Question

question 1-1
drop down: 1 point for correct answer only.
what is the end behavior of the graph of the function $f(x)$?
$f(x) = -2x^2 + 6x^3 - x$
as $x \to -\infty$, $f(x) \to \boxed{}$ and as $x \to +\infty$, $f(x) \to \boxed{}$

Explanation:

Step1: Identify leading term

The leading term of $f(x) = -2x^2 + 6x^3 - x$ is $6x^3$, which has degree 3 (odd) and positive coefficient 6.

Step2: Analyze end behavior for $x\to-\infty$

For odd-degree leading term with positive coefficient, as $x\to-\infty$, $x^3\to-\infty$, so $6x^3\to-\infty$. Thus $f(x)\to-\infty$.

Step3: Analyze end behavior for $x\to+\infty$

For odd-degree leading term with positive coefficient, as $x\to+\infty$, $x^3\to+\infty$, so $6x^3\to+\infty$. Thus $f(x)\to+\infty$.

Answer:

As $x \to -\infty$, $f(x) \to -\infty$ and as $x \to +\infty$, $f(x) \to +\infty$