Chapter 2.2, Problem 47E

Solution

To graph: The function $f\left(x\right)=x^5+x^2-4$ and to describe its end behaviour

Graph is as follows.

The end behavior of the graph is such that $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to +\infty$ as $x\to +\infty$ .

Given information:

Function is $f\left(x\right)=x^5+x^2-4$

Graph:

Make a table of values that satisfy the given function. For this, select some values for $x$ and find the corresponding values of $f\left(x\right)$ or $y$ .

  $\begin{array}{cccccc}x& -2& -1& 0& 1& 2\\ y& -32& -4& -4& -2& 32\end{array}$

Plot the points and connect them using a smooth curve.

The degree of the function is odd and the leading coefficient is positive. The end behavior of the graph is such that $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to +\infty$ as $x\to +\infty$ .