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Parabola Calculator: Vertex, Focus, Directrix, Roots

This free parabola calculator takes y = ax² + bx + c and finds the vertex, focus, directrix, axis of symmetry, roots, y-intercept, discriminant and vertex form, and tells you which way the parabola opens.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A parabola is more than a curve: it has a focus, a directrix and an axis. These show up in satellite dishes, headlights and projectiles.

Enter a, b and c from your equation.

How to Use This Tool

  1. Enter or paste your input in the box or boxes above.
  2. Choose the options, if there are any.
  3. Press the button to run the tool.
  4. Read the result and use the copy button where one is shown.

How the Calculation Works

The vertex is at h = -b / 2a and k = c - b² / 4a. The distance p = 1 / 4a gives the focus (h, k + p) and the directrix y = k - p. The roots come from the discriminant b² - 4ac. The vertex form is y = a(x - h)² + k.

Worked Example

For y = x² - 4x + 3 the vertex is (2, -1), the roots are 1 and 3, the focus is (2, -0.75) and the directrix is y = -1.25.

Common Uses

  • Algebra homework.
  • Projectile paths.
  • Reflector design.
  • Checking graphs.

Tips and Common Mistakes

  • Write 0 for a missing term.
  • A negative a opens the parabola downwards.
  • Complex roots mean the graph never touches the x axis.
  • Check the first result against a case you already know, so you can trust the tool for the rest.
  • Nothing you type leaves your device. The work is done by your browser, so it also works with a slow connection once the page has loaded.

Next, you may need the hyperbola calculator, quadratic equation solver, percentage calculator, average calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

The x value is -b / 2a. Put it back into the equation to get the y value.
For a parabola that opens up or down, the focus is p above or below the vertex and the directrix is the line p on the other side, where p = 1 / 4a.
The values of x where the parabola meets the x axis. They come from the quadratic formula and may be complex.
Then the graph is a straight line, not a parabola.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.