Question

analyze the end behavior of the polynomial function.
$f(x) = -x^3 + 8x^2 - 12x$
$\bigcirc x \
ightarrow \infty, f(x) \
ightarrow \infty$
$x \
ightarrow -\infty, f(x) \
ightarrow \infty$
$\bigcirc x \
ightarrow \infty, f(x) \
ightarrow \infty$
$x \
ightarrow -\infty, f(x) \
ightarrow -\infty$
$\bigcirc x \
ightarrow \infty, f(x) \
ightarrow -\infty$
$x \
ightarrow -\infty, f(x) \
ightarrow -\infty$
$\bigcirc x \
ightarrow \infty, f(x) \
ightarrow -\infty$
$x \
ightarrow -\infty, f(x) \
ightarrow \infty$

Explanation:

Step1: Identify leading term

Leading term: $-x^3$

Step2: Check degree and sign

Degree = 3 (odd), leading coefficient = $-1$ (negative)

Step3: Determine end behavior

For odd degree, negative leading coefficient:
As $x \to \infty$, $f(x) \to -\infty$; as $x \to -\infty$, $f(x) \to \infty$

Answer:

$\boldsymbol{x \to \infty,\ f(x) \to -\infty}$
$\boldsymbol{x \to -\infty,\ f(x) \to \infty}$