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 first rewrite the given polynomial function in standard form. $$\begin{array}{c}f(x)=8x^5-4x^7+6x^2\iff f(x)=\text{-}4x^7+8x^5+6x^2\end{array}$$ We can see above that the leading coefficient is $\text{-}4$ 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 $\text{-}4<0$ and $7$ is an odd number, the end behavior of the given function is up and down. We can see above that as $x$ approaches negative infinity, $f(x)$ approaches positive infinity. As $x$ approaches positive infinity, $f(x)$ approaches negative infinity. $$\begin{array}{c}f(x)\to +\infty \text{ as }x\to \text{-}\infty \\ \text{and}\\ f(x)\to \text{-}\infty \text{ as }x\to +\infty \end{array}$$