Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=9x^6-3x^4+x^2-2

Key Concepts

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as the input values (x) approach positive or negative infinity. This behavior is primarily determined by the leading term of the polynomial, which is the term with the highest degree. For example, in the polynomial f(x) = 9x^6 - 3x^4 + x^2 - 2, the leading term is 9x^6, indicating that as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will approach infinity as well.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial expression. It plays a crucial role in determining the end behavior and the number of turning points of the graph. In the given polynomial f(x) = 9x^6 - 3x^4 + x^2 - 2, the degree is 6, which is even. This means that both ends of the graph will rise or fall together, depending on the sign of the leading coefficient.

Leading Coefficient Test

The Leading Coefficient Test helps predict the end behavior of a polynomial function based on the sign of the leading coefficient and the degree of the polynomial. If the leading coefficient is positive and the degree is even, the graph rises on both ends; if it is negative, the graph falls on both ends. In the case of f(x) = 9x^6 - 3x^4 + x^2 - 2, the leading coefficient is positive (9), confirming that the graph will rise on both ends.