In Exercises 10–13, 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).] f(x) = -x^3 + x^2 + 2x

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of polynomial functions based on the sign and degree of the leading term. For a polynomial of the form f(x) = ax^n, where 'a' is the leading coefficient and 'n' is the degree, the test states that if 'n' is even, the ends of the graph will either both rise or both fall, depending on the sign of 'a'. If 'n' is odd, one end will rise while the other falls, again determined by the sign of 'a'.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It plays a crucial role in determining the shape and end behavior of the graph. For example, a polynomial of degree 3, like f(x) = -x^3 + x^2 + 2x, will have a characteristic 'S' shape, with one end going to positive infinity and the other to negative infinity, influenced by the leading coefficient.

End Behavior of Polynomials

End behavior refers to the behavior of the graph of a polynomial function as the input values (x) approach positive or negative infinity. Understanding end behavior helps predict how the graph will behave far away from the origin. For instance, in the case of f(x) = -x^3 + x^2 + 2x, the negative leading coefficient indicates that as x approaches positive infinity, f(x) will approach negative infinity, and as x approaches negative infinity, f(x) will approach positive infinity.