Polynomial Input Form
Choose the degree, enter coefficients, set the graph range, and evaluate the function at any x-value.
Example Data Table
Example polynomial: P(x) = x3 - 4x + 1.
| x | P(x) | Meaning |
|---|---|---|
| -2 | 1 | Positive output on the left side of the graph. |
| -1 | 4 | The curve stays above the x-axis here. |
| 0 | 1 | This is the y-intercept. |
| 1 | -2 | The graph crosses below the x-axis nearby. |
| 2 | 1 | The function rises again after a local minimum. |
Formula Used
General polynomial: P(x) = anxn + an-1xn-1 + ... + a1x + a0
Derivative: P'(x) = n·anxn-1 + (n-1)·an-1xn-2 + ... + a1
Y-intercept: P(0) = a0
Numerical roots: approximate real roots are detected over the selected interval using sign-change scanning and bisection refinement.
The graph is created by evaluating the polynomial across evenly spaced x-values between your chosen minimum and maximum range limits.
How to Use This Calculator
- Select the polynomial degree from 1 to 8.
- Enter the coefficient for each visible term.
- Set the minimum and maximum x-values for the graph.
- Choose how many sample points to draw.
- Enter any x-value for direct function evaluation.
- Press Graph Polynomial to generate results above the form.
- Review the graph, roots, critical points, and sampled point table.
- Use the CSV and PDF buttons to save your output.
Frequently Asked Questions
1) What does this calculator graph?
It graphs a polynomial function using the coefficients you provide. It also estimates roots, shows turning behavior, evaluates chosen x-values, and lists sampled points.
2) How are roots estimated?
The tool scans the chosen interval, detects sign changes, and refines approximate root locations with bisection. Repeated or tangent roots may need narrower ranges or higher sampling to appear clearly.
3) Why must the leading coefficient stay non-zero?
A zero leading coefficient changes the true degree of the polynomial. The calculator blocks that case so the selected degree and the entered equation stay consistent.
4) What are critical points?
Critical points are locations where the derivative becomes zero. They often mark local maxima, local minima, or stationary points where the curve briefly flattens.
5) Why change the x-range?
A wide x-range shows global shape and end behavior. A tighter x-range helps you inspect local roots, crossings, turning points, and curve detail more clearly.
6) What does the sample count control?
The sample count controls how many points are evaluated for the graph. Higher values usually produce smoother plots but increase computation and export size.
7) Can I export the results?
Yes. Use the CSV button to save graph points and summary details, or use the PDF button to capture the result section as a portable report.
8) Does it support only one variable?
Yes. This calculator is built for single-variable polynomials in x. It does not solve multivariable equations or implicit relations.