Choose a known parabola description
All coordinate values may be positive, negative, or decimal values.
Standard parabola relationships
Vertex: (h, k). Focus: (h, k + p). Directrix: y = k − p.
Vertex: (h, k). Focus: (h + p, k). Directrix: x = h − p.
The value p measures the signed vertex-to-focus distance. Its sign determines the opening direction. The latus rectum length equals |4p|.
Enter known values and read the complete result
- Choose the type of data provided in the problem.
- Enter coordinates, a directrix value, or equation coefficients.
- Select the appropriate orientation when the chosen method requests it.
- Press Find Standard Form to calculate the equation.
- Review the focus, directrix, axis, latus rectum, domain, range, and graph.
- Use CSV or PDF export to save the result.
Worked parabola inputs and outputs
| Known data | Vertex | p | Standard form | Opening |
|---|---|---|---|---|
| Focus (2, 2) | (2, −1) | 3 | (x − 2)² = 12(y + 1) | Upward |
| Directrix x = −1 | (3, 2) | 4 | (y − 2)² = 16(x − 3) | Right |
| Point (6, 3) | (2, −1) | 1 | (x − 2)² = 4(y + 1) | Upward |
Standard form makes parabolas easier to interpret
A parabola is the set of points equally distant from a focus and a directrix. Standard form exposes this geometry immediately. It gives the vertex, the axis of symmetry, and the opening direction without graphing every point.
Start by locating the vertex. It is written as (h, k). In a vertical parabola, the squared expression contains x. The equation is (x − h)² = 4p(y − k). A positive p opens the graph upward. A negative p opens it downward.
A horizontal parabola has y inside the squared expression. Its equation is (y − k)² = 4p(x − h). A positive p opens right. A negative p opens left. This quick sign check prevents many common direction errors.
The number p is more than a coefficient. It is the directed distance from the vertex to the focus. The directrix sits the same distance on the opposite side of the vertex. Therefore, a larger absolute p creates a wider curve. A smaller absolute p creates a narrower curve.
When a focus is known, compare it with the vertex. Matching x-values means a vertical axis. Matching y-values means a horizontal axis. Subtract the aligned coordinates to find p. Then place 4p into the standard equation.
When a directrix is known, measure from the vertex toward the focus. For vertical graphs, p equals k minus the directrix y-value. For horizontal graphs, p equals h minus the directrix x-value. The sign appears naturally from this subtraction.
A point on the curve can also determine p. Substitute the point and vertex into the proper standard form. Solve the remaining equation for p. Check that the point does not lie on the axis through the vertex. Such a point cannot identify a finite parabola.
General equations require completing the square. Group the squared variable and its linear term. Factor the squared coefficient first. Complete the square. Then rearrange until one side contains a shifted square. The calculator performs this conversion for equations with one squared term and no xy term.
Always verify your answer. Substitute the focus or known point. Confirm the directrix location. Finally, inspect the graph. A consistent equation, focus, vertex, and directrix describe the same parabola. Use coordinates carefully. Keep signs visible. Write substitutions on separate lines. This makes checking easier during tests and homework. Check patiently.
Common standard form questions
1. What is the standard form of a vertical parabola?
It is (x − h)² = 4p(y − k). The vertex is (h, k). The parabola opens upward when p is positive and downward when p is negative.
2. What is the standard form of a horizontal parabola?
It is (y − k)² = 4p(x − h). The vertex remains (h, k). Positive p means rightward opening. Negative p means leftward opening.
3. What does p represent?
The value p is the signed distance from the vertex to the focus. Its absolute value also equals the distance from the vertex to the directrix.
4. Why does the equation use 4p?
The factor 4p comes from the equal-distance definition of a parabola. It connects the algebraic equation with the geometric focus and directrix distance.
5. How can I tell whether the axis is vertical?
A vertical axis occurs when the focus and vertex have the same x-coordinate. The squared term will be x, while y appears as the unsquared shifted coordinate.
6. How can I tell whether the axis is horizontal?
A horizontal axis occurs when the focus and vertex have the same y-coordinate. The squared term will be y, while x appears as the unsquared shifted coordinate.
7. Can a point and vertex determine a parabola?
Yes, when the orientation is known and the point is not on the axis through the vertex. Substitute both into the corresponding standard form and solve for p.
8. Can this calculator use a directrix?
Yes. Enter the vertex, choose vertical or horizontal orientation, and enter the directrix coordinate. The calculator derives p, the focus, and the standard equation.
9. Which general equations can this calculator convert?
It converts equations in the form Ax² + Cy² + Dx + Ey + F = 0 when exactly one squared coefficient is nonzero and no xy term appears.
10. What is the latus rectum?
It is the chord through the focus that is perpendicular to the axis of symmetry. Its length is |4p|.
11. Why might the calculator reject my focus?
The focus must lie exactly on the axis through the vertex. For an unrotated parabola, it must share either the vertex x-coordinate or the vertex y-coordinate.