Find vertex, axis of symmetry, focus, directrix, and intercepts. Supports standard and vertex form.
y = ax² + bx + c
y = a(x − h)² + k
What Is a Parabola?
A parabola is a U-shaped curve that is one of the four conic sections — the curves formed by the intersection of a plane with a cone. Mathematically, a parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas appear everywhere in the physical world: the path of a thrown ball, the reflective surface of satellite dishes, the shape of suspension bridge cables, and the mirror in a car headlight all follow parabolic curves.
The simplest parabola is y = x², which opens upward with its vertex at the origin. Changing the coefficient a controls how wide or narrow the parabola is, and whether it opens upward (a > 0) or downward (a < 0).
Standard Form vs Vertex Form
Property
Standard Form: y = ax²+bx+c
Vertex Form: y = a(x−h)²+k
Vertex
(−b/2a, f(−b/2a))
(h, k) — directly visible
Y-intercept
(0, c) — directly visible
(0, ah²+k)
Best for
Finding intercepts quickly
Reading vertex and transformations
How to Find the Vertex
From standard form y = ax² + bx + c, the vertex coordinates are found by:
Step 1: h = −b / (2a)
Step 2: k = a·h² + b·h + c (or use k = c − b²/(4a))
Vertex = (h, k)
From vertex form y = a(x−h)²+k, the vertex (h, k) is read directly — no calculation needed.
Axis of Symmetry Explained
The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two perfectly mirrored halves. Its equation is simply x = h. If you fold the parabola along this line, both sides match exactly. This symmetry is why parabolic antennas focus signals so precisely — every incoming ray parallel to the axis reflects through the same focal point.
Focus and Directrix
The focus is a point inside the parabola at (h, k + 1/(4a)). The directrix is a horizontal line y = k − 1/(4a) outside the parabola. The defining property of a parabola is that every point P on the curve satisfies: distance from P to focus = distance from P to directrix. This property is why parabolic mirrors create perfect reflectors — light from the focus exits parallel to the axis.
Real-World Applications
Parabolas are one of the most practically useful geometric shapes in engineering and physics:
Satellite Dishes: The parabolic reflector focuses incoming signals onto the receiver at the focus point, maximizing signal strength.
Projectile Motion: Under constant gravity, the trajectory of any thrown object follows a parabolic arc (ignoring air resistance).
Bridge Cables: Suspension bridge cables hang in a catenary curve under self-weight, but under uniform road load they approximate a parabola.
Car Headlights: A parabolic reflector with a bulb at the focus converts point-source light into a directed parallel beam.
Solar Collectors: Parabolic troughs concentrate sunlight onto a tube at the focus, heating fluid for solar power generation.
Architecture: Parabolic arches in structures distribute stress efficiently, used in bridges, domes, and stadium roofs.
Frequently Asked Questions
The vertex is the turning point of a parabola — the minimum point when the parabola opens upward (a > 0) and the maximum point when it opens downward (a < 0). It lies on the axis of symmetry and is the point (h, k).
Calculate h = −b/(2a), then substitute into the equation: k = a·h² + b·h + c. Alternatively, use k = c − b²/(4a). The vertex is (h, k).
The axis of symmetry is the vertical line x = h that passes through the vertex. It divides the parabola into two mirror-image halves. For y = ax²+bx+c, this line is x = −b/(2a).
The focus is a fixed point inside the parabola at coordinates (h, k + 1/(4a)). Every point on the parabola is equidistant from the focus and the directrix. The focus is used in optical and antenna design.
The directrix is a horizontal line y = k − 1/(4a) that lies outside the parabola, on the opposite side from the focus. Together with the focus, it provides the geometric definition of a parabola.
Complete the square: y = ax²+bx+c → factor out a from first two terms → y = a(x² + (b/a)x) + c → add and subtract (b/(2a))² inside → y = a(x + b/(2a))² + (c − b²/(4a)). So h = −b/(2a) and k = c − b²/(4a).
Yes. When the discriminant Δ = b²−4ac is negative, the quadratic has no real roots, meaning the parabola never touches the x-axis. It floats entirely above (a > 0) or below (a < 0) the x-axis.