Parabola Properties
Enter the coefficients of a parabola y = ax² + bx + c. The calculator returns the vertex (h, k), the axis of symmetry, the direction it opens, the focus and the directrix, plus the y-intercept and any x-intercepts.
From three coefficients to the whole curve
A parabola can be described two ways that look unrelated. In algebra it is the graph of a quadratic, defined by three coefficients. In geometry it is the set of points equidistant from a fixed point and a fixed line. The two descriptions turn out to be the same curve, and the focus and directrix are where that becomes visible.
This calculator takes the three coefficients and reports both readings: the vertex, axis and intercepts from the algebra, and the focus and directrix from the geometry.
The standard form is compact but says almost nothing at a glance. Where the curve turns, which way it opens and whether it meets the axis are all determined by the coefficients, and none of them can be read off directly.
Completing the square is what unlocks the rest, and every quantity here follows from it. The vertex comes out of that rearrangement, the axis is the vertical line through it, and the focus and directrix sit a fixed distance either side along that axis.
How to use this calculator
- Enter the coefficient of x² It must not be zero — that would make the equation linear rather than quadratic, and the solver rejects it explicitly. Its sign decides which way the curve opens and its size how narrow it is.
- Enter the coefficient of x and the constant Both may be zero. The constant is the y-intercept directly, since setting x to zero leaves it alone.
- Read the vertex and direction first Everything else is positioned relative to the vertex, so confirming it before reading the focus or the intercepts is worth the moment.
- Check the intercept line Two, one or none, according to the discriminant. A single intercept means the curve is tangent to the axis rather than crossing it.
The formula, and where it comes from
h = −b/(2a) k = c − b²/(4a) p = 1/(4a) focus (h, k + p) directrix y = k − p
The vertex coordinates come from completing the square. Rewriting the quadratic as a squared bracket plus a constant puts the turning point in plain sight: the bracket is zero at the value of x that makes it so, and the leftover constant is the height there.
The axis of symmetry is the vertical line through the vertex. A parabola is a mirror image about it, which means the two roots — when they exist — are equally spaced either side, and their midpoint is the vertex's x-coordinate.
The focal distance is the reciprocal of four times the leading coefficient, so a steep parabola has its focus close to the vertex and a shallow one has it far away. The focus sits that distance above the vertex along the axis and the directrix the same distance below, which is why the two always straddle the curve's turning point.
The direction follows from the sign of the leading coefficient alone. A positive value opens upward and the vertex is a minimum; a negative one opens downward and the vertex is a maximum. The focal distance inherits that sign, which is how the focus stays on the inside of the curve in both cases.
What each input means
- a Quadratic coefficient — form field “a (x² coefficient)”
- Must be non-zero. Its sign sets the direction, and its magnitude the narrowness — a larger value gives a tighter curve and a closer focus.
- b Linear coefficient — form field “b (x coefficient)”
- Together with a it fixes the horizontal position of the vertex. On its own it says nothing about the shape.
- c Constant term — form field “c (constant)”
- The y-intercept, since it is the value of the expression when x is zero.
- p Focal distance
- Not entered — computed from the leading coefficient. It measures from the vertex to the focus, and equally to the directrix in the other direction.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
An upward parabola with two roots
The quadratic with coefficients 1, −4 and 3. It opens upward and crosses the axis in two places.
Inputs a (x² coefficient) = 1, b (x coefficient) = -4, c (constant) = 3
- Standard form y = 1x² − 4x + 3
- Vertex (h, k) = (2, -1)
- Axis of symmetry x = 2
- Direction opens upward
- Focal distance p = 1/(4a) = 0.25
- Focus (2, -0.75)
- Directrix y = -1.25
- y-intercept (0, 3)
- x-intercepts (3, 0), (1, 0)
Result vertex (2, -1), focus (2, -0.75), directrix y = -1.25
The vertex sits at x = 2, midway between the two intercepts at 1 and 3. That midpoint relationship holds whenever real roots exist, and it makes the vertex easy to place from the roots or the roots easy to sanity-check from the vertex.
The focal distance is a quarter, so the focus is just above the vertex and the directrix just below. A leading coefficient of 1 gives the standard-width parabola, from which every other width is a scaling.
A parabola tangent to the axis
Coefficients 1, −2 and 1 — a perfect square, so the discriminant is zero and the curve touches the axis at one point.
Inputs a (x² coefficient) = 1, b (x coefficient) = -2, c (constant) = 1
- Standard form y = 1x² − 2x + 1
- Vertex (h, k) = (1, 0)
- Axis of symmetry x = 1
- Direction opens upward
- Focal distance p = 1/(4a) = 0.25
- Focus (1, 0.25)
- Directrix y = -0.25
- y-intercept (0, 1)
- x-intercept (1, 0) (tangent to x-axis)
Result vertex (1, 0), focus (1, 0.25), directrix y = -0.25
The single intercept coincides with the vertex, which is what tangency to the axis means: the curve reaches zero and turns back rather than passing through. The output labels it as tangent rather than reporting one root of two.
A discriminant of exactly zero is the boundary between two intercepts and none. Nudging the constant either way in the third decimal place would move this parabola into one case or the other.
Reading the result
Vertex as minimum or maximum
It is the extreme value of the quadratic, which is what makes it the answer to most optimisation questions built on one. Whether it is the largest or smallest value depends entirely on the sign of the leading coefficient.
What the focus is for
Every point on the curve is equally far from the focus and from the directrix — that equidistance is the geometric definition of a parabola. It is also why a parabolic reflector concentrates parallel incoming rays at a single point.
Reading the intercept count
Two intercepts mean the discriminant is positive, one means it is zero and the curve is tangent, none means it is negative and the parabola misses the axis entirely. The vertex's height relative to the axis says the same thing.
When you would use this
Finding a maximum or minimum
Where a quantity is modelled by a quadratic — projectile height, revenue against price — the vertex is the optimum, and it comes from the coefficients without any calculus.
Designing a reflector or antenna
A parabolic dish focuses incoming parallel signals at its focus, so the focal distance is the design parameter that positions the receiver. It follows directly from the leading coefficient.
Assumptions and limitations
What this calculator assumes
- The parabola opens vertically, as the graph of a function of x.
- The leading coefficient is non-zero; a zero value describes a line and is rejected.
- The axis of symmetry is vertical, so no rotation is represented.
- The focal distance is signed, which keeps the focus inside the curve for both directions of opening.
Where it stops being the right tool
- Vertical parabolas only. One opening left or right is not the graph of a function and is not accepted.
- Rotated parabolas, and general conic equations containing an xy term, are out of scope.
- Real intercepts only: when the discriminant is negative the complex roots are not given.
Common mistakes
Dropping the minus sign in the vertex formula
Why it happens. The x-coordinate is the negative of b over twice a, and the sign is easy to lose — especially when b is itself negative and two negatives are in play.
How to avoid it. Check the vertex against the intercepts. It must sit exactly midway between them, so a vertex outside that span has the wrong sign.
Confusing the focus with the vertex
Why it happens. For a wide parabola the two are close together on the plot, and both are described as points on the axis of symmetry.
How to avoid it. The vertex is on the curve and the focus is not. Any point of the curve is equidistant from the focus and the directrix, which the vertex satisfies only by being halfway between them.
Assuming a parabola must cross the axis
Why it happens. Sketches usually show one that does, and the quadratic formula is taught alongside the assumption that it will produce answers.
How to avoid it. Read the intercept line and the vertex together. A vertex above the axis on an upward-opening parabola means no real intercepts exist.
Frequently asked questions
How is the vertex found?
By completing the square, which gives its x-coordinate as −b/(2a) and its height as c − b²/(4a). Rewriting the quadratic as a squared bracket plus a constant makes the turning point explicit.
What is the focus of a parabola?
A point on the axis of symmetry from which every point of the curve is exactly as far as it is from the directrix. That equidistance is the geometric definition of the parabola, and it sits a distance 1/(4a) from the vertex.
Which way does the parabola open?
Upward when the leading coefficient is positive and downward when it is negative. The magnitude controls the width instead — a larger value gives a narrower curve with its focus closer to the vertex.
What if there are no x-intercepts?
Then the discriminant is negative and the parabola misses the axis entirely, sitting wholly above or below it. The intercept line says so rather than reporting complex roots.