Function Graph
The chart uses the selected x range and step size.
Roots, Turning Points, and Intervals
Roots
| Root |
Value |
Type |
| Root 1 |
3 |
Real |
| Root 2 |
1 |
Real |
| Root 3 |
2 |
Real |
Critical Points
| Type |
x |
f(x) |
f''(x) |
| Local maximum |
1.4226 |
0.3849 |
-3.4641 |
| Local minimum |
2.5774 |
-0.3849 |
3.4641 |
Monotonic Intervals
| Interval |
Behavior |
| (-∞, 1.4226) |
Increasing |
| (1.4226, 2.5774) |
Decreasing |
| (2.5774, ∞) |
Increasing |
Example Data Table
Sample polynomial: f(x)=x³-6x²+11x-6. It has roots near 1, 2, and 3.
| x |
f(x) |
f'(x) |
Meaning |
| 0 |
-6 |
11 |
Y-intercept and positive slope |
| 1 |
0 |
2 |
First real root |
| 2 |
0 |
-1 |
Second real root |
| 3 |
0 |
2 |
Third real root |
Generated Sample Points
| x |
f(x) |
f'(x) |
| -2 |
-60 |
47 |
| -1.9 |
-55.419 |
44.63 |
| -1.8 |
-51.072 |
42.32 |
| -1.7 |
-46.953 |
40.07 |
| -1.6 |
-43.056 |
37.88 |
| -1.5 |
-39.375 |
35.75 |
| -1.4 |
-35.904 |
33.68 |
| -1.3 |
-32.637 |
31.67 |
| -1.2 |
-29.568 |
29.72 |
| -1.1 |
-26.691 |
27.83 |
| -1 |
-24 |
26 |
| -0.9 |
-21.489 |
24.23 |
Formula Used
Cubic form: f(x)=ax³+bx²+cx+d, where a ≠ 0.
Value: substitute the chosen x into the function.
Derivative: f'(x)=3ax²+2bx+c. It gives slope.
Second derivative: f''(x)=6ax+2b. It describes concavity.
Discriminant: Δ=18abcd−4b³d+b²c²−4ac³−27a²d². It describes root behavior.
Inflection point: x=-b/(3a), then calculate f(x).
How to Use This Calculator
- Enter coefficients a, b, c, and d for the cubic function.
- Enter the x value where you want the function evaluated.
- Set graph start, graph end, and step size.
- Choose decimal precision for rounded results.
- Press the calculate button to view results above the form.
- Use CSV export for spreadsheet work.
- Use PDF export for a shareable report.
Understanding Third Degree Polynomial Functions
A third degree polynomial is also called a cubic function. It has the general form f(x)=ax³+bx²+cx+d, where a is not zero. This calculator studies that function from several useful angles. It evaluates a selected x value, finds real and complex roots, checks slope, locates turning points, and draws a graph.
Why Cubic Functions Matter
Cubic functions are important because they can model bending growth. They may rise, fall, flatten, and change direction. A cubic can have one real root or three real roots. It can also have repeated roots. The discriminant helps describe this root pattern. A positive discriminant means three distinct real roots. A zero value means a repeated root. A negative value means one real root and two complex roots.
Derivatives and Shape
The derivative gives the slope of the curve. For f(x)=ax³+bx²+cx+d, the derivative is 3ax²+2bx+c. Solving that derivative gives possible local maximum and minimum points. These are called critical points. The second derivative, 6ax+2b, helps classify each point. A positive value suggests a local minimum. A negative value suggests a local maximum.
Inflection and Graph Review
The inflection point is also useful. It is where the curve changes concavity. For a cubic function, this happens at x=-b/(3a). The calculator reports that point and its function value. It also gives the tangent line at your chosen x value.
Use the graph to inspect behavior visually. Choose a wide x range when roots are unknown. Reduce the step size for a smoother curve. Use more precision when coefficients are small or roots are close together. Export the CSV file when you need spreadsheet data. Export the PDF when you need a quick report.
Practical Use
This tool is useful for algebra, calculus, engineering, and data modeling. It does not replace proof or symbolic reasoning. It gives a fast numeric analysis. Always review inputs, units, and rounding before using results in formal work. When results look unexpected, start with the graph. Then compare the root list, critical points, and function table. These sections often reveal scale problems, sign errors, or coefficient mistakes before they affect the final interpretation during study, review, exams, projects, homework, revision, or tutoring.
FAQs
What is a third degree polynomial?
It is a function with highest power three. Its standard form is f(x)=ax³+bx²+cx+d, where a is not zero.
Why must coefficient a be nonzero?
If a equals zero, the x³ term disappears. The expression becomes quadratic, linear, or constant instead of cubic.
Can a cubic have complex roots?
Yes. A cubic always has three roots when counted with multiplicity. Some roots may be complex conjugates.
What does the discriminant show?
The discriminant describes root behavior. Positive means three distinct real roots. Zero means repeated roots. Negative means one real root.
What are critical points?
Critical points occur where the derivative is zero. They may mark a local maximum, local minimum, or stationary inflection.
What is the inflection point?
It is the point where concavity changes. For any cubic, the x-coordinate is -b divided by 3a.
Why use a smaller graph step?
A smaller step creates more plotted points. This can make the curve smoother, but it also creates more data.
Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable summary.