The given equation is:

$y = -\frac{6x^2}{x^2 - 1}$

This is a rational function, where the numerator is $-6x^2$ and the denominator is $x^2 - 1$.

Key features of the function:

1. Domain: The function is undefined wherever the denominator equals zero. Set the denominator equal to zero to find the points of discontinuity:

   $x^2 - 1 = 0 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1$ Thus, the function is undefined at $x = 1$ and $x = -1$. Therefore, the domain is: $x \in \mathbb{R} \setminus \{-1, 1\}$

2. Vertical Asymptotes: Since the function is undefined at $x = \pm 1$, these are vertical asymptotes. The function approaches infinity as $x$ approaches 1 or -1 from either side.

3. Horizontal Asymptote: To determine the horizontal asymptote, look at the behavior of the function as $x \to \infty$ or $x \to -\infty$. When $x$ becomes very large, both the numerator and the denominator are dominated by $x^2$, so the behavior of the function approaches:

   $y \approx -\frac{6x^2}{x^2} = -6$ Therefore, the horizontal asymptote is $y = -6$.

4. Intercepts:

   • Y-intercept: To find the y-intercept, set $x = 0$: $y(0) = -\frac{6(0)^2}{(0)^2 - 1} = 0$ So, the y-intercept is $(0, 0)$.

   • X-intercept: To find the x-intercepts, set $y = 0$. The numerator must be 0 for this to happen: $-6x^2 = 0 \quad \Rightarrow \quad x = 0$ Therefore, the only x-intercept is also at $(0, 0)$.

Summary of the function's key features:

• Domain: $x \in \mathbb{R} \setminus \{-1, 1\}$
• Vertical asymptotes at $x = -1$ and $x = 1$
• Horizontal asymptote at $y = -6$
• Only intercept at the origin, $(0, 0)$

Would you like to explore a graph or any other aspect of the function? Let me know if you'd like further details on any point!

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Related Questions:

1. How do you find the derivative of this rational function?
2. What happens to the function near the vertical asymptotes?
3. How would you classify the function in terms of limits at $x = \pm 1$?
4. What is the behavior of the function as $x \to 0$?
5. How does this function compare to other rational functions with similar forms?

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Tip: For rational functions like this one, identifying vertical asymptotes and horizontal asymptotes can quickly give you a sense of the graph's general behavior.