1. **State the problem:** We are given the equation $$x^2 - y^2 = 2$$ and asked to analyze it. 2. **Identify the type of curve:** This is a hyperbola because it can be rewritten in the form $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ where $a^2$ and $b^2$ are positive constants. 3. **Rewrite the equation:** Divide both sides by 2 to get $$\frac{x^2}{2} - \frac{y^2}{2} = 1$$ 4. **Explain the hyperbola form:** Here, $a^2 = 2$ and $b^2 = 2$. The hyperbola opens along the x-axis because the $x^2$ term is positive. 5. **Center and vertices:** The center is at the origin $(0,0)$. The vertices are at $(\pm \sqrt{2}, 0)$. 6. **Asymptotes:** The asymptotes of the hyperbola are given by $$y = \pm \frac{b}{a} x = \pm x$$ 7. **Summary:** The graph of $$x^2 - y^2 = 2$$ is a hyperbola centered at the origin, opening left and right, with vertices at $(\pm \sqrt{2}, 0)$ and asymptotes $y = \pm x$.