College Algebra, Chapter 4, 4.2, Section 4.2, Problem 42

Determine the end behaviour of the function
$\displaystyle P(x) = \frac{-1}{8} x^3+ \frac{1}{4} x^2 + 12x$. Compare the graphs of $P$ and $\displaystyle Q(x) = \frac{-1}{8} x^3$ on large and small viewing rectangle.
The function $P(x)$ has an odd degree and a negative leading coefficient. Thus, its end behaviour is $y \rightarrow \infty \text{ as } x \rightarrow -\infty \text{ and } y \rightarrow -\infty \text{ as } x \rightarrow \infty$.


The graph shows the function $P$ and $Q$ in progressively larger viewing rectangle. The larger the viewing rectangle, the more the graphs aline. This confirms that they have the same end behaviour.