Quadratic Function Input
Enter coefficients and graph settings. The form keeps a single-column page flow while using a responsive field grid.
Example Data Table
This sample uses y = x² - 3x + 2 to illustrate typical outputs and interpretation.
| Example Item | Value | Meaning |
|---|---|---|
| Coefficients | a = 1, b = -3, c = 2 | Defines the quadratic equation. |
| Roots | x = 1 and x = 2 | The curve crosses the x-axis twice. |
| Vertex | (1.5, -0.25) | This is the minimum point. |
| Axis of symmetry | x = 1.5 | The parabola mirrors across this vertical line. |
| Y-intercept | (0, 2) | The graph meets the y-axis at 2. |
| Opening | Upward | Because coefficient a is positive. |
Formula Used
These relationships let the calculator classify roots, locate the turning point, rewrite the equation in multiple forms, and generate the graph precisely.
How to Use This Calculator
- Enter the coefficients a, b, and c from your quadratic function.
- Choose the graph range using x start, x end, and step size.
- Add any x-value you want to evaluate directly.
- Select decimal places for cleaner or more precise output.
- Tick the complex-root option if you want non-real answers displayed.
- Press the calculate button to show the result section above the form.
- Review the graph, value table, forms, and geometric features.
- Use the CSV or PDF buttons to export your results.
Frequently Asked Questions
1) What does coefficient a control?
Coefficient a controls the parabola’s opening direction and vertical stretch. A positive value opens upward, a negative value opens downward, and larger absolute values make the curve narrower.
2) Why is the discriminant important?
The discriminant tells you the root type. Positive means two real roots, zero means one repeated real root, and negative means the roots are complex.
3) What is the vertex of a quadratic function?
The vertex is the turning point of the parabola. It gives the minimum value when the curve opens upward and the maximum value when it opens downward.
4) What is the axis of symmetry?
The axis of symmetry is the vertical line passing through the vertex. The left and right sides of the parabola mirror each other across this line.
5) Can this calculator show complex roots?
Yes. Enable the complex-root option and the calculator will display roots in a ± bi style when the discriminant is negative.
6) What does the y-intercept represent?
The y-intercept is the point where the graph crosses the y-axis. For a quadratic in standard form, it always equals c because f(0) = c.
7) Why might the step size change automatically?
Very small steps across a large range can create too many points. The calculator widens the step automatically to keep plotting and export reliable.
8) When is factored form unavailable?
Real factored form is unavailable when the quadratic has no real roots. In that case, the function still has a valid vertex form and complex solutions.