Polynomial Grapher

Graph a polynomial with its real roots and turning points

About this calculator

The Polynomial Grapher takes a coefficient list (highest degree first, up to degree 8) and analyzes the whole curve: the exact y-intercept, every real root found by scanning the Cauchy bound interval, and each turning point classified as a local minimum or maximum by the exact derivative. Tangent (repeated) roots that only touch the x-axis are recovered too. It renders the polynomial in clean mathematical notation and suggests a viewing window that frames all the interesting features.

How to use the Polynomial Grapher calculator

  1. Enter the values for your problem into the input fields.
  2. Read the result — it updates instantly as you type.
  3. Check the formula and the visual explanation to follow how the answer was found.
  4. Copy the page URL to share the exact calculation.

Common examples

  • 1, −6, 11, −6 (x³ − 6x² + 11x − 6) → roots 1, 2, 3; max (1.4226…, 0.3849…), min (2.5774…, −0.3849…)
  • 1, 0, −2, 0 (x³ − 2x) → roots 0 and ±1.414213562, turning points at x = ±0.8164965809
  • 1, 0, −4 (x² − 4) → roots ±2 with a single minimum at (0, −4)

Frequently asked questions

How does the Polynomial Grapher calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Polynomial Grapher calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.