The graph in the image appears to depict a polynomial function with two distinct arms extending to infinity. Let’s analyze the graph in terms of end behavior.

1. Left end behavior (as $x \to -\infty$):

   • The graph rises toward positive infinity as $x \to -\infty$. This suggests that as $x$ approaches negative infinity, the function’s values go to positive infinity.

2. Right end behavior (as $x \to +\infty$):

   • The graph falls toward negative infinity as $x \to +\infty$. This suggests that as $x$ approaches positive infinity, the function’s values go to negative infinity.

Now, let’s match this with the given multiple-choice options:

• Option 1: As the x-values go to positive infinity, the function's values go to negative infinity. – This is correct based on the right end behavior.
• Option 4: As the x-values go to negative infinity, the function's values go to positive infinity. – This is correct based on the left end behavior.

Thus, the correct answer is:

• Option 1 (for positive infinity)
• Option 4 (for negative infinity)

Would you like more details or have any other questions? Here are some follow-up questions to consider:

1. What is the degree of the polynomial shown in the graph?
2. How does the leading coefficient of a polynomial affect its end behavior?
3. Can you determine the roots of the function from the graph?
4. What would happen to the graph if the leading coefficient were negative?
5. How does symmetry affect the graph of polynomials?

Tip: When analyzing end behavior, always check how the function behaves as $x$ approaches both positive and negative infinity to understand the nature of the function.