Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -77x^2 + 224 - 7x^3 - 140x$
answer attempt 1 out of 2
$\circ$ as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to -\infty$
$\circ$ as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to \infty$
$\circ$ as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to -\infty$
$\circ$ as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to \infty$
submit

Explanation:

Step1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x) = -77x^2 + 224 - 7x^3 - 140x \), we reorder the terms by degree: \( f(x) = -7x^3 - 77x^2 - 140x + 224 \). The leading term is \( -7x^3 \), with degree 3 (odd) and leading coefficient \( -7 \) (negative).

Step2: Analyze End Behavior

For a polynomial, the end behavior is determined by the leading term:

• If the degree is odd and the leading coefficient is negative:
• As \( x \to \infty \), \( x^3 \to \infty \), so \( -7x^3 \to -\infty \) (since the coefficient is negative). Thus, \( f(x) \to -\infty \) as \( x \to \infty \).
• As \( x \to -\infty \), \( x^3 \to -\infty \) (because odd power of negative is negative), so \( -7x^3 = -7 \times (-\infty) = \infty \) (negative times negative is positive). Thus, \( f(x) \to \infty \) as \( x \to -\infty \).

Answer:

as \( x \to \infty, y \to -\infty \) and as \( x \to -\infty, y \to \infty \) (the last option in the list, which is: as \( x \to \infty, y \to -\infty \) and as \( x \to -\infty, y \to \infty \))