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?

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.