2
-1
2
-0.75
-1.25
2
4
3
2
-1
2
-0.75
-1.25
2
4
3
The parabola is one of the four conic sections, formed by the intersection of a right circular cone with a plane parallel to one of its generating lines. Defined as the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix), the parabola has a unique reflective property that makes it indispensable in optics, telecommunications, and engineering. The Parabola Calculator analyzes the quadratic function y = ax² + bx + c and extracts its key geometric properties.
Every quadratic function y = ax² + bx + c (with a ≠ 0) defines a parabola. The coefficient a controls the direction and width of the parabola: positive a opens upward, negative a opens downward, and larger |a| produces a narrower curve. The vertex—the turning point of the parabola—occurs at h = −b/(2a) and k = ah² + bh + c, giving the vertex form y = a(x − h)² + k.
The focus of the parabola lies on the axis of symmetry at a distance p = 1/(4a) above the vertex (for upward-opening parabolas) or below it (for downward-opening ones). The focus coordinates are (h, k + 1/(4a)). The directrix is the horizontal line y = k − 1/(4a), positioned at the same distance from the vertex as the focus but on the opposite side. Every point on the parabola is equidistant from the focus and the directrix—this is the defining property.
The reflective property of the parabola states that any ray parallel to the axis of symmetry reflects off the parabolic surface and passes through the focus. Conversely, any ray from the focus reflects off the parabola and travels parallel to the axis. This property is the principle behind satellite dishes, radio telescopes, flashlight reflectors, and solar concentrators. The 305-meter Arecibo radio telescope (operational until 2020) and the 500-meter FAST telescope in China both use parabolic reflectors to focus incoming radio waves onto a receiver at the focus.
In physics, the trajectory of a projectile under uniform gravity (neglecting air resistance) is a parabola. Galileo Galilei first established this in his Dialogues Concerning Two New Sciences (1638). The vertex of the trajectory corresponds to the maximum height, and the axis of symmetry passes through this point. The parabolic shape also appears in suspension bridge cables (under uniform load), the cross-section of reflector antennas, and the shape of water flowing from a horizontal spout.
The discriminant Δ = b² − 4ac determines the nature of the parabola's x-intercepts (roots): when Δ > 0, the parabola crosses the x-axis at two points; when Δ = 0, it touches the x-axis at exactly one point (the vertex); when Δ < 0, the parabola does not intersect the x-axis. The y-intercept is simply c, the value of the function at x = 0.
This calculator is essential for students studying quadratic functions, engineers designing parabolic reflectors, physicists analyzing projectile motion, and anyone working with second-degree polynomial curves.
Given coefficients a, b, c of the quadratic y = ax² + bx + c, the calculator computes:
$$h = -\frac{b}{2a}, \quad k = ah^2 + bh + c$$
$$\text{Vertex} = (h, k)$$
$$\text{Focus} = \left(h,\; k + \frac{1}{4a}\right)$$
$$\text{Directrix:}\; y = k - \frac{1}{4a}$$
$$\text{Axis of Symmetry:}\; x = h$$
$$\Delta = b^2 - 4ac$$
The vertex form of the parabola is y = a(x − h)² + k. The parameter p = 1/(4a) is the signed distance from the vertex to the focus. When a > 0, the parabola opens upward and the focus is above the vertex; when a < 0, it opens downward and the focus is below.
The vertex (h, k) is the minimum point (if a > 0) or maximum point (if a < 0) of the parabola. The focus is the point where reflected parallel rays converge. The directrix is the reference line such that every point on the parabola is equidistant from the focus and directrix. The axis of symmetry is the vertical line through the vertex dividing the parabola into mirror-image halves. The discriminant tells you how many x-intercepts exist (2 if positive, 1 if zero, 0 if negative). The y-intercept is where the parabola crosses the y-axis.
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Vertex at (2, −1), focus at (2, −0.75), opens upward. Discriminant = 4 > 0, so two x-intercepts (at x = 1 and x = 3).
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Opens downward (a < 0), vertex at (2, 3) is the maximum. Focus at (2, 2.875) is below the vertex. Two x-intercepts since Δ = 24 > 0.
In y = ax² + bx + c: a controls the direction (up if positive, down if negative) and width (larger |a| = narrower); b affects the horizontal position of the vertex; c is the y-intercept (where the parabola crosses the y-axis). Together, they fully determine the parabola's shape and position.
The focus is where parallel rays reflecting off the parabolic surface converge. In satellite dishes, the receiver is placed at the focus to collect signals. In flashlights and headlights, the bulb is placed at the focus so light reflects outward in a parallel beam. Solar concentrators focus sunlight to the focal point to generate heat or electricity.
The discriminant Δ = b² − 4ac determines the number of real roots (x-intercepts): Δ > 0 means two distinct real roots, Δ = 0 means one repeated root (the vertex touches the x-axis), and Δ < 0 means no real roots (the parabola doesn't cross the x-axis). The roots, when they exist, are x = (−b ± √Δ) / (2a).
The vertex form is y = a(x − h)² + k, where h = −b/(2a) and k is the y-value at the vertex. This form directly reveals the vertex coordinates and the direction/width of the parabola. The calculator provides h and k, so you can write the vertex form immediately.
Yes. A horizontal parabola has the form x = ay² + by + c, opening left or right instead of up or down. This calculator handles the standard vertical form (y = ax² + bx + c). For horizontal parabolas, swap the roles of x and y in all formulas: the axis of symmetry becomes horizontal, and the directrix becomes a vertical line.
Under uniform gravity with no air resistance, a projectile's trajectory is a parabola described by y = −(g/(2v₀²cos²θ))x² + (tanθ)x + y₀, where g is gravitational acceleration, v₀ is initial speed, θ is launch angle, and y₀ is initial height. The vertex of this parabola gives the maximum height, and the x-intercepts give the range.
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