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

Key Concepts

End Behavior of Polynomials

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) = 5x^5 + 2x^3 - 3x + 4, the leading term is 5x^5, indicating that as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will approach negative infinity.

Leading Coefficient Test

The leading coefficient test helps predict the end behavior of a polynomial function based on the sign and degree of the leading term. 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 and the degree is odd, the graph will fall to the right and rise to the left. This test is crucial for sketching the overall shape of the polynomial graph.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It plays a significant role in determining the number of turning points and the overall shape of the graph. For instance, a polynomial of degree 5, like f(x) = 5x^5 + 2x^3 - 3x + 4, can have up to 4 turning points and will exhibit specific end behaviors based on its degree and leading coefficient.