Polynomial Functions

Degree, roots, and end behavior — all visible on one graph

Roller coaster curves, economic models, signal processing — when a straight line or parabola is not enough, polynomials step in. A polynomial is a sum of terms, each being a constant times a power of x: f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0. The highest power is called the degree, and its coefficient an is the leading coefficient.

The degree and leading coefficient control the end behavior — what happens as x goes to positive or negative infinity. An odd-degree polynomial rises on one end and falls on the other; an even-degree polynomial rises (or falls) on both ends.

In this lesson you'll explore polynomials of degree 2, 3, and 4 with sliders, discover how the number of roots connects to the degree, and see how the leading coefficient flips the curve.

Related Topics

Quadratic EquationsPower FunctionsFactoring & Roots

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What is a polynomial function?
A polynomial function is a function of the form f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, where each ai is a constant and n is a non-negative integer called the degree. Examples: x² + 3x − 1 (degree 2), 2x³ − x (degree 3).
What is the degree of a polynomial?
The degree is the highest power of x that appears with a non-zero coefficient. It determines the maximum number of roots and the general shape: degree 2 makes a parabola, degree 3 makes an S-curve, degree 4 can have a W-shape.
What is end behavior?
End behavior describes what happens to f(x) as x → ∞ and x → −∞. For even degree with positive leading coefficient, both ends go UP. For odd degree with positive leading coefficient, the left end goes DOWN and the right end goes UP. A negative leading coefficient flips everything.
How many roots can a polynomial have?
A polynomial of degree n has at most n real roots (x-intercepts). It can have fewer if some roots are complex (non-real). For example, x² + 1 has degree 2 but zero real roots because x² + 1 > 0 for all real x.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.