Cubic functions

Definition

A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$. These functions can have up to three real roots and exhibit distinct characteristics such as points of inflection.

5 Must Know Facts For Your Next Test

1. Cubic functions can have one or three real roots.
2. The graph of a cubic function has at most two turning points.
3. There is always one point of inflection in the graph of a cubic function.
4. The end behavior of cubic functions depends on the sign of the leading coefficient $a$; if $a > 0$, it rises to the right and falls to the left, and if $a < 0$, it falls to the right and rises to the left.
5. Cubic functions are invertible on intervals where they are strictly increasing or decreasing.

Review Questions

• What is the general form of a cubic function?
• How many turning points can a cubic function have at most?
• What determines the end behavior of a cubic function?

Related terms

Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Roots: The values for which a given polynomial equates to zero. Also known as solutions or zeros.

Inflection Point: A point on the curve at which the sign of the curvature (the concavity) changes.