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How are cubic functions identified from their equations?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

A cubic function is identified by its status as a polynomial equation in which the highest exponent of the variable is three.12

In standard form, these equations are expressed as:
$f (x) = a x^{3} + b x^{2} + c x + d$

To identify a cubic function from an equation, look for the following characteristics:
* Highest Degree: The term with the highest exponent must be exactly 3.
* Variable Power: No variable in the equation can be raised to a power greater than 3.
* Coefficient Constraint: The coefficient of the $x^{3}$ term (represented by $a$ ) must be non-zero, otherwise the equation would be of a lower degree (such as quadratic or linear).

These functions represent cubic curves when graphed, typically characterized by their "S" shape and at least one inflection point.2