Parabola Calculator
Convert common parabola definitions into equations, vertex, focus, directrix, intercepts, and a live coordinate graph.
Define the parabola
Choose the information you know. Values use a decimal point; scientific notation is not accepted.
Select the known equation form or geometric data.
Vertical opens up/down; horizontal opens right/left.
Required nonzero quadratic coefficient.
Required linear coefficient; zero is allowed.
Required constant term; zero is allowed.
Required nonzero scale coefficient.
Required x-coordinate of the vertex.
Required y-coordinate of the vertex.
Required vertex x-coordinate.
Required vertex y-coordinate.
Required coordinate of the directrix line; it cannot pass through the vertex.
Required vertex x-coordinate.
Required vertex y-coordinate.
Required focus x-coordinate.
Required focus y-coordinate.
Required vertex x-coordinate.
Required vertex y-coordinate.
Required x-coordinate of a second point.
Required y-coordinate of the same point.
Required focus x-coordinate.
Required focus y-coordinate.
Required coordinate of the directrix line; it cannot contain the focus.
Required x-coordinate.
Required y-coordinate.
Required x-coordinate.
Required y-coordinate.
Required x-coordinate.
Required y-coordinate.
y = 2x² + 3x - 4
y = 2(x + 0.75)² - 5.125
(x + 0.75)² = 0.5(y + 5.125)
Coordinate graph
The curve, focus, vertex, and directrix all use the same current model.
The positive quadratic coefficient makes this vertical parabola open upward. The focus lies 0.125 unit above the vertex.
Key coordinates and sampled points
Geometry rows identify the defining features; sampled rows verify the plotted equation.
| Item | x | y | Meaning |
|---|---|---|---|
| Vertex | -0.75 | -5.125 | Turning point |
| Focus | -0.75 | -5 | Fixed point inside the opening |
| Latus endpoint 1 | -1 | -5 | Endpoint through the focus |
| Latus endpoint 2 | -0.5 | -5 | Endpoint through the focus |
Coordinates are calculated from the unrounded model. Display values are rounded to six decimal places.
How to use this parabola calculator
What this calculator does
This calculator turns several common descriptions of an axis-aligned parabola into one consistent geometric model. It reports the vertex, focus, directrix, axis of symmetry, opening direction, focal length, latus rectum length, discriminant, real intercepts, three equivalent equations, a coordinate graph, and a table of defining points. It handles vertical parabolas whose equation is based on x² and horizontal parabolas whose equation is based on y². It does not fit rotated parabolas containing an xy term, and it does not solve arbitrary higher-degree polynomials.
Use it when you need to convert a quadratic from standard form to vertex form, sketch a conic from its focus and directrix, recover an equation from three points, check a completed-square calculation, or prepare reliable values for a graphing or geometry exercise. The geometric meaning follows the focus-directrix definition explained in the OpenStax Precalculus key concepts for parabolas.
How to calculate
- The calculator opens with a complete demonstration: Standard form, Vertical: y depends on x², a = 2, b = 3, and c = – 4. The results and Excel workbook are ready immediately.
- Choose What to input. The visible fields change to match Standard form, Vertex form, Vertex and directrix, Vertex and focus, Vertex and point, Focus and directrix, or Three points.
- Choose Parabola orientation. Vertical means y is the dependent coordinate and the axis is x = h. Horizontal means x is the dependent coordinate and the axis is y = k.
- Replace the example values with decimal numbers. A leading minus sign and correctly placed thousands separators are accepted; decimal commas and scientific notation are rejected to prevent ambiguous interpretation.
- Read the live result cards, equivalent equations, graph, and coordinate table. Select Download Excel to export the current canonical values to a validated .xlsx workbook.
- Select Reset to clear the demonstration and every calculated result. The export becomes unavailable until a complete valid definition is entered again.
Input guide
What to input is required and identifies the data you already know. Parabola orientation is also required. In Standard form, enter required coefficients a, b, and c; a must be nonzero, while b and c may be zero. For a vertical parabola these create y = ax² + bx + c, while a horizontal parabola uses x = ay² + by + c. Increasing |a| narrows the curve, and changing the sign of a reverses the opening.
In Vertex form, enter required a, h – vertex x, and k – vertex y. The example a = 2, h = – 0.75, k = – 5.125 reproduces the startup curve. A common mistake is reversing the sign inside the parenthesis: h = – 0.75 produces (x + 0.75)², not (x – 0.75)².
For Vertex and directrix, enter h – vertex x, k – vertex y, and Directrix coordinate d. In vertical orientation d defines y = d; in horizontal orientation it defines x = d. The directrix may not pass through the vertex. For Vertex and focus, enter the vertex plus Focus x and Focus y. The focus must lie on the selected axis through the vertex and cannot equal the vertex.
For Vertex and point, enter the vertex and one additional Point x/Point y pair on the curve. The point must determine a nonzero quadratic coefficient; a point placed on the wrong axis or at the vertex cannot define the scale. For Focus and directrix, enter Focus x, Focus y, and the directrix coordinate. The focus and directrix must be separated.
For Three points, provide Point 1 x/y, Point 2 x/y, and Point 3 x/y. A vertical fit requires three distinct x-values; a horizontal fit requires three distinct y-values. Collinear or duplicate independent coordinates do not define a nondegenerate quadratic.
Output guide
Vertex is the turning point (h, k). Focus is the fixed point inside the opening, and Directrix is the fixed line on the opposite side of the vertex. Axis of symmetry passes through the vertex and focus. Opening direction is Up, Down, Right, or Left, based on orientation and the sign of a. Focal length |p| is the positive distance from vertex to focus, while Latus rectum length is 4|p|. The geometric relationship between these features is illustrated in the LibreTexts treatment of the parabola.
Discriminant is b² – 4ac. It controls the number of real intercepts where the dependent coordinate is zero: positive gives two, zero gives one repeated intercept, and negative gives none. Real axis intercepts lists those independent-coordinate values. Standard form, Vertex form, and Focus-directrix form are exact identities from the same coefficients. The live summary pills repeat Opening, Vertex, Focus, and Focal length for quick scanning. The graph series labeled Parabola, Vertex, Focus, and Directrix is a visual comparison. In the Key coordinates and sampled points table, Item names the feature or sample, x and y give its coordinates, and Meaning explains the row. The note below the table states the six-decimal display policy.
Worked example
For y = 2x² + 3x – 4, the startup inputs are a = 2, b = 3, c = – 4. The vertex x-coordinate is h = – b/(2a) = – 3/4 = – 0.75. Then k = c – b²/(4a) = – 4 – 9/8 = – 5.125. The directed focal parameter is p = 1/(4a) = 0.125, so the focus is ( – 0.75, – 5), the directrix is y = – 5.25, and the latus rectum length is 4|p| = 0.5. The displayed vertex form is y = 2(x + 0.75)² – 5.125. These values match the first-open cards, graph, table, and workbook.
How the model works
For a vertical parabola, the vertex equation is y = a(x – h)² + k and the conic equation is (x – h)² = 4p(y – k), with p = 1/(4a). The focus is (h, k + p), the directrix is y = k – p, and the axis is x = h. A horizontal parabola swaps the coordinate roles: x = a(y – k)² + h, focus (h + p, k), directrix x = h – p, and axis y = k.
Interpreting width and direction
A small |a| produces a wide curve because |p| is large; a large |a| produces a narrow curve because the focus sits close to the vertex. The sign of a controls direction. This directed-distance convention and the meaning of p are discussed in the LibreTexts guide to directed focal distance.
Common mistakes
- Using a = 0. That creates a line or constant relation, not a parabola.
- Confusing the vertex with the focus. They share the symmetry axis, but they are separated by |p|.
- Forgetting that a horizontal parabola squares y rather than x.
- Reading rounded display values as the internal calculation. The model and workbook retain full finite numeric precision.
- Entering three points with repeated independent coordinates. The resulting system is singular and has no unique quadratic fit.
For another concise explanation of the equal-distance definition, see the Khan Academy focus and directrix review.