In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE> f(x)=(x−3)^2

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of polynomial functions based on the degree and leading coefficient of the polynomial. If the leading coefficient is positive and the degree is even, the ends of the graph rise; if the leading coefficient is negative, the ends fall. For odd degrees, the ends behave oppositely: they rise on one side and fall on the other, depending on the sign of the leading coefficient.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer, and 'a_n' is the leading coefficient. Understanding the structure of polynomial functions is crucial for analyzing their graphs and behaviors.

End Behavior of Graphs

End behavior refers to the behavior of the graph of a function as the input values approach positive or negative infinity. For polynomial functions, this behavior is determined by the degree and leading coefficient, which dictate whether the graph rises or falls at the ends. Analyzing end behavior helps in predicting how the graph will look and is essential for matching polynomial functions to their corresponding graphs.