From the graph shown in the uploaded image, the end behavior of the graph can be determined by observing what happens to $f(x)$ as $x \to \infty$ (positive infinity) and $x \to -\infty$ (negative infinity).

• As $x \to \infty$: The graph is increasing towards $+\infty$, meaning $f(x) \to +\infty$.
• As $x \to -\infty$: The graph is decreasing towards $-\infty$, meaning $f(x) \to -\infty$.

Correct Answer:

"As $x \to \infty$, $f(x) \to +\infty$. As $x \to -\infty$, $f(x) \to -\infty$."

This matches the option:

• "As $x \to \infty$, $f(x)$ approaches positive infinity. As $x \to -\infty$, $f(x)$ approaches negative infinity."

Let me know if you'd like further clarification or additional questions related to this!

Questions to Expand Understanding:

1. What are some real-world applications where understanding end behavior of functions is essential?
2. How does the degree of a polynomial affect its end behavior?
3. What happens to the end behavior if a leading coefficient of the function changes its sign?
4. How can the end behavior of exponential or logarithmic functions differ from polynomial functions?
5. Can end behavior be determined solely by looking at the highest power term in a polynomial? Why?

Tip:

To determine end behavior quickly, focus on the term with the highest degree in the function—this dominates as $x$ approaches infinity or negative infinity.