A parabola is the set of points equally distant from a fixed point, the focus, and a fixed line, the directrix. Every other named feature follows from those two objects. This parabola calculator takes the equation in standard, vertex or sideways form and returns the vertex, focus, directrix, axis of symmetry, latus rectum, roots and direction of opening, with the completed square written out so the algebra is visible rather than assumed.
Arb Digital publishes this because most parabola pages stop at the vertex. The vertex is the easy half. The focus and directrix are where the geometry lives, and they are the reason parabolas appear in satellite dishes, headlight reflectors and projectile paths rather than being an arbitrary shape from a textbook.
What This Parabola Calculator Does
It reads three coefficients and identifies the curve completely. For an upward or downward parabola it reports the vertex (h, k), the focus at (h, k + 1/(4a)), the directrix as the horizontal line y = k − 1/(4a), the vertical axis of symmetry x = h, and the latus rectum length |1/a|. It also solves for the roots, reports the discriminant, gives the y-intercept, and states whether the curve opens up, down, left or right. The sideways form swaps the roles of x and y throughout.
That is a wider job than our completing the square calculator, which converts a quadratic into vertex form and stops there without producing the focus or directrix; and it is a different question from our quadratic equation solver, which finds where the curve crosses the axis rather than describing the curve itself.
How to Use It
- Choose the form your equation is already in. Converting by hand first is unnecessary and is where sign errors creep in.
- Enter the coefficients. In vertex form the second and third fields are h and k, and the minus sign in (x − h) is part of the form — a vertex at x = 3 means h = 3.
- Check the direction of opening in the summary line. A positive a opens upward for the standard form and rightward for the sideways form.
- Read the focus and directrix together. They sit the same distance either side of the vertex, so one is a check on the other.
- Use the roots panel. It reports two real roots, one repeated root, or no real roots, together with the discriminant that decided which.
The Formulas and How It's Calculated
Start from y = ax² + bx + c and complete the square. Factor a out of the first two terms, add and subtract (b/2a)², and the equation becomes y = a(x + b/2a)² + c − b²/4a. Reading off the vertex form gives h = −b/(2a) and k = c − b²/(4a). The focal distance p satisfies 4p = 1/a, so p = 1/(4a), and the focus sits at (h, k + p) with the directrix at y = k − p. The latus rectum — the chord through the focus parallel to the directrix — has length |1/a|, or equivalently |4p|.
Work through the default equation y = 2x² − 8x + 5. Here h = 8/4 = 2 and k = 5 − 64/8 = −3, so the vertex is (2, −3) and the vertex form is y = 2(x − 2)² − 3. The focal distance is 1/(4×2) = 0.125, putting the focus at (2, −2.875) and the directrix at y = −3.125. The axis of symmetry is x = 2 and the latus rectum is 0.5 units long. The discriminant is 64 − 40 = 24, so there are two real roots at x = (8 ± √24)/4, which is 2 ± 1.2247 — approximately 0.7753 and 3.2247.
The check that ties it together is the defining property. Take any point on the curve, say (3, −1) since 2×9 − 24 + 5 = −1. Its distance to the focus (2, −2.875) is √(1 + 1.875²) = √4.5156 = 2.125, and its vertical distance to the directrix y = −3.125 is −1 − (−3.125) = 2.125. Equal, as the definition requires. The vertex and axis of symmetry route through completing the square is set out in the parabolas section of Paul's Online Math Notes at Lamar University.
The Focus and Directrix Are the Definition
Most courses introduce a parabola as the graph of a quadratic, which makes the focus look like an extra fact bolted on afterwards. It is the other way round. The parabola is defined as the locus of points equidistant from the focus and the directrix, and the quadratic equation is what that condition looks like when you write it in coordinates and simplify.
The reflective property follows directly and is why the shape is used in hardware. Any ray arriving parallel to the axis reflects off the curve straight through the focus, and any ray leaving the focus reflects out parallel to the axis. That is a satellite dish and a headlight reflector, respectively — the same curve run in two directions. No other shape does this exactly, which is why parabolic reflectors are manufactured to that precise profile rather than approximated with a circle.
What the Coefficient a Actually Controls
The sign of a decides which way the curve opens: upward for positive, downward for negative in the standard form. Its size controls the width, but in an inverse way that is easy to get backwards. A larger |a| means a narrower parabola, because the focal distance 1/(4a) shrinks and the arms climb faster. An a of 10 gives a tight curve with the focus 0.025 units from the vertex; an a of 0.1 gives a broad one with the focus 2.5 units away.
This is also why every parabola is geometrically similar to every other one — they differ only in scale, never in shape. Zoom in far enough on a wide parabola and it looks exactly like a narrow one. The same is not true of ellipses or hyperbolas, which have a genuine shape parameter in their eccentricity, and our ellipse calculator and circle calculator handle those relatives of the parabola in the conic family.
Roots, Vertex and Discriminant Are One Story
The discriminant b² − 4ac decides how many times the curve crosses the axis, and it is directly tied to the vertex. Rewrite k as −(b² − 4ac)/(4a), and the connection is exact: the vertex is above the axis when the discriminant and a have opposite signs, on the axis when the discriminant is zero, and below when they share a sign. A negative discriminant with a positive a means the whole curve sits above the axis and never crosses it.
That is worth knowing because it turns a root question into a picture. If a is positive and the vertex is above the axis, there are no real roots — you can see it without solving anything. When roots do exist they are symmetric about the axis of symmetry, so their average is always h, which is the fastest possible check on a pair of roots you have computed by hand. Our discriminant calculator isolates that single value if it is all you need.
Sideways Parabolas and Why They Are Not Functions
The form x = ay² + by + c produces a parabola opening left or right. Everything transfers with x and y exchanged: the axis of symmetry is horizontal, the focus sits left or right of the vertex, and the directrix is a vertical line. Positive a opens rightward.
The consequence that catches people is that a sideways parabola is not a function of x. A vertical line crosses it twice, so no expression of the form y = f(x) can describe it in one piece — you need two branches, y = k ± √((x − h)/a). This matters when plotting software refuses the curve, and it explains why the roots this calculator reports for the sideways form are y-values where the curve meets the y-axis, not x-intercepts.
Where Parabolas Show Up Outside the Textbook
Projectile motion under constant gravity with no air resistance traces an exact parabola, which is why the shape is the standard model for a thrown ball or an unpowered trajectory. The vertex is the highest point, the axis of symmetry is the vertical through it, and the symmetry of the curve is the reason ascent and descent take equal time in the idealised case.
The quadratic also describes any quantity with a single optimum and diminishing returns either side of it: cost against production volume, response against dose, yield against a setting. Fitting a parabola to three or more measurements and reading its vertex is a standard way to locate an optimum you have not directly measured. Where the relationship is straight rather than curved, a linear model is the correct choice, and the difference between the two is visible in the residuals rather than in the raw data. The algebraic machinery for the roots themselves, including the discriminant and the explicit quadratic formula, is set out in section 1.11 of the NIST Digital Library of Mathematical Functions.
Arb Digital models diminishing returns properly instead of extrapolating a straight line past the point where it stopped being true.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Getting the sign of h wrong — vertex form contains (x − h), so a vertex at x = −4 means h = −4 and the bracket reads (x + 4).
- Assuming a large a means a wide curve — it is the opposite; a larger coefficient narrows the parabola and pulls the focus toward the vertex.
- Putting the directrix on the same side as the focus — they sit on opposite sides of the vertex, always the same distance from it.
- Forgetting to factor a out before completing the square — the correction term is b²/(4a), not b²/4, and skipping the factor is the most common algebra slip here.
- Treating a sideways parabola as a function — it fails the vertical line test and needs two branches to plot.
Related Free Tools From Arb Digital
Solve for the roots directly with the quadratic equation solver, convert to vertex form step by step with the completing the square calculator, factorise the quadratic with the factoring trinomials calculator, or study the related conics with the ellipse calculator. The full free online tools hub lists every mathematics tool we publish.
Frequently Asked Questions
For y = ax² + bx + c the vertex is at x = −b/(2a), and substituting that back gives the y-coordinate as c − b²/(4a). In vertex form the coordinates are read straight off as (h, k).
The focus is a fixed point inside the curve and the directrix is a fixed line outside it. Every point on the parabola is exactly the same distance from both, which is the definition the equation comes from.
The focal distance is 1/(4a). A larger coefficient gives a narrower curve with the focus close to the vertex; a smaller one widens the curve and moves the focus away.
It is the chord through the focus running parallel to the directrix. Its length is the absolute value of 1/a, and it gives you two extra plotting points level with the focus.
The parabola never crosses the axis, so there are no real roots. The vertex sits entirely above the axis when a is positive, or entirely below it when a is negative.
Yes. The form x = ay² + by + c opens left or right, with a horizontal axis of symmetry and a vertical directrix. It is not a function of x, because a vertical line crosses it twice.
Because the vertex is the turning point, and the curve is a mirror image on either side of it. That also means the two roots, when they exist, are equally spaced either side and average to the vertex x-coordinate.
This page explains a mathematical method for study and for checking your own working. It is not a substitute for showing your method, and it is not medical, legal, or financial advice.
