Question: Describe the end behavior of the graph of the polynomial function. \[ f(x) = 5 + 4x - 3x^2 -…

Describe the end behavior of the graph of the polynomial function.

\[ f(x) = 5 + 4x - 3x^2 - 7x^{10} \]

Choose the correct answer below.

A. \(\nearrow \searrow\)

B. \(\searrow \nearrow\)

C. \(\searrow \searrow\)

D. \(\nearrow \nearrow\)

Solution

To determine the end behavior of the polynomial function \( f(x) = 5 + 4x - 3x^2 - 7x^{10} \), we consider the term with the highest degree, which is \(-7x^{10}\). Since the highest degree term is \(-7x^{10}\): 1. The degree of the polynomial is 10, which is even. 2. The leading coefficient is \(-7\), which is negative. For polynomials with an even degree and a negative leading coefficient: - As \( x \to \infty \), \( f(x) \to -\infty \). - As \( x \to -\infty \), \( f(x) \to -\infty \). Thus, the graph of the polynomial falls in both directions. The correct answer is option C.