**What is the vertex of:  f(x) = 6/x – 1/ - 2**

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.

What is the vertex of: f(x) = 6/x – 1/ - 2

Explore how to find the vertical and horizontal asymptotes, and the critical point for the rational function f(x) = 6/(x - 1) - 2. This problem highlights key concepts in rational functions and is suitable for high school students in Grades 10-12.

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