Parabola Geometry Explorer

Common questions

How do you find the equation of a parabola from the focus and directrix?

The vertex sits halfway between the focus and the directrix, and p is the signed distance from the vertex to the focus. With the focus at (2, 0) and the directrix x = -2, the vertex is (0, 0) and p = 2, so (y - k)² = 4p(x - h) becomes y² = 8x, a parabola opening to the right. Choose Focus and directrix, say whether the directrix is a horizontal line (y = c) or a vertical one (x = c), and the page writes out those steps with your numbers. A focus that lies on the directrix is refused: the points equally far from both form a straight line, not a parabola.

What is p in (x - h)² = 4p(y - k)?

p is the signed distance from the vertex to the focus. Its sign sets the direction: positive p opens upward, or to the right in (y - k)² = 4p(x - h), and negative p opens downward or to the left. Its size sets the width, because the chord through the focus, the latus rectum, is 4|p| long. With the vertex at (2, -1) and the focus at (2, 1), p = 2, the equation is (x - 2)² = 8(y + 1), the directrix is y = -3, and the latus rectum runs from (-2, 1) to (6, 1), a length of 8.

How do I find the parabola through three points?

Choose Three points and enter them. Each point is substituted into y = ax² + bx + c and the three equations are solved exactly, so (0, 0), (1, 1) and (2, 4) give y = x² with focus (0, 1/4) and directrix y = -1/4, while (0, 1), (1, 0) and (3, 2) give y = (2/3)x² - (5/3)x + 1 with vertex (5/4, -1/24). Switch the axis to horizontal to fit x = ay² + by + c instead. Three points on one line are refused and the line is named, as are a repeated point and two points with the same x, which no parabola with a vertical axis can pass through.

Can I enter an equation or convert from vertex, general or factored form?

This explorer takes geometric inputs, not a typed equation or its coefficients. It calculates standard, vertex, polynomial and general forms together after you enter supported points, a vertex, a focus or a directrix. For example, vertex (2, -1) and point (0, 3) with a vertical axis give y = x² - 4x + 3, focus (2, -3/4) and directrix y = -5/4. Converting an existing equation to these geometric inputs is a separate step that this page does not perform. The quadratic formula calculator accepts a, b and c for roots and vertex form, including a working reverse conversion from a, h and k; it does not supply the focus and directrix.

Why is a point on a parabola the same distance from the focus and the directrix?

That is the definition of a parabola: the set of points exactly as far from one fixed point, the focus, as from one fixed line, the directrix. The drawing marks a check point with dashed lines to both, and the working beneath it computes both distances exactly. On (x - 2)² = 8(y + 1) the check point starts at (6, 1), the end of the latus rectum, which is √((6 - 2)² + (1 - 1)²) = 4 from the focus and |1 - (-3)| = 4 from the directrix. Drag the point, move it with the arrow keys once it has focus, or type its x, and the two distances stay equal.

Which inputs does it refuse, and why?

Anything that does not make a parabola with a vertical or horizontal axis is refused with the reason, rather than drawn as something else. A focus placed at the vertex, or a vertex on the directrix, gives a focal length of 0. A focus on the directrix gives a straight line. A focus that is neither directly above or below nor beside the vertex would need a tilted axis, which this page does not build. Three points on one line, a repeated point, or two points sharing x when the axis is vertical (y when it is horizontal) have no parabola through them. Coordinates longer than 30 digits are declined too. In Vertex and point mode, the point must differ from the vertex in both coordinates. A repeated vertex leaves the width undetermined, and sharing just one coordinate is incompatible with the chosen nondegenerate parabola.

Can I type fractions and decimals, and is my work saved?

Coordinates can be whole numbers, decimals such as -1.5, or fractions such as 3/4, up to 30 digits each, and they are read exactly: 0.5 is treated as 1/2. Results are exact fractions, and the vertex, focus, directrix, axis and focal length also show a decimal when they are not whole numbers. Once you change something, the page saves your inputs in this browser's own storage so it reopens where you left it, and clearing your browser data clears them. Copy results puts every form and feature, plus the distance check when the current check point is valid, on your clipboard, and Download SVG saves the diagram as a file.

How do I find the equation of a parabola from its vertex and a point?

Choose Vertex and point, select a vertical or horizontal axis, then enter the vertex and one other point on the curve. For a vertical axis, the page solves p = (x - h)² / (4(y - k)) using the point, then calculates the focus, directrix and all four equation forms. Vertex (2, -1) and point (0, 3) give p = 1/4, y = (x - 2)² - 1 and focus (2, -3/4). A horizontal axis uses p = (y - k)² / (4(x - h)) instead. Two ordinary points alone do not uniquely determine a parabola: one of them must be its vertex, and the axis must be chosen.

Can I graph a parabola online and save the diagram?

The diagram shows the parabola constructed from your geometric inputs, with its vertex, focus, directrix and axis of symmetry. Use Fit to the parabola or set the plot window by hand. The check point evaluates a point on the current curve and shows equal distances to the focus and directrix. Download SVG saves the labelled diagram; Copy results copies the equations and features, plus the distance check when the current check point is valid. This page does not import a graph image, fit more than three points, or export a PDF or a table of values.