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write an equation for the function graphed below

The graph shown in the image depicts a rational function with vertical asymptotes and horizontal asymptotes. Here's how we can determine the equation of the function:

### Observations:
1. **Vertical Asymptotes:**
   - The graph has vertical asymptotes at \( x = -3 \) and \( x = 2 \).
   
2. **Horizontal Asymptote:**
   - The graph approaches a horizontal line as \( x \to \infty \). It appears to be approaching \( y = 1 \).

3. **Intercepts and Behavior:**
   - The graph crosses the x-axis around \( x = -1 \) and the y-axis around \( y = -2 \).

Based on these observations, the function can be modeled by a rational function of the form:
\[ y = \frac{a(x + 1)}{(x + 3)(x - 2)} + b \]

Here, \(a\) is a constant that scales the function vertically and adjusts the intercept, and \(b\) is a constant that shifts the function vertically.

### Equation Form:
1. **Vertical Asymptotes:** 
   The factors in the denominator \((x + 3)(x - 2)\) ensure the vertical asymptotes at \(x = -3\) and \(x = 2\).

2. **Horizontal Asymptote:** 
   For the horizontal asymptote \(y = 1\), the function should be shifted up by 1. Hence, \(b = 1\).

3. **Intercepts:**
   To match the intercept at \(x = -1\) and behavior, we need to find the correct value of \(a\).

### Function Derivation:
Let’s start with:
\[ y = \frac{a(x + 1)}{(x + 3)(x - 2)} + 1 \]

To find the value of \(a\):
- We can check the y-intercept by substituting \(x = 0\):
\[ y = \frac{a(0 + 1)}{(0 + 3)(0 - 2)} + 1 = \frac{a}{-6} + 1 \]
If the y-intercept is \(-2\), then:
\[ -2 = \frac{a}{-6} + 1 \]
\[ -3 = \frac{a}{-6} \]
\[ a = 18 \]

So, the equation for the function is:
\[ y = \frac{18(x + 1)}{(x + 3)(x - 2)} + 1 \]

### Final Answer:
\[ y = \frac{18(x + 1)}{(x + 3)(x - 2)} + 1 \]

Would you like further details or have any questions?

Here are some related questions to explore further:
1. What is a rational function, and how do its asymptotes affect its graph?
2. How do you determine the vertical asymptotes of a rational function?
3. What is the significance of the horizontal asymptote in a rational function?
4. How do intercepts help in determining the equation of a function?
5. Can you explain the behavior of the function near its asymptotes?
6. What are the general steps to finding the equation of a rational function given its graph?
7. How does the constant in the numerator affect the graph of a rational function?
8. How do you verify the equation of a function by comparing it to its graph?

**Tip:** Always check both the intercepts and asymptotic behavior when deriving the equation of a rational function from its graph.

Learn how to write an equation for a rational function graphed with vertical asymptotes at x = -3 and x = 2, and a horizontal asymptote at y = 1. This problem covers key concepts in rational functions, including vertical and horizontal asymptotes, and intercepts. Suitable for advanced high school students.

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