Cubic functions

Definition

Cubic functions are polynomial functions of degree three, typically written in the form $f(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$. These functions can have up to three real roots and exhibit various shapes including an S-curve.

5 Must Know Facts For Your Next Test

1. Cubic functions can have one, two, or three real roots.
2. The end behavior of a cubic function depends on the leading coefficient: if $a > 0$, as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$; if $a < 0$, the behavior is reversed.
3. Cubic functions always have at least one real root due to the Intermediate Value Theorem.
4. The graph of a cubic function can have up to two turning points (local maxima and minima).
5. Inverses of cubic functions exist if the function is strictly increasing or decreasing.

Review Questions

• What is the general form of a cubic function?
• How many turning points can a cubic function's graph have?
• Describe how you determine the end behavior of a cubic function based on its leading coefficient.

Related terms

Polynomial Functions: Functions that involve terms consisting of variables raised to non-negative integer powers and coefficients. They are expressed in the form $P(x) = a_n x^n + ... + a_1 x + a_0$.

Degree: The highest power of the variable in a polynomial. For example, in a cubic function, the degree is three.

Intermediate Value Theorem:

A theorem stating that for any continuous function on an interval [a, b], if a value y lies between f(a) and f(b), then there exists some c in (a, b) such that f(c) = y.