Math
Parabola Calculator
Enter a parabola as y = ax^2 + bx + c, as vertex form y = a(x - h)^2 + k, or as a focus and directrix. mathcheck returns the vertex, focus, directrix, axis of symmetry, focal length, latus rectum, x- and y-intercepts, and the opening direction with exact fractions and radicals, converts between the forms, and graphs the curve with its focus and directrix.
Coefficient of x^2. Positive opens up, negative opens down.
Coefficient of x.
Constant term (the y-intercept).
Try an example
Result
Vertex
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- Focus
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- Directrix
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- Axis of symmetry
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- Focal length p
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- X-intercepts
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- Standard form
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More details (1 more)
- Vertex form
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Study path
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What a parabola is
A parabola is the set of points equally distant from a fixed point (the focus) and a fixed line (the directrix). The The turning point of a parabola, or a corner point of a shape. sits halfway between them on the The vertical line that splits a parabola into two mirror-image halves.. Every quadratic function y = ax^2 + bx + c graphs as a vertical parabola, and each of its features can be read from a, b, and c.
Parabola features at a glance
| Feature | Meaning | For y = x^2 - 4x + 3 |
|---|---|---|
| Vertex | turning point (min if a > 0, max if a < 0) | (2, -1) |
| Axis of symmetry | vertical line through the vertex | x = 2 |
| Focus | point inside the curve, p above the vertex | (2, -3/4) |
| Directrix | line outside the curve, p below the vertex | y = -5/4 |
| Latus rectum | chord through the focus parallel to the directrix | length 1 |
| X-intercepts | where y = 0 | x = 1 and x = 3 |
From focus (0, 1) and directrix y = -1
The vertex is halfway: (0, 0). p = 1, so a = 1/(4p) = 1/4 and the equation is y = (1/4)x^2, or x^2 = 4y in conic form.
How to use this calculator
- Choose the form you have: standard, vertex, or focus and directrix.
- Enter the The number multiplying a variable, like the 3 in 3x. or the focus point and directrix line.
- Read the vertex in the header, and the focus, directrix, axis, focal length, and A point where a graph crosses an axis. in the grid.
- Use the formula list to copy the standard, vertex, and conic forms.
- Open Show the work for the h, k, and p calculations.
How to read the answer
The vertex is the turning point and the axis of symmetry passes through it. The focus lies inside the curve and the directrix outside, each at distance |p| from the vertex; if a is negative, p is negative and the focus is below the vertex. Two x-intercepts mean the The b² − 4ac part of the quadratic formula; its sign tells you how many real solutions exist. is positive, one means the vertex touches the x-axis, and none means the whole curve stays on one side of it.
Common mistakes and edge cases
- Using h = b/(2a) instead of -b/(2a) for the vertex x-coordinate.
- Mixing up p and 4p: the focal length is p = 1/(4a), and the conic form uses 4p.
- Reading the vertex form sign backwards: y = (x - 2)^2 has vertex at x = 2, not -2.
- Putting the focus outside the curve. It is always on the concave side, above the vertex when the parabola opens up.
- a = 0 is not a parabola; the equation collapses to a line.
Worked examples
Standard form with two roots
y = x^2 - 4x + 3
Vertex
(2, -1)
Opens downward
y = -2x^2 + 8
Vertex
(0, 8)
Fraction vertex
y = 3x^2 + 2x - 1
Vertex
(-1/3, -4/3) ≈ (-0.3333, -1.3333)
Vertex form
y = 2(x - 1)^2 - 3
Vertex
(1, -3)
From focus and directrix
Focus (0, 1), directrix y = -1
Vertex
(0, 0)
No real roots
y = x^2 + 1
Vertex
(0, 1)
Decimal coefficients
y = 0.5x^2 - x + 2.5
Vertex
(1, 2)
Not a parabola
a = 0
Vertex
—
Focus on the directrix
Vertex
—
Frequently asked questions
How do I find the vertex of a parabola?+
For y = ax^2 + bx + c, the vertex x-coordinate is h = -b/(2a). Substitute h back in to get k, or use k = c - b^2/(4a). In vertex form y = a(x - h)^2 + k the vertex is simply (h, k).
What are the focus and directrix?+
The focus is a point inside the curve and the directrix is a line outside it; every point on the parabola is equally distant from both. For y = ax^2 + bx + c they sit p = 1/(4a) above and below the vertex.
How do I convert vertex form to standard form?+
Expand: a(x - h)^2 + k = ax^2 - 2ahx + ah^2 + k. So b = -2ah and c = ah^2 + k. The calculator shows both forms whichever one you enter.
What is the latus rectum?+
The chord through the focus parallel to the directrix. Its length is |4p| = |1/a|, which tells you how wide the parabola is at the height of the focus.
How do I find the equation from the focus and directrix?+
The vertex is the midpoint between the focus and the directrix, and p is the distance from the vertex to the focus (negative if the focus is below). Then a = 1/(4p) and the equation is y = a(x - h)^2 + k.
Does the calculator handle sideways parabolas?+
This page handles vertical parabolas of the form y = ax^2 + bx + c. For a parabola that opens left or right, swap the roles of x and y in the formulas: x = ay^2 + by + c with the focus at (h + p, k).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026