Parabola calculator
Enter the coefficients a, b and c of a parabola that opens up or down (y = ax² + bx + c) or sideways (x = ay² + by + c). You get the vertex, focus, directrix, axis of symmetry, intercepts, focal length and the vertex form, plus a sketch.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose whether the parabola opens up/down or left/right.
- Enter a, b and c. Fractions and constants such as 1/4 or sqrt(2) work.
- Read the vertex and focus at the top, the rest below, and check the graph.
For y = ax² + bx + c:
- vertex (h, k): h = −b / (2a), k = c − b² / (4a)
- focal length p = 1 / (4a); focus (h, k + p); directrix y = k − p
- axis of symmetry x = h; latus rectum length |1/a|
- x-intercepts from the quadratic formula; y-intercept (0, c)
For x = ay² + by + c the same formulas hold with x and y swapped.
Worked example
y = 2x² − 4x + 1: the vertex is (1, −1). p = 1/8 = 0.125, so the focus is (1, −0.875) and the directrix is y = −1.125. The vertex form is y = 2(x − 1)² − 1, and the roots are x = 1 ± √2 / 2, about 0.2929 and 1.7071.
Tips
- The sign of a decides the direction: positive opens up (or right), negative down (or left).
- A larger |a| makes a narrower parabola and moves the focus closer to the vertex.
- For the roots alone, the quadratic equation solver shows the working step by step.
Frequently asked questions
How do I find the vertex of a parabola?
For y = ax² + bx + c the vertex is at x = −b / (2a), and its y is c − b² / (4a). For y = 2x² − 4x + 1 that is x = 4 / 4 = 1 and y = 1 − 16 / 8 = −1, so the vertex is (1, −1).
How do I find the focus and directrix?
The focal length is p = 1 / (4a). The focus lies p from the vertex along the axis, in the direction the parabola opens, and the directrix is the line p from the vertex on the other side. For y = x², p = 1/4: focus (0, 0.25), directrix y = −0.25.
What is the vertex form?
y = a(x − h)² + k, where (h, k) is the vertex. It shows the vertex directly and is the same parabola as the standard form ax² + bx + c.
What if the parabola opens sideways?
Choose x = ay² + by + c. The roles of x and y swap: the axis is horizontal, the directrix is a vertical line x = …, and the intercepts are on the y-axis. Positive a opens to the right.
Why are no intercepts shown?
When b² − 4ac is negative the parabola does not cross the axis, so there are no real roots. The vertex, focus and directrix still exist.