Let's address both parts of the image you uploaded:

1. End Behavior of the Function:

The function given is $f(x) = -3x^5 + 2x^3 - 7x + 2$. To determine the end behavior, we focus on the term with the highest degree (the leading term), which is $-3x^5$.

• As $x \to +\infty$, the term $-3x^5$ dominates and becomes very negative, so $f(x) \to -\infty$.
• As $x \to -\infty$, the term $-3x^5$ also becomes negative because $(-x)^5 = -x^5$, so $f(x) \to +\infty$.

Thus, the end behavior is:

• $f(x) \to -\infty$ as $x \to +\infty$
• $f(x) \to +\infty$ as $x \to -\infty$

This corresponds to the second option in the list.

2. Evaluating the Function:

The second problem asks to evaluate the function $f(x) = -2x^3 - 3x^2 + x$ at $x = 2$.

Substitute $x = 2$ into the function: $f(2) = -2(2)^3 - 3(2)^2 + 2$ $f(2) = -2(8) - 3(4) + 2$ $f(2) = -16 - 12 + 2$ $f(2) = -26$

So, the value of the function at $x = 2$ is $-26$.

Summary:

1. The correct end behavior is: $f(x) \to +\infty$ as $x \to -\infty$, and $f(x) \to -\infty$ as $x \to +\infty$.
2. The value of the function at $x = 2$ is $-26$.

Would you like more details on any of these steps or additional questions? Here are some related questions for practice:

1. What would the end behavior be for $f(x) = 3x^4 + x^2 - 5$?
2. How does the degree and leading coefficient of a polynomial affect its end behavior?
3. If $f(x) = -x^6 + 4x^3 - x$, what are the end behaviors as $x \to \infty$ and $x \to -\infty$?
4. Evaluate $f(x) = x^3 - 4x^2 + 7x - 5$ at $x = -1$.
5. How do higher-degree terms in a polynomial influence its graph's shape?

Tip:

When determining the end behavior of a polynomial, always focus on the leading term (the term with the highest degree) because it dominates as $x$ grows large in either direction.