To find a possible formula for the rational function based on the given properties, we can proceed as follows:

Step 1: Understand the requirements

1. Zeros at $x = 5$ and $x = 6$:

   • The function has factors $(x - 5)$ and $(x - 6)$ in the numerator.

2. Vertical asymptote at $x = 4$:

   • The function has a factor $(x - 4)$ in the denominator.

3. Horizontal asymptote at $y = -8$:

   • The degree of the numerator and denominator must be the same, and the leading coefficient of the numerator divided by the leading coefficient of the denominator must be $-8$.

Step 2: Write the general form of the rational function

The general form of the function can be expressed as: $f(x) = \frac{A(x - 5)(x - 6)}{(x - 4)}$ where $A$ is a constant.

Step 3: Ensure the horizontal asymptote is $y = -8$

For the horizontal asymptote, the degree of the numerator and denominator are both 2 (matching degrees), and the ratio of the leading coefficients must equal $-8$. This implies: $A = -8$

Step 4: Write the final formula

Substitute $A = -8$ into the function: $f(x) = \frac{-8(x - 5)(x - 6)}{(x - 4)}$

Final Answer:

A possible formula for the rational function is: $f(x) = \frac{-8(x - 5)(x - 6)}{(x - 4)}$

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Here are five related questions to deepen understanding:

1. What happens to the function's graph as $x$ approaches 4 from the left and right?
2. How would the function change if the horizontal asymptote were $y = 2$ instead of $y = -8$?
3. Can there be additional vertical asymptotes in a rational function? Why or why not?
4. How can the multiplicity of zeros affect the graph of the function?
5. What are the implications if the numerator's degree is higher than the denominator's degree?

Tip: Always check the degrees of the numerator and denominator to understand the horizontal asymptote behavior.