The function you've written, $f(x) = \frac{6}{x} - 1 - 2$, appears to have a typo. If you intended it to be a rational function, it might look something like:

$f(x) = \frac{6}{x - 1} - 2$

If this is the correct function, let's find its vertex and other relevant characteristics.

Understanding the Function

1. Form of the Function:
   The function $f(x) = \frac{6}{x - 1} - 2$ is a rational function, which typically does not have a "vertex" like a quadratic function. Instead, it has asymptotes (both vertical and horizontal) and a center point that is critical for understanding its behavior.

2. Vertical Asymptote:
   The vertical asymptote occurs where the denominator is zero. For this function: $x - 1 = 0 \implies x = 1.$ So, there is a vertical asymptote at $x = 1$.

3. Horizontal Asymptote:
   For a function of the form $\frac{a}{x - h} + k$, the horizontal asymptote is $y = k$. Here, $k = -2$, so the horizontal asymptote is: $y = -2.$

4. Critical Point or "Center":
   While not a "vertex," the rational function is centered around the point $(h, k) = (1, -2)$. This is the point from which the graph appears to approach its asymptotes.

Conclusion

• Vertical Asymptote: $x = 1$
• Horizontal Asymptote: $y = -2$
• Center (critical point around which the function is symmetric): $(1, -2)$

So, while the function does not have a "vertex" in the traditional sense, the point $(1, -2)$ serves as the key reference for the graph.

Would you like more details or have any questions about this function? Here are some related questions to consider:

1. What happens to the function as $x \to \infty$ or $x \to -\infty$?
2. How would the graph change if the function were $f(x) = \frac{6}{x} - 2$?
3. What are the intercepts of the function?
4. How does the value of $a$ (here, 6) affect the shape of the rational function?
5. What is the range of this function?

Tip: Rational functions are defined by their asymptotes and critical points, rather than vertices, due to their unique shapes and behaviors.