Polynomial Graphing Calculator
Enter the coefficients of any polynomial up to degree 4, pick an x-value to evaluate, and instantly see the graph, real roots, and (for quadratics) the vertex.
Degree
Value of the polynomial at the given x
- 1
Term 1x²
1 × 2² = 4 - 2
Term -4
-4 = -4 - 3
y = f(x)
0
How does this calculator work?
Enter coefficients for a polynomial up to degree 4 and a point x. The calculator evaluates y = f(x), plots the curve from −8 to 8, finds real roots by bisection, and shows the vertex for quadratics. Leading coefficient must be non-zero. Roots are located in the visible window only.
Formula
How this is calculated
A polynomial is a sum of terms of the form a·xⁿ. This calculator lets you define up to a quartic (degree 4) by entering each coefficient in descending power order, then evaluates the function at any point x and draws the curve over the range −8 to 8.
Real roots — where the curve crosses the x-axis — are found by scanning for sign changes and then refining each crossing with bisection to four-decimal accuracy. For quadratics the exact vertex is derived analytically from xᵥ = −b⁄(2a), which gives the minimum or maximum of the parabola. For higher-degree polynomials, multiple local minima and maxima exist but are not individually labelled.
The graph is plotted with 120 evenly spaced sample points, so very sharp turns (e.g. narrow cusps near roots) may appear slightly smoothed. Values larger than ±1 000 000 are clipped to keep the axis scale readable. Change any coefficient and the curve, roots, and evaluation result update instantly.
Frequently asked questions
Roots are marked with red dots where the polynomial crosses y = 0. They are also listed numerically below the chart; if the polynomial has no real roots (e.g. x² + 1), the field shows "None in [−10, 10]".
Yes — you can still graph it; the curve simply never crosses the x-axis. The calculator only shows real roots; complex roots occur in conjugate pairs and require the quadratic formula with a negative discriminant.
For a quadratic ax² + bx + c, the vertex is the single turning point. Its x-coordinate is −b⁄(2a) and its y-coordinate is f(−b⁄(2a)). If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
Also known as
TG we-Calculate Editorial Team. (2026). Polynomial Graphing Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polynomial-graphing-calculator
TG we-Calculate Editorial Team. "Polynomial Graphing Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/polynomial-graphing-calculator.
TG we-Calculate Editorial Team, "Polynomial Graphing Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polynomial-graphing-calculator
@misc{wecalculate_polynomial_graphing_calculator, title = {Polynomial Graphing Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polynomial-graphing-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?