Focus and Directrix Calculator
This focus and directrix calculator takes a parabola in any form and returns the focus, the directrix, the vertex, the axis and 4p together — and it never asks you which way the curve opens.
Type y = 2x² − 8x + 3 and you get focus (2, −39/8), directrix y = −41/8, vertex (2, −5). Type y² − 6y = 4x − 13 and it works out for itself that this one opens sideways.
What the Focus and Directrix Calculator Does Differently
Most pages that do this job ask you two questions before you can type anything: which orientation is your parabola, and which form is the equation in. One well-known tool goes further and tells the reader that for a sideways parabola they should swap x and y themselves before using it.
Those two questions are the exercise. Deciding whether y² − 6y = 4x − 13 opens sideways is the thing being tested, and a calculator that makes you answer it first has handed the hard part back. Paste the equation exactly as your book writes it and this page reads both facts out of it.
What a Parabola Actually Is
A parabola is every point that is the same distance from one fixed point as it is from one fixed line. The point is the focus. The line is the directrix. Nothing else defines it.
Everything else follows from that sentence. The vertex is the point on the curve closest to both, so it sits exactly halfway between them. The axis of symmetry is the line through the focus at right angles to the directrix. And the curve bends around the focus because moving away from it costs distance that has to be paid back by moving away from the line.
Complete the Square on One Letter Only
Whichever letter is squared gets the treatment; the other one stays linear and ends up on the right-hand side. That gives one of these two forms:
(x − h)² = 4p(y − k) opens up or down
(y − k)² = 4p(x − h) opens left or right
Take y = 2x² − 8x + 3. Group the x terms, factor out the 2, halve the −4 and square it:
y = 2(x² − 4x) + 3 = 2[(x − 2)² − 4] + 3 = 2(x − 2)² − 5
which rearranges to (x − 2)² = 1/2(y + 5). Vertex (2, −5), and 4p = 1/2.
One Number Does All Three Jobs
Once you have p, there is nothing left to derive:
focus — from the vertex, p along the axis
directrix — from the vertex, p the other way
width at the focus — |4p|, the latus rectum
In the example above 4p = 1/2, so p = 1/8. The focus is 1/8 above the vertex at (2, −39/8) and the directrix is 1/8 below at y = −41/8. Three answers, one number — which is why getting p right matters more here than anything else.
Watch the Sign of p, Not Just Its Size
A negative p does not mean a negative distance. It means the curve opens the other way, so the focus and the directrix swap sides of the vertex.
This is the commonest way to lose a mark on the topic: taking |p| for both and putting the focus above the vertex on a parabola that opens downwards. If the coefficient of the squared letter is negative, the focus is on the same side the curve bends towards, and the directrix is on the empty side.
Expect a Fraction, and Keep It
From y = ax² you get 4p = 1/a. So unless a happens to be 1, you are in fractions from the first line. With a = 3 the focus sits 1/12 from the vertex; with a = 12 it is 1/48.
Those stay tidy if you leave them alone. Turn 1/12 into 0.083 and it is already wrong in the fourth place, and the error grows when you add it to a vertex that is itself a fraction. This page carries them exactly to the end.
Sideways Parabolas Are Not a Special Case
If y is the squared letter, the curve opens left or right, the directrix is a vertical line, and the focus moves along a horizontal axis. That is the whole difference — the same formula with the letters exchanged.
Take y² − 6y = 4x − 13. Completing the square on y gives (y − 3)² = 4(x − 1), so the vertex is (1, 3), 4p = 4 and p = 1. The focus is one to the right at (2, 3) and the directrix is the vertical line x = 0.
When It Is Not a Parabola
Two things can go wrong, and they are different from each other.
If both x² and y² are present you have a circle, an ellipse or a hyperbola, and this page names which. If the squared letter is there but the other letter is missing entirely — as in x² = 9 — there is nothing for the curve to open along, and what you have is a pair of parallel lines.
There is also a third case worth knowing. An equation with an xy term, such as x² + 2xy + y² = 4x, can still be a parabola — a turned one, with its axis at an angle. Calling that "not a parabola" would be a wrong answer rather than a refusal, so this page names it for what it is and says why it stops there.
Check It Against the Definition
Pick any point on the curve and measure both distances. They must come out equal, because that equality is what a parabola means.
On y = 2x² − 8x + 3 the point (0, 3) is 8.125 from the focus and 8.125 from the directrix. So is (5, 13), at 18.125 each way. The check costs a minute and catches a wrong p, a wrong sign and a wrong vertex all at once — which is more than re-reading your working will do.
Using the Focus and Directrix Calculator
Put the equation in the box, however it is written, and tap the button. You get the focus, the directrix, the vertex, the axis, 4p and the focal width, a full working that shows the square being completed, and a drawing of the curve with the two equal distances marked on it — so you can see the definition rather than take it on trust.
Focus and Directrix Calculator FAQ
How do you find the focus and directrix of a parabola?
What is p in a parabola?
How do you find the directrix from the focus and vertex?
Does a sideways parabola have a different formula?
What happens when p is negative?
Find the squared letter, complete the square on it alone, then let p give you all three answers — and keep the fraction to the end.
