Calculator Inputs
Enter quadratic coefficients and graph settings. The page returns the axis, vertex, roots, focus, directrix, and a parabola plot.
Example Data Table
| Quadratic Function | a | b | c | Axis of Symmetry | Vertex |
|---|---|---|---|---|---|
| y = x² - 6x + 8 | 1 | -6 | 8 | x = 3 | (3, -1) |
| y = 2x² + 4x - 3 | 2 | 4 | -3 | x = -1 | (-1, -5) |
| y = -x² + 8x - 7 | -1 | 8 | -7 | x = 4 | (4, 9) |
| y = 0.5x² - 3x + 1 | 0.5 | -3 | 1 | x = 3 | (3, -3.5) |
| y = 3x² + 12x + 9 | 3 | 12 | 9 | x = -2 | (-2, -3) |
Formula Used
For a quadratic written as y = ax² + bx + c, the axis of symmetry is the vertical line passing through the vertex.
- Axis of symmetry: x = -b / (2a)
- Vertex y-coordinate: k = f(-b / 2a)
- Discriminant: Δ = b² - 4ac
- Roots: x = (-b ± √Δ) / (2a)
- Focus parameter: p = 1 / (4a)
Once the axis is known, the parabola mirrors evenly across that line. Equal horizontal distances from the axis produce equal y-values.
How to Use This Calculator
- Enter the quadratic coefficients a, b, and c from your expression y = ax² + bx + c.
- Set graph minimum x, maximum x, total graph points, and decimal precision.
- Click the calculate button to show the result above the form.
- Review the axis, vertex, roots, focus, directrix, and symmetry table.
- Use the CSV or PDF buttons to export the results.
FAQs
1) How to find the axis of symmetry in a quadratic function?
Identify a and b in y = ax² + bx + c, then use x = -b / 2a. That x-value gives the vertical mirror line passing through the parabola’s vertex.
2) What is the equation of the axis of symmetry?
The equation is x = -b / 2a for any quadratic in standard form. It is always a vertical line because the parabola opens upward or downward.
3) Why can’t coefficient a be zero?
If a equals zero, the x² term disappears. The function becomes linear, not quadratic, so it no longer has a parabola or a quadratic axis of symmetry.
4) Does the axis of symmetry always pass through the vertex?
Yes. The axis of symmetry is defined as the line that splits the parabola into two matching halves, and that line always goes through the vertex.
5) Can the two roots lie on opposite sides of the axis?
Yes. When a quadratic has two real roots, they appear at equal horizontal distances from the axis. The axis lies exactly midway between them.
6) What happens when the discriminant is negative?
The parabola has no real x-intercepts, but the axis of symmetry still exists. The graph remains symmetric, and the roots become complex numbers.
7) Does changing c affect the axis of symmetry?
No. The axis depends only on a and b through x = -b / 2a. Changing c shifts the graph up or down without moving the axis horizontally.
8) Why is vertex form useful for symmetry?
Vertex form y = a(x - h)² + k shows the axis immediately as x = h. It also makes the turning point and opening direction easy to interpret.