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To create a rational function with the specified properties, let’s analyze each requirement:

1. **Vertical asymptotes at \( x = 6 \) and \( x = -5 \):**  
   Vertical asymptotes are created by the zeros in the denominator. Thus, the denominator should have factors of \( (x - 6) \) and \( (x + 5) \).

2. **X-intercepts at \( x = -3 \) and \( x = -2 \):**  
   X-intercepts occur when the numerator is zero. Therefore, the numerator should have factors of \( (x + 3) \) and \( (x + 2) \).

3. **Y-intercept at 4:**  
   To find the y-intercept, we substitute \( x = 0 \) into the function. This will allow us to determine any constant factor required in the function.

### General Form of the Function
Given the above factors, we can start with the following form:

\[
y = \frac{k (x + 3)(x + 2)}{(x - 6)(x + 5)}
\]

where \( k \) is a constant that we will determine using the y-intercept condition.

### Finding \( k \)
To satisfy the y-intercept at \( y = 4 \), substitute \( x = 0 \) and set \( y = 4 \):

\[
4 = \frac{k (0 + 3)(0 + 2)}{(0 - 6)(0 + 5)}
\]

Simplify this expression:

\[
4 = \frac{k \cdot 3 \cdot 2}{-6 \cdot 5}
\]

\[
4 = \frac{6k}{-30}
\]

\[
4 = \frac{-k}{5}
\]

Multiply both sides by \( -5 \):

\[
k = -20
\]

### Final Equation
Now that we have \( k = -20 \), the equation for the rational function is:

\[
y = \frac{-20 (x + 3)(x + 2)}{(x - 6)(x + 5)}
\]

This function has the specified vertical asymptotes, x-intercepts, and y-intercept.

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. How would the equation change if the vertical asymptotes were at different values?
2. What happens to the graph if we change the x-intercepts or the y-intercept?
3. How would adding a horizontal asymptote affect the equation?
4. What impact does the value of \( k \) have on the function's graph?
5. Can we create a similar function with horizontal asymptotes as well?

**Tip:** When finding constants in rational functions, use given points (like y-intercepts) to solve for any unknowns.

Write an equation for a rational function with:

Vertical asymptotes at x = 6 and x = -5

x-intercepts at x = -3 and x = -2

y-intercept at 4

Learn how to write an equation for a rational function with vertical asymptotes at x = 6 and x = -5, x-intercepts at x = -3 and x = -2, and a y-intercept at 4. This problem covers concepts in rational functions, including asymptotes, intercepts, and solving for unknown constants.

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