Polynomial Graphing Calculator

Enter your polynomial function using standard notation (e.g. 3x^3 - 2x^2 + x - 5) and set your x-axis range to plot the curve. The calculator evaluates the polynomial across your range, identifies real roots (x-intercepts), finds critical points (local maxima and minima), and displays a line chart of the curve. Adjust the x minimum and x maximum values to zoom in or out on any region of interest.

Select the highest power of x in your polynomial.

Coefficient of x⁵ (used for degree 5 only).

Coefficient of x⁴ (used for degree 4+).

Leading coefficient for the highest degree term.

Left boundary of the graph x-axis.

Right boundary of the graph x-axis.

Results

Number of Real Roots Found

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Real Roots (x-intercepts)

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Critical Points Found

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Y-Intercept f(0)

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Polynomial Expression

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Polynomial Function Graph

Results Table

Frequently Asked Questions

How do I enter a polynomial into this calculator?

Select your polynomial degree from the dropdown, then fill in the coefficient fields (a, b, c, d, e) corresponding to each power of x. For example, for f(x) = 2x³ - 3x + 1, set degree to 3, a = 2, b = 0, c = -3, d = 1. Leave coefficients as 0 if that term does not appear.

What is a root (zero) of a polynomial?

A root or zero of a polynomial f(x) is a value of x where f(x) = 0 — the points where the curve crosses or touches the x-axis. A polynomial of degree n has at most n real roots. The calculator finds these by scanning the plotted range for sign changes.

What are critical points of a polynomial?

Critical points are x-values where the derivative f'(x) = 0. They correspond to local maxima (peaks) and local minima (valleys) of the curve. The calculator numerically approximates these turning points within your specified x-range.

How accurate are the roots and critical points?

The calculator uses numerical methods scanning 10,000 sample points across your x-range, then applies bisection to refine root locations to about 4 decimal places. For precise symbolic results, a CAS (computer algebra system) such as Wolfram Alpha may be preferred.

Can I graph polynomials of degree higher than 5?

Currently the calculator supports polynomials up to degree 5. For higher-degree polynomials, tools like Desmos or Symbolab offer free-form expression entry that supports arbitrary degrees.

Why does the graph look flat or all zeros?

This usually means the leading coefficient is set to 0 while the degree is set higher. Make sure the coefficient for the highest-degree term is non-zero. Also check that your x-range is wide enough to show the interesting behavior of the polynomial.

What is the y-intercept and how is it calculated?

The y-intercept is the value of f(0) — the point where the curve crosses the y-axis. For any polynomial, this equals the constant term (d for cubic, e for quartic/quintic). It tells you the function's value when x = 0.

What is the difference between a local and global maximum?

A local maximum is a point higher than its immediate neighbors, while a global maximum is the highest point over the entire domain. For polynomials with odd degree, there is no global maximum or minimum. Critical points found here are local extrema within your plotted x-range.

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