Question

the table shows ordered pairs of the function, $f(x) = -2x^3 - 3x^2 + 36x$.

x	f(x)
-6	108
-4	-64
-2	-68
0	0
2	44
4	-32
6	-324

which option accurately describes the ending behavior of $f(x)$?

• as $x$ approaches negative infinity, $f(x)$ approaches negative infinity. as $x$ approaches positive infinity, $f(x)$ approaches negative infinity.
• as $x$ approaches negative infinity, $f(x)$ approaches positive infinity. as $x$ approaches positive infinity, $f(x)$ approaches negative infinity.
• as $x$ approaches negative infinity, $f(x)$ approaches positive infinity. as $x$ approaches positive infinity, $f(x)$ approaches positive infinity.
• as $x$ approaches negative infinity, $f(x)$ approaches negative infinity. as $x$ approaches positive infinity, $f(x)$ approaches positive infinity.

Explanation:

Step1: Identify leading term

The function is $f(x) = -2x^3 - 3x^2 + 36x$. The leading term is $-2x^3$, which determines end behavior.

Step2: Analyze degree and leading coefficient

The degree (3) is odd, and the leading coefficient (-2) is negative.

Step3: Evaluate as $x\to-\infty$

For odd degree, negative leading coefficient: as $x\to-\infty$, $x^3\to-\infty$, so $-2x^3\to -2(-\infty) = +\infty$. Thus $f(x)\to+\infty$.

Step4: Evaluate as $x\to+\infty$

As $x\to+\infty$, $x^3\to+\infty$, so $-2x^3\to -2(+\infty) = -\infty$. Thus $f(x)\to-\infty$.

Step5: Match to options

This matches the behavior: As $x$ approaches negative infinity, $f(x)$ approaches positive infinity. As $x$ approaches positive infinity, $f(x)$ approaches negative infinity.

Answer:

As $x$ approaches negative infinity, $f(x)$ approaches positive infinity. As $x$ approaches positive infinity, $f(x)$ approaches negative infinity.