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 find the focus and directrix of y = x² - 4x + 3?

The page builds a parabola from points and lines rather than reading a typed equation, but any three points on the curve pin it down. Put x = 0, 1 and 2 into y = x² - 4x + 3 to get (0, 3), (1, 0) and (2, -1), enter those under Three points, and the result is vertex (2, -1), focus (2, -3/4) and directrix y = -5/4, with the standard form (x - 2)² = y + 1. For the roots of the equation, use the quadratic formula calculator: this tool does not solve for them.

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.

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 for the current check point, on your clipboard, and Download SVG saves the diagram as a file.

Parabola Geometry Explorer

Choose what you know about the parabola: its vertex and focus, its focus and directrix, its vertex and directrix, or three points it passes through. The page works out the rest in exact fractions, so a vertex at 1/3 stays 1/3: the standard form (x - h)² = 4p(y - k), the vertex form, the expanded y = ax² + bx + c and a general form with whole-number coefficients, along with the focus, directrix, axis of symmetry, focal length and latus rectum. Parabolas that open sideways are built the same way, from a vertical directrix or a horizontal axis. The drawing keeps equal scale on both axes and carries a check point you can drag or move with the arrow keys, with its distance to the focus and to the directrix worked out beneath, so the rule that defines a parabola is on screen with your numbers in it. Inputs that cannot make a parabola, such as a focus on the directrix or three points on one line, are refused with the reason. It does not solve for roots.

Your input

Build the parabola from

VertexVertex xVertex y

FocusFocus xFocus y

Plot window

Result

Example values: edit any coordinate to build your own

(x - 2)² = 8(y + 1)

An upward-opening parabola, (x - 2)² = 8(y + 1), with vertex (2, -1), focus (2, 1), dashed directrix y = -3 and axis of symmetry x = 2. The check point (6, 1) is 4 from the focus and 4 from the directrix. The window runs from x = -4 to 8 and y = -5 to 3 at equal scale on both axes.

Equation: (x - 2)² = 8(y + 1)

Vertex: (2, -1)

Focus: (2, 1)

Directrix: y = -3

Focal length |p|: 2 (p = 2)

Opens: upward

Every form and feature

Exact fractions throughout. The vertex, focus, directrix, axis and focal length also show a decimal when they are not whole numbers.

Standard form	(x - 2)² = 8(y + 1)
Vertex form	y = (1/8)(x - 2)² - 1
Polynomial form	y = (1/8)x² - (1/2)x - 1/2
General form	x² - 4x - 8y - 4 = 0
Vertex	(2, -1)
Focus	(2, 1)
Directrix	y = -3
Axis of symmetry	x = 2
Opens	upward
Focal length \|p\|	2 (p = 2)
Latus rectum	length 8, from (-2, 1) to (6, 1)

Check the focus and directrix property

Every point on a parabola is exactly as far from the focus as from the directrix. Drag the check point on the drawing, move it with the arrow keys once it has focus, or type its x here.

Check point x

Point on the parabola: (6, 1)
Distance to the focus: √((6 - 2)² + (1 - 1)²) = √16 = 4
Distance to the directrix: |1 - (-3)| = 4
Equal, as the definition of a parabola requires.

How it was built

The focus (2, 1) has the same x as the vertex (2, -1), so the axis of symmetry is vertical: x = 2.
p = focus y - vertex y = 1 - (-1) = 2.
Focus: (2, 1). Directrix: y = k - p = -1 - 2 = -3.
Standard form (x - h)² = 4p(y - k): (x - 2)² = 8(y + 1).
With a = 1/(4p) = 1/8, the vertex form is y = (1/8)(x - 2)² - 1, which expands to y = (1/8)x² - (1/2)x - 1/2.

Worked cases

Each result below is computed by the same construction the tool runs.

Vertex and focusvertex (0, 0), focus (0, 1)
x² = 4y, directrix y = -1
Three points(0, 0), (1, 1), (2, 4)
y = x², focus (0, 1/4)
Focus and directrixfocus (2, 0), directrix x = -2
y² = 8x, opens right
Collinear points(0, 0), (1, 1), (2, 2)
Degenerate: the three points are collinear, all on the line y = x, so no parabola passes through them.

A parabola is a distance rule. It is every point exactly as far from one point, the focus, as from one line, the directrix. The vertex sits halfway between them, and p, the signed distance from the vertex to the focus, sets both the direction it opens and its width: the chord through the focus, the latus rectum, is 4|p| long.

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 find the focus and directrix of y = x² - 4x + 3?: The page builds a parabola from points and lines rather than reading a typed equation, but any three points on the curve pin it down. Put x = 0, 1 and 2 into y = x² - 4x + 3 to get (0, 3), (1, 0) and (2, -1), enter those under Three points, and the result is vertex (2, -1), focus (2, -3/4) and directrix y = -5/4, with the standard form (x - 2)² = y + 1. For the roots of the equation, use the quadratic formula calculator: this tool does not solve for them.
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.
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 for the current check point, on your clipboard, and Download SVG saves the diagram as a file.

Every equation, focus, directrix, vertex and focal length is an exact fraction computed from the numbers you enter, for nondegenerate parabolas with a vertical or horizontal axis; the diagram is drawn in floating point. Tilted parabolas, other conics and roots are outside what it computes.