Parabola Calculator
Find the vertex, focus, directrix, and axis of symmetry of a parabola.
This tool is for informational and educational purposes only. It is not a substitute for professional financial, medical, legal, or engineering advice. See Terms of Service.
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Request a ToolHow to Use the Parabola Calculator
This tool analyzes a parabola given its standard form equation y = ax^2 + bx + c.
- Enter coefficients. Provide a (non-zero), b, and c.
- View the properties. The vertex, focus, directrix, axis of symmetry, and opening direction are displayed.
- Copy or share. Use the buttons to save or share the result.
About the Parabola
A parabola is the graph of a quadratic function y = ax^2 + bx + c. The vertex is at (-b/(2a), f(-b/(2a))), which is the minimum point if a > 0 (opens upward) or the maximum point if a < 0 (opens downward). The focus is a point inside the parabola at distance 1/(4|a|) from the vertex, and the directrix is a horizontal line the same distance on the opposite side. Every point on the parabola is equidistant from the focus and the directrix.
Frequently Asked Questions
How do you find the vertex of a parabola?
The vertex x-coordinate is -b/(2a). Plug that into the equation to get the y-coordinate. Together they form the vertex (h, k).
What is the focus of a parabola?
The focus is a point inside the parabola such that every point on the curve is equidistant from the focus and the directrix. It is located at distance 1/(4a) from the vertex along the axis of symmetry.
What determines if a parabola opens up or down?
The sign of the coefficient a. If a > 0, the parabola opens upward. If a < 0, it opens downward.