Sketching Polynomial Functions

We want to determine the end behavior of the graph of the given polynomial function. To do so, we will pay close attention to the leading term $a{x}^n,$ where $a$ is the leading coefficient and $n$ is the degree of the polynomial. Let's consider the given polynomial function. Note that it is already written in standard form. $$\begin{array}{c}f(x)=5x^7+11x^4-7x^2-4x-17\end{array}$$ We can see above that the leading coefficient is $5$ and the degree is $7.$ Let's now see how the leading coefficient and degree affect the end behavior of the graph of a polynomial function.

Since $5>0$ and $7$ is an odd number, the end behavior of the given function is down and up. $$\begin{array}{lcl}f(x)\to \text{-}\infty & \text{as}& x\to \text{-}\infty \\ f(x)\to \infty & \text{as}& x\to \infty \end{array}$$