College Algebra, Chapter 4, 4.2, Section 4.2, Problem 46

Determine the end behaviour of the function
$\displaystyle P(x) = 2x^2 - x^{12}$. Compare the graphs of $P$ and $\displaystyle Q(x) = -x^{12}$ on large and small viewing rectangle.
The function $P(x)$ has an even degree of 12 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.