How this instrument works
A parabola isn't defined by its algebra first — it's defined by distance. Take a fixed point, the focus, and a fixed line, the directrix, and the parabola is the set of every point exactly as far from one as from the other. Algebra catches up afterward: writing that equidistance condition out for the curve y = ax² and simplifying lands you on x² = y ⁄ a, which matches the standard conic form x² = 4py once 4p = 1 ⁄ a — so p, the distance from vertex to focus, works out to 1 ⁄ (4a).
That p does double duty. It is the distance from the vertex up to the focus, and it is also the distance from the vertex down to the directrix — two distances that must, by definition, stay equal for every point on the curve, not merely at the vertex. This shared distance is also why parabolic mirrors and dish antennas work: a beam travelling parallel to the axis strikes the curve and reflects toward the focus no matter where it lands, because the focus-directrix equidistance forces every reflected path to the same length.
The coefficient controls how tightly the curve curls around its axis, and the focus distance moves the opposite way from what intuition suggests: a large coefficient pulls the parabola narrow and drags the focus in close to the vertex, while a small one spreads the curve wide and pushes the focus and directrix far out. As the coefficient approaches zero the parabola flattens toward a line and the focus recedes toward infinity; this sheet requires a positive coefficient, so it only plots the upward-opening branch — for a negative value the same distances apply below the vertex by mirror symmetry.
- Enter your equation's coefficient into the a, in y = ax² field — this is the only input the sheet needs.
- Read Focus: distance from vertex for the y-coordinate of the focus point, located at (0, that value).
- Read Directrix: distance from vertex for the line's position — the directrix sits at y equal to that negative number.
- Confirm the built-in symmetry: Focus and Directrix should always match in size and sit on opposite sides of zero.
- Change the coefficient to see the focus pull toward the vertex as it grows, or spread away as it shrinks.
Worked example — the parabola y = 2x²
Take the parabola y = 2x², so the coefficient is 2. The focus sits at 1 ⁄ (4 × 2) = 1 ⁄ 8 = 0.125 above the vertex, at the point (0, 0.125), and the directrix is the horizontal line y = −0.125, the same distance on the opposite side of the vertex.
Check it against a point on the curve itself: at x = 1, y = 2(1)² = 2, so (1, 2) lies on this parabola. Its distance to the focus is √(1² + (2 − 0.125)²) = √4.515625 = 2.125, and its distance straight down to the directrix line y = −0.125 is 2 − (−0.125) = 2.125 — the same number, exactly, as the definition demands for every point on the curve.
Questions
What is the focus of a parabola?
The focus is the fixed point that, together with the directrix, defines the curve: every point on the parabola sits exactly as far from the focus as from the directrix. For y = ax² the focus is the point (0, 1 ⁄ 4a), on the axis of symmetry directly above the vertex.
What is a directrix and why does it matter?
The directrix is the fixed line paired with the focus in the parabola's definition — for y = ax² it is y = −1 ⁄ 4a, a mirror image of the focus distance on the opposite side of the vertex. Without it the equidistance condition has nothing to measure against, and the curve has no definition at all.
Why do satellite dishes and headlight reflectors use a parabolic shape?
Because the equidistance property forces every ray travelling parallel to the axis to reflect off the curve and land at one point, the focus. A dish shaped this way concentrates a weak signal onto a single receiver; a headlight run in reverse sends a bulb's light out as a parallel beam instead of scattering it.
How does the focus distance change as the coefficient changes?
Inversely: focus = 1 ⁄ (4a), so doubling the coefficient halves the focus distance and pulls it toward the vertex, while shrinking the coefficient pushes the focus and directrix further away. A narrow, tightly curved parabola has a close focus; a wide, flat one has a distant focus.
Why does this calculator require a positive coefficient?
A negative coefficient simply flips the parabola to open downward, moving the focus below the vertex and the directrix above it — the same 1 ⁄ 4a distance applies, just mirrored. Restricting to positive values keeps the sheet's upward-opening convention consistent; swap the sign yourself to read the downward case.
Is the vertex always at the origin here?
Yes — y = ax² has no linear or constant term, so its vertex sits fixed at (0, 0) with the axis of symmetry running along the y-axis. A shifted parabola like y = a(x − h)² + k moves the vertex to (h, k) and carries the focus and directrix along with it, but the 1 ⁄ 4a distance between them stays the same.