Enter cubic function values
Formula used
y = ax³ + bx² + cx + d
y(x) = ax³ + bx² + cx + d
y′(x) = 3ax² + 2bx + c
y″(x) = 6ax + 2b
x = -b / 3a
Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²
Roots are obtained with the depressed cubic transformation and Cardano-based real-root rules. Turning points come from solving y′(x) = 0.
How to use this calculator
- Enter coefficients a, b, c, and d for your cubic function.
- Provide an x-value to evaluate the function and its derivatives.
- Choose the table range start, range end, and step size.
- Press the calculate button to show results above the form.
- Review real roots, turning points, inflection point, and behavior summary.
- Download the generated table as CSV or save the result block as PDF.
Example data table
Sample function: y = x³ − 6x² + 11x − 6
| x | y | y′ | y″ |
|---|---|---|---|
| 0 | -6 | 11 | -12 |
| 1 | 0 | 2 | -6 |
| 2 | 0 | -1 | 0 |
| 3 | 0 | 2 | 6 |
| 4 | 6 | 11 | 12 |
Frequently asked questions
1. What does this calculator solve?
It evaluates a cubic function, finds real roots, reports turning points, identifies the inflection point, calculates derivatives, and generates a value table.
2. Why must coefficient a be nonzero?
A zero value removes the x³ term. That changes the equation into a quadratic, linear, or constant expression instead of a cubic function.
3. What is the meaning of the discriminant?
The discriminant helps classify root behavior. Its sign indicates whether the cubic has three distinct real roots, repeated roots, or one real root with complex partners.
4. What are turning points in a cubic?
Turning points are locations where the slope becomes zero and the function changes direction. Some cubics have two, while others have none.
5. What is an inflection point?
It is the point where curvature changes sign. For a cubic, it always exists and separates the concave-down region from the concave-up region.
6. Why does the table include y′ and y″?
The first derivative shows slope direction and steepness. The second derivative shows curvature, helping you inspect concavity and local shape changes quickly.
7. What does the PDF button save?
It saves the visible result section as a PDF document, including summary values, roots, behavior details, and the generated value table.
8. Can I use decimal inputs?
Yes. The calculator accepts decimal coefficients, decimal x-values, and decimal step sizes, which is useful for scaled models and curve analysis.