Parabola Calculator
Enter a parabola in standard form (y = ax² + bx + c) or switch to vertex form (y = a(x-h)² + k) or a horizontal parabola (x = ay² + by + c). The calculator instantly finds the vertex, focus, directrix, axis of symmetry, latus rectum, focal length, x-intercepts, y-intercept, domain and range - with every step shown.
What is a parabola?
A parabola is the set of all points in a plane equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric definition gives rise to the U-shaped curve familiar from quadratic equations. Parabolas appear in physics as the path of a projectile under constant gravity, in engineering as the ideal reflector shape for satellite dishes and car headlights, and in architecture as the natural curve of suspension cables under uniform load. Every parabola has exactly one axis of symmetry passing through its vertex and focus, one directrix perpendicular to that axis, and a latus rectum chord through the focus whose length controls how narrow or wide the curve is.
Vertex, focus and directrix explained
The vertex is the point where the parabola turns - the minimum for upward-opening curves and the maximum for downward-opening ones. For y = ax² + bx + c the vertex x-coordinate is h = -b/(2a) and the y-coordinate is k = c - b²/(4a). The focal length p = 1/(4a) is the distance from the vertex to the focus and also from the vertex to the directrix. When a is large the parabola is narrow because p is small; when a is small the curve is wide. The focus sits inside the parabola at (h, k + p) for a vertical parabola and at (h + p, k) for a horizontal one. Any ray parallel to the axis reflects off the parabola and passes through the focus, which is why parabolic mirrors concentrate parallel signals (satellite, radio, solar) at one point.
Standard form vs vertex form
Standard form y = ax² + bx + c is the natural output of multiplying out brackets and is easy to read off the y-intercept (just set x = 0 to get y = c). Vertex form y = a(x - h)² + k puts the vertex coordinates front and centre and makes it trivial to shift the parabola or compare widths. You can convert between them by completing the square: group the x² and x terms, factor out a, add and subtract (b/2a)² inside the bracket, then absorb the spare term into k. This calculator performs the conversion automatically in both directions so you always see both forms.
Latus rectum and its uses
The latus rectum is the chord drawn through the focus perpendicular to the axis of symmetry. Its length is always |4p| = |1/a|. In optics the latus rectum determines the aperture: a reflector captures all incoming rays within half the latus rectum length on each side of the axis. In calculus the latus rectum length appears in integration shortcuts for parabolic areas. The two endpoints of the latus rectum lie on the parabola itself, each at a distance of 2|p| from the axis, so they serve as useful reference points when sketching the curve by hand.
Parabola properties by form
| Property | Vertical (y = ax² + bx + c) | Horizontal (x = ay² + by + c) |
|---|---|---|
| Vertex | (-b/2a, c - b²/4a) | (c - b²/4a, -b/2a) |
| Focal length p | 1/(4a) | 1/(4a) |
| Focus | (h, k + p) | (h + p, k) |
| Directrix | y = k - p | x = h - p |
| Axis of symmetry | x = h | y = k |
| Latus rectum length | |4p| | |4p| |
| Opens | Up if a > 0, Down if a < 0 | Right if a > 0, Left if a < 0 |
| Domain | All real x | x >= h (a > 0) or x <= h (a < 0) |
| Range | y >= k (a > 0) or y <= k (a < 0) | All real y |
Key formulas for vertical and horizontal parabolas in standard and vertex form.
Frequently asked questions
What is the difference between vertex form and standard form of a parabola?
Standard form is y = ax² + bx + c. It shows the y-intercept directly as c and makes finding roots easy with the quadratic formula. Vertex form is y = a(x - h)² + k, where (h, k) is the vertex. Vertex form makes it immediately clear where the parabola turns and in which direction, and it simplifies comparing parabolas or applying transformations. Both forms describe exactly the same curve - this calculator shows you both simultaneously.
How do I find the focus and directrix of a parabola?
First find the vertex (h, k) and compute the focal length p = 1/(4a). For a vertical parabola (y = ax² + ...) the focus is at (h, k + p) and the directrix is the horizontal line y = k - p. For a horizontal parabola (x = ay² + ...) the focus is at (h + p, k) and the directrix is the vertical line x = h - p. If a is positive the focus is above the vertex (or to the right) and if a is negative the focus is below (or to the left).
What does the focal length p tell me about the parabola shape?
The focal length p = 1/(4a) controls how narrow or wide the parabola is. A large |a| gives a small |p| and a narrow, steep curve. A small |a| gives a large |p| and a wide, flat curve. The latus rectum length is always |4p|, so a wider parabola has a longer latus rectum. In engineering, p is the working distance of a parabolic reflector: signals arriving parallel to the axis all converge at the focus, which is p units from the vertex.
Can a parabola open sideways?
Yes. A horizontal parabola has the form x = ay² + by + c and opens to the right when a > 0 or to the left when a < 0. Its axis of symmetry is a horizontal line y = k and its vertex, focus and directrix are computed the same way as for a vertical parabola but with x and y roles swapped. Note that a horizontal parabola is not a function of x (it fails the vertical line test) but it is a function of y.
How do I find the x-intercepts of a parabola?
Set y = 0 and solve ax² + bx + c = 0 using the quadratic formula: x = (-b +/- sqrt(b² - 4ac)) / (2a). If the discriminant b² - 4ac is positive there are two real intercepts; if it is zero there is exactly one (the vertex touches the x-axis); if it is negative the parabola does not cross the x-axis at all. This calculator evaluates the discriminant automatically and reports all real intercepts.