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^4+x^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 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.

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 4, like f(x) = -x^4 + x^2, indicates that the graph will have specific characteristics, such as symmetry and the potential for multiple turning points, which are influenced by the degree.

End Behavior of a Polynomial

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^4 + x^2, the negative leading coefficient and even degree suggest that both ends of the graph will fall, indicating that as x approaches ±∞, f(x) will approach -∞.