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Polynomial functions are mathematical expressions that consist of variables raised to whole number powers and coefficients. These functions are foundational in algebra and appear in many different contexts within mathematics and science. A polynomial function can be written in the form
\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0 \)
where
\( a_0, a_1, \ldots, a_n \) are constants with
\( a_n \eq 0 \) (except in the case of the zero polynomial), and
\( n \) is a non-negative integer. The term \( a_nx^n \) is called the leading term because it has the highest power, and \( a_n \) is the leading coefficient. Polynomial functions are continuous everywhere and are differentiable wherever they are defined. Their graphs paint a picture that helps us understand the behavior of these functions, especially how they rise and fall, which is critical in understanding the nature of these functions.

The end behavior of graphs refers to the tendency of the plots of functions to approach certain bounds or directions as
\( x \) heads towards positive or negative infinity. This behavior is particularly important when analyzing polynomial functions because it helps provide a visual snapshot of the function's long-term tendencies. For instance, in the example
\( f(x)=-5x^{4}+7x^{2}-x+9 \),
the Leading Coefficient Test is used to determine the graph's behavior at the extreme ends. In general, if a polynomial's degree is even, its ends will either both rise or both fall. If the degree is odd, one end will rise while the other falls. The actual direction in each case also depends on the sign of the leading coefficient. Thus, by understanding the end behavior, we can predict and graph the overall shape of polynomial functions. This concept is integral to accurately sketching the curves of these functions without having to plot every single point.

The degree of a function, in polynomials referred to as the highest power of the variable, gives important clues about the function's graph. It plays a pivotal role in determining many characteristics of the function, including the number of potential real roots, the shape of the graph, and the end behavior of the function. The degree can inform us how complex the function might be due to the fact that, generally, a higher degree indicates more fluctuations between rises and falls in the graph. In the given example
\( f(x)=-5x^{4}+7x^{2}-x+9 \),
the degree is 4, which is an even number, suggesting that the end behavior would be symmetric, with both ends of the graph either going up or down. Knowing the degree of the polynomial function also assists in potential factoring, finding derivatives, and integrals of the function. In practice, identifying the degree is a fundamental step in graph sketching and in the subsequent application of calculus tools.