Cubic function

Definition

A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$. The graph of a cubic function can have up to three real roots and two critical points.

5 Must Know Facts For Your Next Test

1. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$.
2. Cubic functions can have one, two, or three real roots.
3. The graph of a cubic function typically has an S-shaped curve with one inflection point.
4. Critical points of a cubic function occur where the first derivative $f'(x)$ equals zero.
5. The end behavior of a cubic function depends on the leading coefficient $a$: if $a > 0$, the graph rises to the right and falls to the left; if $a < 0$, it falls to the right and rises to the left.

Review Questions

• What is the general form of a cubic function?
• How many real roots can a cubic function have?
• Describe the end behavior of a cubic function when the leading coefficient is positive.

Related terms

Polynomial Function:

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

Critical Point: A point on a graph where the first derivative is zero or undefined, indicating potential local maxima, minima, or inflection points.

Inflection Point: A point on the curve where the concavity changes from concave up to concave down or vice versa.