Question: Describe the end behavior of the graph of the polynomial function. \[ f(x) = -8x^3 + 4x^2 - 1 \]…

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

\[ f(x) = -8x^3 + 4x^2 - 1 \]

Choose the correct answer below.

A. \(\uparrow \downarrow\)

B. \(\downarrow \uparrow\)

C. \(\downarrow \downarrow\)

D. \(\uparrow \uparrow\)

Solution

To determine the end behavior of the polynomial function \( f(x) = -8x^3 + 4x^2 - 1 \), we look at the leading term, which is \(-8x^3\). Since the leading coefficient is negative and the degree of the polynomial (3) is odd, the end behavior is as follows: - As \( x \to \infty \), \( f(x) \to -\infty \) - As \( x \to -\infty \), \( f(x) \to \infty \) Thus, the correct end behavior is: as \( x \) goes to positive infinity, \( f(x) \) goes down, and as \( x \) goes to negative infinity, \( f(x) \) goes up. The correct answer is C: \(\downarrow\uparrow\).