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Parabola Calculator

Get a parabola's turning point (vertex), focus, directrix, axis, vertex form and intercepts. Or build its equation from a focus and directrix.

Level: Grades 10–11

Printable worksheet

Parabola Calculator

Inputs
Start from

Can't be 0. Its sign sets which way the parabola opens, and its size sets how narrow it is.

Result

Vertex

(2, −1)

The parabola y = 0.5x² − 2x + 1 has its vertex at (2, −1), focus at (2, −0.5) and directrix y = −1.5.

Standard form
y = 0.5x² − 2x + 1
Focus
(2, −0.5)
Directrix
y = −1.5
Axis of symmetry
x = 2
Vertex form
y = 0.5(x − 2)² − 1
Opens
Upward
Focal length p (vertex to focus)
0.5
Focal width (latus rectum length, |1 ÷ a|)
2
x-intercepts
(2 − √2, 0) and (2 + √2, 0)
y-intercept
(0, 1)

See the step-by-step solution ↓

Step-by-step solution

  1. Find the middle line of the curve (the axis of symmetry). It passes through the vertex at x = −b ÷ 2a

    xᵥ=22×0.5=2

  2. Put that back into the equation (or use c − b² ÷ 4a) to get the vertex's y-coordinate

    yᵥ=1−(−2)24×0.5=−1

  3. Find the focal length p = 1 ÷ 4a. It is the distance from the vertex to the focus, and from the vertex to the directrix

    p=14×0.5=0.5

  4. Move a distance p from the vertex along the axis to reach the focus

    focus = (2, −0.5)

  5. Move p the other way to reach the directrix, a line at a right angle to the axis

    y = −1.5

  6. Write the vertex form, which shows the vertex directly

    y = 0.5(x − 2)² − 1

  7. So the vertex is

    (2, −1)

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find every key point and line of a U-shaped curve (a parabola) from its equation. Or go the other way. A parabola is made of every point that's the same distance from a fixed point and a fixed line. The point is called the focus, and the line is the directrix. The same curve is the graph of any quadratic, an equation whose highest power is x². That's why it shows up in thrown balls, satellite dishes and headlight reflectors. It also shows up in the cables of suspension bridges. This calculator turns an equation into the parabola's key features. It can also turn a focus and directrix into the equation.

For a parabola that opens up or down, enter the numbers a, b and c from y = ax² + bx + c. For one that opens sideways, use x = ay² + by + c. You get the vertex (turning point), focus, directrix and axis of symmetry. You also get the vertex form, focal length, focal width and the points where it crosses the axes (intercepts). Values are worked out exactly. So a focus at 1/12 shows as 1/12, not a rounded decimal.

Where it's used

Why this math matters outside the classroom:

  • Satellite dishes. An installer places the receiver of a dish shaped like y = 0.0625x² at its focus, 1 ÷ (4 × 0.0625) = 4 feet above the center.
  • Headlight reflectors. A lamp designer puts the bulb at the focus of a parabolic mirror so its light leaves in a beam parallel to the axis.
  • Maximum height. A coach models a kick as y = −4.9x² + 14.7x and finds the vertex at x = 1.5 s, where the ball peaks at 11.025 m.
  • Solar cookers. Students building a parabolic solar cooker with a 0.5 m focal length shape it as y = x² ÷ 2, since a = 1 ÷ (4 × 0.5).
  • Bridge cables. An engineer models a suspension cable sagging between towers as a parabola and uses its vertex to find the lowest point of the cable.

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.GPE.A.2, HSF.IF.C.8.a. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose your starting point: an equation in y, an equation in x, or a focus and directrix.
  2. Enter the numbers a, b and c (a can't be 0). Or enter the focus point and the directrix line.
  3. Read the vertex first. (If you started from a focus and directrix, you'll see the equation first.) The focus, directrix, axis and vertex form are below.
  4. Open the working to see h = −b ÷ 2a and p = 1 ÷ 4a with your numbers.

What the inputs mean

a
a sets the direction and width. A positive a opens up (or right, for the sideways form). A negative a opens down (or left). The larger |a| (a without its sign) is, the narrower the parabola. The focus also sits closer to the vertex.
Directrix
A flat, side-to-side directrix (y = d) gives a parabola that opens up or down. An up-and-down one (x = d) gives one that opens left or right. The parabola always opens toward the focus and away from the directrix.

Formula and method

vertex: h = −b ÷ 2a,  k = c − b² ÷ 4a
vertex form: y = a(x − h)² + k
focal length: p = 1 ÷ 4a
focus = (h, k + p),  directrix: y = k − p,  axis: x = h
from a focus (h, f) and directrix y = d: k = (f + d) ÷ 2,  p = (f − d) ÷ 2,  a = 1 ÷ 4p
focal width (latus rectum) = |1 ÷ a| = 4|p|

Here a, b and c are the numbers in your equation. (h, k) is the vertex, p is the focal length and f is the focus height. A method called completing the square rewrites y = ax² + bx + c as y = a(x − h)² + k, with h = −b ÷ 2a and k = c − b² ÷ 4a. In that form, you can read the vertex (h, k) straight off. The squared part is zero at x = h. That's the lowest point of an upward parabola, or the highest point of a downward one.

Now put the focus a distance p above the vertex, and the directrix the same distance below. A point (x, y) is equally far from both when (x − h)² + (y − k − p)² = (y − k + p)². That simplifies to (x − h)² = 4p(y − k). Comparing it with y = a(x − h)² + k gives a = 1 ÷ 4p, or p = 1 ÷ 4a. A negative p means the focus is below the vertex, and the parabola opens downward.

The sideways form x = ay² + by + c is the same parabola with x and y swapped. So every formula carries over, with x and y trading places. The focal width (also called the latus rectum) is |1 ÷ a|. It's the width of the parabola measured straight across through the focus, side by side with the directrix.

Worked example

A satellite dish has a cross-section shaped like y = 0.0625x², measured in feet from the bottom of the dish. Where should the receiver go so that it sits at the focus?

  1. Here a = 0.0625, b = 0 and c = 0. So the vertex is at h = −0 ÷ (2 × 0.0625) = 0 and k = 0. That's the bottom of the dish, (0, 0).
  2. Focal length: p = 1 ÷ (4 × 0.0625) = 4.
  3. Focus: (0, 0 + 4) = (0, 4), and the directrix is y = −4.
  4. Focal width: |1 ÷ 0.0625| = 16 feet across at the height of the focus.

Result: The receiver belongs 4 feet above the center of the dish, at the focus (0, 4). Signals that arrive straight in, along the axis, all bounce to that point.

Understanding the result

The vertex is the turning point. It's the lowest point of an upward parabola, or the highest point of a downward one. The axis of symmetry is the up-and-down (or side-to-side) line through it. The parabola is a mirror image across that line.

The focus always sits inside the curve, on the axis. The directrix sits outside it, the same distance p from the vertex on the other side. A small |p| means a tight, narrow parabola. A large |p| means a wide, shallow one.

Assumptions and limitations

  • The parabola's axis is straight up and down, or side to side. Tilted parabolas (with an xy term) aren't covered.
  • The numbers you enter are exact decimals. Results are exact fractions. They show as decimals when the decimal ends, and as fractions otherwise.
  • The focus-and-directrix mode only accepts directrix lines that are flat or straight up and down.
  • Intercepts that can't be written as fractions are shown with square roots (radical form). With very large numbers, they may show as decimals instead.

Frequently asked questions

How do I find the focus of a parabola from its equation?

Find the vertex (h, k), then p = 1 ÷ 4a. For y = ax² + bx + c, the focus is (h, k + p). For y = x², a = 1, so p = 0.25 and the focus is (0, 0.25).

What is the directrix of a parabola?

It's the line the same distance from the vertex as the focus, on the opposite side. Every point of the parabola is exactly as far from the directrix as from the focus. For y = x², the directrix is y = −0.25.

How do I convert y = ax² + bx + c to vertex form?

Work out h = −b ÷ 2a and k = c − b² ÷ 4a. Then write y = a(x − h)² + k. For y = 0.5x² − 2x + 1, h = 2 and k = −1. So the vertex form is y = 0.5(x − 2)² − 1.

How can I tell which way a parabola opens?

In y = ax² + bx + c, it opens up when a > 0 and down when a < 0. In x = ay² + by + c, it opens right when a > 0 and left when a < 0. It always opens toward the focus.

How this calculator is checked

Method: Standard analytic geometry of the parabola: completing the square for the vertex, the focus–directrix definition giving a = 1/(4p), and the quadratic formula for intercepts. Exact rational arithmetic throughout. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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