Parabola Calculator
Find the vertex, focus, directrix, axis of symmetry and intercepts of a parabola y = ax² + bx + c.
Vertex
(2, −1)
- Focus
- (2, −0.75)
- Directrix
- y = −1.25
- Axis of symmetry
- x = 2
- x-intercepts
- x = 1 and x = 3
- y-intercept
- (0, 3)
- Vertex form
- y = (x − 2)² − 1
- Opens
- Upward
The vertex of the parabola is (2, −1).
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| x-coordinate of the vertex | −-4 ÷ (2 × 1) | 2 |
| Discriminant | -4² − 4 × 1 × 3 | 4 |
| Distance from the vertex to the focus | 1 ÷ (4 × 1) | 0.25 |
How it works
Enter a, b and c for the parabola y = ax² + bx + c.
Worked example
y = x² − 4x + 3:
- Vertex: h = 2, k = 3 − 16 ÷ 4 = −1, so (2, −1)
- Focus: (2, −0.75). Directrix: y = −1.25
- Discriminant: 16 − 12 = 4, so x-intercepts at x = 1 and x = 3. The y-intercept is (0, 3)
Sources
Frequently asked questions
How do you find the vertex of a parabola?
For y = ax² + bx + c, the x-coordinate of the vertex is −b ÷ (2a). Put that back into the equation to get the y-coordinate. For y = x² − 4x + 3, x = 4 ÷ 2 = 2 and y = 4 − 8 + 3 = −1, so the vertex is (2, −1).
How do you find the focus and directrix?
Let p = 1 ÷ (4a). The focus is (h, k + p), and the directrix is the line y = k − p, where (h, k) is the vertex. For y = x² − 4x + 3, p = 0.25, so the focus is (2, −0.75) and the directrix is y = −1.25.
How do you find the x-intercepts?
Solve ax² + bx + c = 0 with the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ (2a). The part under the root is the discriminant. If it is positive there are two intercepts, if it is zero there is one, and if it is negative there are none.
What is the axis of symmetry?
It is the vertical line x = −b ÷ (2a) through the vertex. The two halves of the parabola are mirror images across it.
Which way does the parabola open?
Upward when a is positive and downward when a is negative. A larger absolute value of a makes the parabola narrower.