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.
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The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$.
Cubic functions can have one, two, or three real roots.
The graph of a cubic function typically has an S-shaped curve with one inflection point.
Critical points of a cubic function occur where the first derivative $f'(x)$ equals zero.
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.