Question

which describes the end behavior of the graph of the function $f(x) = -8x^4 - 2x^3 + x$? (1 point) $\bigcirc$ $f(x) \to \infty$ as $x \to -\infty$ and $f(x) \to \infty$ as $x \to \infty$ $\bigcirc$ $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to -\infty$ as $x \to \infty$ $\bigcirc$ $f(x) \to \infty$ as $x \to -\infty$ and $f(x) \to -\infty$ as $x \to \infty$ $\bigcirc$ $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to \infty$ as $x \to \infty$

Explanation:

Step1: Identify the leading term

The function is \( f(x) = -8x^4 - 2x^3 + x \). The leading term is the term with the highest degree, which is \( -8x^4 \).

Step2: Analyze the degree and leading coefficient

The degree of the leading term is 4 (even), and the leading coefficient is -8 (negative).
For a polynomial function with an even degree:

• If the leading coefficient is positive, as \( x \to \pm\infty \), \( f(x) \to \infty \).
• If the leading coefficient is negative, as \( x \to \pm\infty \), \( f(x) \to -\infty \).

Since the degree is even (4) and the leading coefficient is negative (-8), as \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to \infty \), \( f(x) \to -\infty \).

Answer:

\( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to -\infty \) as \( x \to \infty \) (the second option: \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to -\infty \) as \( x \to \infty \))