A polynomial function is a function that can be expressed in the form f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are coefficients, and n is a non-negative integer representing the degree of the polynomial.
Polynomial functions have several key properties:
Degree: The degree of a polynomial is the highest power of x in the function. It determines the overall shape and behavior of the graph.
Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree. It affects the steepness of the graph.
End Behavior: The end behavior of a polynomial function is determined by the leading term. If the leading term has a positive coefficient, the graph will rise on both ends. If the leading term has a negative coefficient, the graph will fall on both ends.
Let's consider an example. Consider the polynomial function f(x) = 2x^3 - 5x^2 + 3x - 1. This is a cubic function with a degree of 3. The leading coefficient is 2, and since it is positive, the graph will rise on both ends.
In summary, polynomial functions are characterized by their degree, leading coefficient, and end behavior. Understanding these properties helps us analyze and graph polynomial functions effectively.