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Calculus · Function analysis

Polynomial Function Study Calculator

Study quadratic and cubic polynomials through real zeros, derivatives, critical points, monotonic intervals, intercepts, and end behavior.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Quadratic and cubic real polynomials

Critical points

f′(x)=0

Monotonicity

f′(x)>0 increasingf′(x)<0 decreasing

Intercepts

f(x)=0y-intercept=f(0)
Calculus · Function study

Study a quadratic or cubic polynomial

Inputs and displayed decimal results use at most three decimal places. Full internal precision is retained during calculation.

f′(x)=0 identifies critical points

Enter a real polynomial with a nonzero leading coefficient.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The controlled expression set contains real quadratic and cubic polynomials with a nonzero leading coefficient. Critical points come from the derivative, and monotonicity is classified between consecutive critical values.

Worked examples

See the method in practice

01

Quadratic

For f(x)=x²−4, the real zeros are −2 and 2, and the minimum occurs at (0,−4).

02

Cubic

For f(x)=x³−3x, critical points occur at x=−1 and x=1, separating three monotonic intervals.

Questions

Frequently asked

Why only degrees two and three?

They cover the standard derivative-based study while keeping every classification readable and verifiable.

What is a critical point?

Here it is a point in the polynomial domain where f′(x)=0 and monotonic behavior may change.

How is end behavior determined?

It follows from the polynomial degree parity and the sign of its leading coefficient.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Calculus Volume 1 — Derivatives and the Shape of a Graph

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology