Parabola Calculator
Analyze a parabola from standard form or vertex form. Find the vertex, focus, directrix, axis of symmetry, intercepts, domain, range, and graph instantly.
Use standard form y = ax² + bx + c or vertex form y = a(x − h)² + k.
Provide the three coefficients that define the parabola.
See the vertex, focus, directrix, intercepts, opening direction, and graph.
h = −(−2) ÷ 2(1) = 1, so the vertex is (1, −4).
A positive value of a makes the parabola open upward, while a negative value makes it open downward. Larger absolute values make the graph narrower; values closer to zero make it wider.
Every point on a parabola is equally distant from its focus and directrix. For y = a(x − h)² + k, the focal distance is p = 1 ÷ 4a.
What is the difference between standard and vertex form?
Standard form is y = ax² + bx + c. Vertex form is y = a(x − h)² + k, where (h, k) is visible directly.
What happens when a equals zero?
The equation becomes linear and no longer represents a parabola, so a must be non-zero.
Can a parabola have no x-intercepts?
Yes. When the discriminant b² − 4ac is negative, the parabola does not cross the x-axis.
How is the focus calculated?
For a vertical parabola, p = 1 ÷ 4a. The focus is (h, k + p), and the directrix is y = k − p.
Results are estimates generated locally in your browser using standard mathematical formulas. They are for informational and educational purposes only and are not a substitute for verification in coursework, examinations or professional contexts requiring certified accuracy. VisionVix accepts no liability for decisions based on this tool's output.
This calculator is intended for educational and general mathematical use. Numerical results may be rounded and should be independently verified for critical work.
