Calculator Inputs
Example Data Table
This sample uses f(x) = x² - 3x + 2 to show how values change across several x positions.
| x | f(x) | Comment |
|---|---|---|
| -1 | 6 | Left side of the parabola |
| 0 | 2 | Y-intercept |
| 1 | 0 | First real root |
| 1.5 | -0.25 | Vertex value |
| 2 | 0 | Second real root |
| 3 | 2 | Right side mirror point |
Formula Used
Standard form: f(x) = ax² + bx + c, where a ≠ 0.
Discriminant: D = b² - 4ac. It tells whether the roots are distinct, repeated, or complex.
Roots: x = (-b ± √D) / (2a). Real roots exist when D is zero or positive.
Vertex: h = -b / (2a), k = f(h). The vertex gives the turning point and extreme value.
Axis of symmetry: x = -b / (2a). Points at equal horizontal distance from this axis share the same y-value.
Sum and product of roots: r₁ + r₂ = -b/a and r₁r₂ = c/a.
Vertex form: f(x) = a(x - h)² + k. It highlights shifts, stretching, and opening direction.
How to Use This Calculator
- Enter coefficients a, b, and c from your quadratic expression.
- Set an x-range and step size for the graph and generated values.
- Choose the x-value you want to evaluate directly.
- Pick the number of decimal places and a variable label.
- Press Submit to view the result block above the form.
- Review roots, discriminant, vertex, intervals, and the table of values.
- Use the CSV and PDF buttons to export the displayed result.
FAQs
1. What does the discriminant tell me?
It classifies the roots. A positive value gives two real roots, zero gives one repeated root, and a negative value gives complex conjugate roots.
2. Why must coefficient a be nonzero?
If a equals zero, the expression becomes linear, not quadratic. A genuine quadratic always has a squared term with a nonzero coefficient.
3. What is the vertex used for?
The vertex marks the turning point of the parabola. It also gives the minimum value when the graph opens upward or the maximum value when it opens downward.
4. Can the calculator show complex roots?
Yes. When the discriminant is negative, the output shows the real part and imaginary part of both complex conjugate roots.
5. How is the factored form generated?
The calculator builds real linear factors from real roots. If the roots are complex, it reports that a real factored form is unavailable.
6. Why do I need an x-range?
The x-range controls the graph window and the value table. A wider range shows more behavior, while a smaller one highlights details near the vertex.
7. What are the intervals shown in results?
They describe where the polynomial increases or decreases. The switch happens at the x-coordinate of the vertex because that is the turning point.
8. When should I use vertex form?
Vertex form is helpful when analyzing shifts, graphing quickly, or identifying maximum or minimum values without expanding the expression again.