Parabola Calculator

The vertex is the highest or lowest point of a vertical parabola (or the leftmost/rightmost point of a horizontal one). It is given by the coordinates (h, k) in vertex form. If the coefficient a is positive, the vertex is the minimum point; if negative, it is the maximum point.

The focus is a point inside the parabola related to its focal length p = 1/(4a). For a vertical parabola y = a(x−h)² + k, the focus is at (h, k + p). For a horizontal parabola x = a(y−k)² + h, the focus is at (h + p, k).

The directrix is a line perpendicular to the axis of symmetry, located on the opposite side of the focus from the parabola. For a vertical parabola, the directrix is y = k − p. Every point on the parabola is equidistant from the focus and the directrix.

The axis of symmetry is the line that divides the parabola into two mirror-image halves, passing through the vertex. For a vertical parabola y = a(x−h)² + k, it is the vertical line x = h. For a horizontal parabola, it is the horizontal line y = k.

The latus rectum is the chord through the focus that is perpendicular to the axis of symmetry. Its length equals |4p| = |1/a|, where p is the focal length. A longer latus rectum indicates a wider parabola.

The coefficient a controls the width and direction of opening. If a > 0, a vertical parabola opens upward (or a horizontal parabola opens rightward). If a < 0, it opens downward (or leftward). The larger the absolute value of a, the narrower the parabola.

Vertex form is y = a(x−h)² + k and directly reveals the vertex (h, k). Standard form is y = ax² + bx + c. You can convert from standard to vertex form by completing the square. The vertex in standard form is at x = −b/(2a), y = c − b²/(4a).

Yes. When the equation is written as x = a(y−k)² + h, the parabola is horizontal. If a > 0 it opens rightward; if a < 0 it opens leftward. The vertex, focus, and directrix are calculated differently than for vertical parabolas.