https://file.aiquickdraw.com/mathbot/file/1727281355860n5fiz0cn.jpg

The graph you uploaded represents a rational function with both vertical and horizontal asymptotes. From the image, we can gather the following key features of the function:

1. **Vertical Asymptotes**:
   - At \( x = -3 \)
   - At \( x = 1 \)
   
   This means the function approaches infinity or negative infinity as \( x \) approaches these values, but never crosses them.

2. **Horizontal Asymptote**:
   - At \( y = 0 \) (the x-axis)

   This suggests that as \( x \) approaches positive or negative infinity, the function approaches zero.

These asymptotes typically occur in functions where the denominator of a rational function becomes zero at certain points for vertical asymptotes, and where the degree of the numerator is less than or equal to the degree of the denominator for horizontal asymptotes.

Would you like help finding the function equation or further details on this graph?

Here are five related questions to deepen your understanding:
1. How do you determine vertical asymptotes algebraically?
2. What does the horizontal asymptote \( y = 0 \) tell you about the long-term behavior of the function?
3. How would the graph change if one of the vertical asymptotes was removed?
4. How are asymptotes related to the domain of a rational function?
5. What happens to the function’s graph near the asymptotes?

**Tip**: When sketching rational functions, always plot the asymptotes first. They help guide the behavior of the function in different regions of the graph.

Analyze the graph of a rational function with vertical asymptotes at x = -3 and x = 1, and a horizontal asymptote at y = 0.

Explore the graph of a rational function with vertical asymptotes at x = -3 and x = 1, and a horizontal asymptote at y = 0. This problem focuses on identifying the asymptotes, understanding the function’s behavior, and applying key algebraic concepts. Suitable for high school students in Grades 10-12.

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