Chapter 3.5, Problem 21E

To determine

To find: the end behavior of the given function.

Answer to Problem 21E

  $\begin{array}{l}y\to -\infty \text{ as }x\to \infty \\ y\to -\infty \text{ as }x\to -\infty \end{array}$

Explanation of Solution

Given information:

  $y=8-2x^3-2x^4$

Calculation:

Make a table for $x$ and y as follows:

$x$	$y$
$-10000$	$-1.9998\times {10}^{16}$
$-1000$	$-1.998\times {10}^{12}$
$-100$	$-1.98\times {10}^8$
0	8
$100$	$-2.02\times {10}^8$
$1000$	$-2.002\times {10}^{12}$
$10000$	$-2.0002\times {10}^{16}$

The highest power of x is even and is negative, so as x approaches infinity the function approaches negative infinity and as x approaches negative infinity the graph approaches negative infinity. Thus, the end behavior is $y\to -\infty \text{ as }x\to \infty$ and $y\to -\infty \text{ as }x\to -\infty$ .