https://file.aiquickdraw.com/mathbot/file/1728968746174stbdssce.jpg

The graph you've shared appears to represent a rational function, based on the vertical asymptotes at \(x = -3\) and \(x = 2\), as well as the shape of the graph, which includes sections approaching infinity and negative infinity. To write an equation for this function, we need to consider the asymptotes and the general behavior.

### Key observations:
1. **Vertical Asymptotes**:
   - There are vertical asymptotes at \(x = -3\) and \(x = 2\), which suggests that the denominator of the rational function has factors \((x + 3)\) and \((x - 2)\).

2. **General Shape**:
   - The graph seems to resemble a hyperbola, implying that the function is likely of the form:
     \[
     y = \frac{k}{(x + 3)(x - 2)}
     \]
   - Where \(k\) is a constant that adjusts the stretch or orientation of the graph.

3. **Horizontal behavior**:
   - Since the graph does not show any horizontal asymptotes within the visible range, the degrees of the numerator and denominator should be the same, confirming that this is a rational function with no constant horizontal asymptote.

### General Equation:
Based on these observations, the function should take the form:
\[
y = \frac{k}{(x + 3)(x - 2)}
\]

The value of \(k\) can be determined by finding a point on the graph and solving for it. Do you want to further explore this with a specific point from the graph or have any other questions?

Here are 5 follow-up questions to deepen your understanding:
1. How do vertical asymptotes affect the shape of rational functions?
2. What are horizontal asymptotes, and how are they determined for rational functions?
3. Could the graph have any horizontal or oblique asymptotes? Why or why not?
4. How would the function change if we added a constant term in the numerator?
5. What role does the constant \(k\) play in determining the graph's behavior?

**Tip**: Always check the asymptotes and intercepts of a graph to help identify the function's form!

Write an equation for the function graphed below with vertical asymptotes at x = -3 and x = 2.

Understand how to write the equation of a rational function with vertical asymptotes at x = -3 and x = 2. This problem demonstrates key concepts of rational functions and asymptotes, ideal for high school students (Grades 10-12).

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