https://file.aiquickdraw.com/mathbot/file/1726156321192k8dfrti2.jpg

Let's break down the problem to find the appropriate form of the function \( f(x) \).

### Given information:
1. **Vertical asymptotes** at \( x = 2 \) and \( x = -3 \).
2. **Horizontal asymptote** at \( y = 2 \).
3. **x-intercepts** at \( x = -4 \) and \( x = 1 \).
4. The graph passes through the point \( (-1, 2) \).

### Step-by-step solution:

1. **Vertical asymptotes**:
   - These occur where the denominator of a rational function equals zero.
   - Thus, the factors \( (x - 2) \) and \( (x + 3) \) should be in the denominator.

2. **Horizontal asymptote**:
   - The horizontal asymptote is \( y = 2 \), which suggests the degree of the numerator and denominator must be the same.
   - To have a horizontal asymptote of \( y = 2 \), the leading coefficient of the numerator must be 2 times that of the denominator.

3. **x-intercepts**:
   - These occur when the numerator equals zero.
   - The x-intercepts are at \( x = -4 \) and \( x = 1 \), so the numerator must include the factors \( (x + 4) \) and \( (x - 1) \).

4. **Equation form**:
   - The equation should resemble the third option: 
     \[
     f(x) = \frac{a (x - b)(x - c)}{(x - d)(x - e)}
     \]
   - Substituting the known intercepts and asymptotes:
     \[
     f(x) = \frac{a (x + 4)(x - 1)}{(x - 2)(x + 3)}
     \]

5. **Determine \( a \)**:
   - Since the function passes through the point \( (-1, 2) \), substitute \( x = -1 \) and \( f(x) = 2 \) to find \( a \):
     \[
     2 = \frac{a (-1 + 4)(-1 - 1)}{(-1 - 2)(-1 + 3)}
     \]
     Simplifying:
     \[
     2 = \frac{a(3)(-2)}{(-3)(2)}
     \]
     \[
     2 = \frac{-6a}{-6}
     \]
     \[
     2 = a
     \]

### Final equation:
\[
f(x) = \frac{2(x + 4)(x - 1)}{(x - 2)(x + 3)}
\]

Let me know if you would like more details, and here are 5 related questions:

1. How do you determine horizontal asymptotes in rational functions?
2. What conditions lead to a vertical asymptote in a rational function?
3. How can you find the x-intercepts of a rational function?
4. What happens to the graph of a rational function at points where the numerator equals zero?
5. Why is the horizontal asymptote affected by the degrees of the numerator and denominator?

**Tip**: Always check for points where the function is undefined to determine vertical asymptotes!

The figure shows the graph of a rational function f. It has vertical asymptotes x = 2 and x = -3, and a horizontal asymptote y = 2. The graph has x-intercepts -4 and 1, and it passes through the point (-1, 2). Choose the appropriate form of f(x) and write the equation in simplest form.

Learn how to determine the equation of a rational function given vertical and horizontal asymptotes, x-intercepts, and a point on the graph. This problem involves understanding how to use the vertical asymptotes, horizontal asymptotes, and intercepts to write the function in simplest form. Suitable for high school students in Grades 10-12.

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