Final answer:
A cubic function is represented by the equation f(x) = ax^3 + bx^2 + cx + d, known as the standard form. It can have up to three real roots, up to two turning points, and an 'S' shaped or symmetrical graph. The leading coefficient determines the end behavior of its graph.
Step-by-step explanation:
A cubic function generally has the form of the mathematical expression f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is not equal to 0. This form is called the standard form of a cubic function. Cubic functions are a type of polynomial function, specifically, a third-order polynomial. Other characteristics of cubic functions include the fact that they can have up to three real roots, can change direction up to two times (these are called turning points), and their graphs have a characteristic 'S' shape or a symmetric curve about a point, depending on the nature of the coefficients.
Cubic functions are different from quadratic functions, which are second-order polynomials having the form quadratic function = at^2 + bt + c. While a quadratic function's graph is a parabola that opens upwards or downwards, a cubic function's graph can have more complex behavior and inflection points.
When analyzing the graph of a cubic function, one might consider the function's leading coefficient (a), which influences the end behavior of the graph. If 'a' is positive, the graph will extend to positive infinity as x approaches positive infinity and to negative infinity as x approaches negative infinity. Conversely, if 'a' is negative, the graph will extend to negative infinity as x increases and positive infinity as x decreases.