Chapter 2.2, Problem 31E

Solution

To Describe: The end behaviour of the graph of the polynomial function $f\left(x\right)=x^7+3x^4-x^2$ .

The end behaviour of the polynomial $f\left(x\right)=x^7+3x^4-x^2$ is $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to \infty$ .

Given:

The polynomial is $f\left(x\right)=x^7+3x^4-x^2$ .

Description:

Consider the polynomial, $f\left(x\right)=x^7+3x^4-x^2$ .

The leading term of the polynomial is $x^7$ .

The degree is 7 which is odd and the leading coefficient is $1$ which is positive.

This indicates that $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to \infty$ .

Conclusion:

Therefore, the end behaviour of the polynomial $f\left(x\right)=x^7+3x^4-x^2$ is $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to \infty$ .