Vertex of a Parabola Calculator: Focus
This vertex of a parabola calculator finds the turning point of any quadratic function. Enter the coefficients a, b and c of y = ax² + bx + c, or switch to vertex form y = a(x − h)² + k, and the tool returns the vertex (h, k), the axis of symmetry, the focus, the directrix, the y-intercept and the roots, including complex roots when the parabola never crosses the x-axis. It also tells you whether the parabola opens up or down and whether the vertex is a minimum or a maximum. Every result is shown with the formula, so you can check homework or plan a projectile or design problem.
Vertex of a Parabola Calculator
Find the vertex, axis of symmetry, focus, directrix, roots and vertex form of any parabola from standard or vertex form.
How to use the Vertex of a Parabola Calculator
- Choose standard form (a, b, c) or vertex form (a, h, k).
- Type the coefficients. The values must have a not equal to zero.
- Read the vertex, axis of symmetry, focus, directrix and roots.
- Copy the result or use the vertex form to graph the curve.
Features
- Standard form to vertex form and back
- Vertex, axis of symmetry, focus and directrix
- Real and complex roots
- Minimum or maximum value
- Works with decimals and negative numbers
- Formulas shown with each answer
Frequently asked questions
How do you find the vertex of a parabola?
For y = ax² + bx + c the x-coordinate is h = −b / (2a). Put h back into the equation to get k. The vertex is the point (h, k).
What is the vertex form of a parabola?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. It is found from standard form by completing the square.
How do I know if the vertex is a maximum or minimum?
If a is positive the parabola opens up and the vertex is the lowest point, a minimum. If a is negative it opens down and the vertex is a maximum.
What are the focus and directrix?
The focus is a point inside the curve at distance 1/(4a) from the vertex along the axis. The directrix is the line the same distance on the other side. Every point of the parabola is equally far from both.
Why are my roots complex?
When the discriminant b² − 4ac is negative the parabola does not touch the x-axis, so the roots are a pair of complex numbers.