Describe the end behavior of the graph of the polynomial function.
\[
f(x)=2+3 x-4 x^{2}-5 x^{10}
\]
Answer
\(\boxed{\text{The end behavior of the graph of the polynomial function } f(x)=2+3 x-4 x^{2}-5 x^{10} \text{ is that as } x \text{ approaches positive or negative infinity, } f(x) \text{ approaches negative infinity.}}\)
Step 1 ：The end behavior of a polynomial function is determined by the degree and the leading coefficient of the polynomial. The degree of the polynomial is the highest power of x, and the leading coefficient is the coefficient of the term with the highest power of x.
Step 2 ：In the given polynomial function \(f(x)=2+3 x-4 x^{2}-5 x^{10}\), the degree of the polynomial is 10 and the leading coefficient is -5.
Step 3 ：If the degree of the polynomial is even, as it is in this case, then the ends of the graph will either both point upwards or both point downwards. If the leading coefficient is positive, then the ends of the graph will point upwards. If the leading coefficient is negative, then the ends of the graph will point downwards.
Step 4 ：Since the degree of this polynomial is even and the leading coefficient is negative, the ends of the graph will both point downwards.
Step 5 ：\(\boxed{\text{The end behavior of the graph of the polynomial function } f(x)=2+3 x-4 x^{2}-5 x^{10} \text{ is that as } x \text{ approaches positive or negative infinity, } f(x) \text{ approaches negative infinity.}}\)
