Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=4x^7-x^5+x^3-1

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) = 4x^7 - x^5 + x^3 - 1, the leading term is 4x^7, indicating that as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will approach negative infinity.

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 polynomial's end behavior and the number of possible turning points. In the given function f(x) = 4x^7 - x^5 + x^3 - 1, the degree is 7, which is odd, suggesting that the ends of the graph will go in opposite directions: one end will rise to infinity while the other will fall to negative infinity.

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 odd, the graph will rise to the right and fall to the left. Conversely, if the leading coefficient is negative, the graph will fall to the right and rise to the left. In the case of f(x) = 4x^7, the positive leading coefficient confirms that the graph will rise to the right and fall to the left.