Parabola Calculator

A parabola is the set of points in a plane that are equally distant from a fixed point, called the focus, and a fixed line, called the directrix. This geometric definition produces the familiar U-shaped curve of a quadratic function. Every vertical parabola has a vertical axis of symmetry through its vertex. When the leading coefficient is positive the arms open upward, and when it is negative they open downward. Standard form writes the function as y = ax² + bx + c, where a cannot be zero. The vertex x-coordinate is -b divided by 2a, and substituting that coordinate into the equation gives the vertex y-coordinate. This calculator uses the equivalent expression k = c - b² divided by 4a. Standard form makes the y-intercept immediately visible as c, while the sign and size of a control the direction and width of the curve. Vertex form writes the same family as y = a(x - h)² + k. The vertex is visible directly at (h, k), and the axis of symmetry is x = h. This form is especially convenient after completing the square or when a graph has been translated from y = x². A large absolute value of a creates a narrower graph, while an absolute value between zero and one creates a wider graph. Changing the sign reflects the curve across a horizontal line through its vertex. The focus and directrix depend on the focal parameter p, which equals 1 divided by 4a for a vertical function parabola. The focus is (h, k + p), and the directrix is the horizontal line y = k - p. The absolute value of p is the focal length, the distance from the vertex to either the focus or the directrix. A narrow parabola has a short focal length; a wide parabola has a longer one. Parabolas appear whenever a quadratic relationship or reflection property is involved. Projectile paths are approximately parabolic when air resistance is neglected. Satellite dishes, radio antennas, telescope mirrors, automobile headlights, and solar concentrators use parabolic reflectors because rays parallel to the axis reflect toward the focus. In analytic geometry, knowing the vertex, axis, focus, and directrix gives a complete description of the curve's position and shape. This calculator handles vertical parabolas represented as functions of x. A sideways parabola instead has an equation based on x as a function of y and requires a horizontal axis, vertical directrix, and differently placed focus. For the supported forms, enter a nonzero a and the remaining coefficients. The result reports all principal geometric properties with rounded decimal values suitable for graphing and verification.