To determine the end behavior of the function $f(x)$ from the graph provided, we observe how the function behaves as $x$ approaches positive infinity ($+\infty$) and negative infinity ($-\infty$).

Observations from the graph:

• As $x \to -\infty$, the graph is heading downward, which means $f(x) \to -\infty$.
• As $x \to +\infty$, the graph is heading upward, meaning $f(x) \to +\infty$.

Conclusion:

• The left-end behavior is $\lim_{x \to -\infty} f(x) = -\infty$.
• The right-end behavior is $\lim_{x \to +\infty} f(x) = +\infty$.

Thus, the end behavior of the function is:

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

Would you like further details on this explanation, or do you have any other questions?

Here are some follow-up questions to deepen understanding:

1. How can you determine the degree of a polynomial based on its end behavior?
2. What does the leading coefficient of the polynomial tell you about its graph's direction?
3. How do turning points help identify key features of polynomial functions?
4. How do you differentiate between even-degree and odd-degree polynomial end behaviors?
5. What are some common end behaviors for cubic or quartic functions?

Tip: The degree and the sign of the leading coefficient of a polynomial control its end behavior. For odd-degree polynomials, the ends go in opposite directions, while for even-degree polynomials, they go in the same direction.