Polynomial Graphing Calculator
Enter the coefficients of any polynomial from linear (degree 1) through quintic (degree 5) and this calculator graphs the curve, finds all real roots, identifies local maxima and minima, locates inflection points, and describes the end behavior. Results update as you type, with a step-by-step breakdown of the key properties.
What is a polynomial and why graph it?
A polynomial is a sum of terms of the form a times x raised to a non-negative integer power. The highest power present is called the degree. Graphing a polynomial gives an immediate picture of where the function is positive or negative, where it crosses the x-axis (its real roots), where it peaks or dips (its local extrema), and how it behaves for very large or very small values of x. These properties are central to pre-calculus, calculus, and many applied fields from physics to economics.
How to read the results
Real roots are the x-values where the curve crosses or touches the x-axis, found by solving P(x) = 0. The y-intercept is simply P(0), the constant term of your polynomial. Critical points are where the first derivative P'(x) equals zero; the second derivative test tells you whether each one is a local maximum (P''(x) < 0) or a local minimum (P''(x) > 0). Inflection points are where the second derivative changes sign, marking a shift from concave-up to concave-down curvature or vice versa. End behavior describes whether the two "arms" of the curve rise or fall as x goes to positive and negative infinity, which depends only on the degree and the sign of the leading coefficient.
How roots are computed
This calculator finds real roots numerically using a two-stage method. First it samples the polynomial at 800 evenly spaced points across the display range and detects every sign change, which brackets a root. Then it refines each bracket with 50 steps of bisection, achieving precision to roughly 12 decimal places. This approach reliably finds all real roots within the displayed window even for higher-degree polynomials, though complex (imaginary) roots are not shown because they do not appear on a real-number graph.
End behavior rules
For even-degree polynomials both arms point in the same direction: up if the leading coefficient is positive, down if it is negative. For odd-degree polynomials the arms point in opposite directions: a positive leading coefficient means the left arm falls to negative infinity and the right arm rises to positive infinity, while a negative leading coefficient flips that. These rules follow from the fact that for very large x the highest-power term completely dominates the value of the polynomial.
Polynomial degree quick-reference
| Degree | Name | Max real roots | Max turning points | Max inflection points |
|---|---|---|---|---|
| 1 | Linear | 1 | 0 | 0 |
| 2 | Quadratic | 2 | 1 | 0 |
| 3 | Cubic | 3 | 2 | 1 |
| 4 | Quartic | 4 | 3 | 2 |
| 5 | Quintic | 5 | 4 | 3 |
Maximum number of roots, turning points and inflection points by degree.
Frequently asked questions
How do I enter a polynomial like 2x^3 - 3x + 1?
Select degree 3, then set a3 to 2, a2 to 0, a1 to -3, and a0 to 1. Each coefficient field corresponds to the term with that power of x. Unused powers should be set to 0.
Why are some roots not shown?
The calculator searches for real roots in the range [-20, 20]. Roots outside that range will not appear. Also, complex roots (involving the square root of a negative number) are not shown because they do not correspond to points on a standard x-y graph.
What is the difference between a critical point and a root?
A root is where the curve crosses or touches the x-axis, meaning P(x) = 0. A critical point is where the curve has a horizontal tangent, meaning the first derivative P'(x) = 0. A critical point can be a local peak, a local valley, or occasionally a flat inflection point. The two concepts are independent and can occur at the same x-value but usually do not.
What does the end behavior tell me?
End behavior describes what happens to P(x) as x grows without bound in either direction. For an even-degree polynomial with a positive leading coefficient both ends rise to positive infinity, giving a U-shaped or W-shaped curve overall. An odd-degree polynomial always has one end rising and the other falling, producing an S-shaped overall shape.
Can I graph a polynomial with fractional coefficients?
Yes. Each coefficient field accepts decimal values. For example, to graph (1/2)x^2 - (3/4)x + 2, enter 0.5 for a2, -0.75 for a1, and 2 for a0 with degree 2 selected.
Why does a degree-4 polynomial sometimes look like a degree-2?
If the degree-4 coefficient (a4) is very small relative to the others, the quartic term has little visible effect in the displayed window. Similarly, if a4 is 0 the polynomial is effectively quadratic even though degree 4 is selected. The calculator flags this case and you should adjust the degree or increase a4.