https://file.aiquickdraw.com/mathbot/file/1727452713964tlni1wym.jpg

The graph you provided appears to represent a rational function, based on the curve's behavior and the presence of asymptotes.

### End behavior description:

1. **As \( x \to -\infty \)**:
   The graph moves downward as \( x \) approaches \( -\infty \). It appears to be approaching the vertical asymptote on the left side, while the curve extends downward towards negative infinity. This suggests that the function approaches \( -\infty \) as \( x \to -\infty \).

2. **As \( x \to +\infty \)**:
   The graph shows that as \( x \) increases, the curve flattens out and approaches a horizontal asymptote at \( y = 5 \). This suggests that the function approaches 5 as \( x \to +\infty \).

3. **Vertical asymptote at \( x = -2 \)**:
   The graph seems to have a vertical asymptote at \( x = -2 \). As the function approaches this value from the left, it tends toward negative infinity, and from the right, it tends toward positive infinity.

4. **Horizontal asymptote at \( y = 5 \)**:
   The graph appears to have a horizontal asymptote at \( y = 5 \), indicating that as \( x \) becomes very large (positively), the function approaches \( y = 5 \).

### Summary of the End Behavior:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to +\infty \), \( f(x) \to 5 \).
- The graph has a vertical asymptote at \( x = -2 \) and a horizontal asymptote at \( y = 5 \).

Would you like more details, or do you have any questions?

Here are some related questions you could explore:
1. What is the general form of a rational function with similar asymptotic behavior?
2. How does the vertical asymptote affect the function's domain?
3. What is the difference between horizontal and vertical asymptotes in terms of limits?
4. Can we determine the exact equation of the function based on this graph?
5. What are the possible real-world applications of this type of graph?

**Tip:** Asymptotes provide critical information about the behavior of functions, especially at extreme values of \( x \). Understanding them can help analyze limits and end behavior effectively.

Describe the end behavior of the graph shown below.

Explore the end behavior of a rational function graph, focusing on its approach to vertical and horizontal asymptotes. This page covers the implications of asymptotes for function behavior as x approaches infinity or negative infinity, suitable for advanced high school mathematics students.

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