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Precalculus serves as the foundation that bridges basic algebra and calculus, preparing students for the more complex and abstract concepts encountered in calculus. It generally involves the study of functions, their properties, and how they can be manipulated. A key concept within precalculus is the average rate of change, which quantifies how much a function's output changes on average over a specified interval.

The average rate of change is particularly useful because it represents the steepness of a secant line connecting two points on a graph of the function. In simpler terms, it's akin to the average speed of a car over a road trip: even if the car speeds up and slows down, you can calculate its overall performance between two points. Mastering this concept is vital, as it lays the groundwork for understanding instantaneous rates of change in calculus, which are represented by derivatives.

Understanding function behavior is an essential part of precalculus, and it involves examining how functions behave as their inputs change. This can refer to the increasing and decreasing nature of the function, the concavity, and the presence of any maximum or minimum points.

In the context of the exercise, determining the average rate of change is a method for analyzing the function's behavior over an interval, which in this case is from \(x_1\) to \(x_2\). While the average rate of change is a singular value, it gives insight into the overall trend during the interval — if it's positive, the function is generally increasing, and if negative, it's decreasing. The pronounced negative average rate of change of -18 indicates that the function \(f(x)=-x^{3}+6x^{2}+x\) is decreasing quite rapidly on the interval between \(x_1 = 1\) and \(x_2 = 6\).

Polynomial functions, such as \(f(x)=-x^{3}+6x^{2}+x\) used in the exercise, are comprised of one or more terms with non-negative integer exponents and real coefficients. The degree of the polynomial, indicated by the highest exponent, provides information about the function’s shape and the number of its roots.

For a cubic polynomial like the one in our example, which is a third-degree polynomial, we can expect up to three real roots and two turning points. The coefficients of the terms, such as the '-1' in front of \(x^{3}\), influence the rate at which the function grows or decreases. Notably in this exercise, the negative leading coefficient indicates that the function's ends go in opposite directions, with the right end turning downwards which is a typical behavior of negative leading coefficient cubics.