solar concentrators reflect sunlight toward the receiver tube positioned at the focus.
Architectural Arches
Many concrete or steel arches follow parabolic profiles to optimally distribute structural forces.
Headlights & Lighthouses
The reflector of a lighthouse or projector is parabolic: a source at the focus produces a parallel beam.
Simulate a ballistic trajectory
Launch a projectile and observe the shape it traces through space. Change speed and angle. Do you notice something familiar in the curve?
The secret of the parabolic focus
Watch what happens to light rays that hit a parabola-shaped mirror. They all converge to the exact same point.
๐ก All rays meet at the same point. That point is called the focus.
Interactive Graph
Move the sliders to change the coefficients and see how the parabola changes in real time.
Key Properties
The Vertex
The vertex is the point where the parabola changes direction. It is the minimum if a > 0, the maximum if a < 0.
where ฮ = bยฒ โ 4ac is the discriminant.
Vertex form: y = a(x โ h)ยฒ + k with V = (h, k)
What is a Parabola?
Geometric definition
A parabola is the locus of points equidistant from a fixed point called the focus F and a fixed line called the directrix d.
|PF| = |Pd|
Standard equation
The standard form of a parabola with a vertical axis of symmetry is:
y = axยฒ + bx + c
where a โ 0 determines opening and width.
Key elements
- โธVertex V โ minimum/maximum point
- โธFocus F โ interior point of the curve
- โธAxis of symmetry โ x = โb/2a
- โธDirectrix โ line outside the curve
Test your knowledge!
Qual รจ la definizione geometrica della parabola?