College Algebra, Chapter 8, 8.3, Section 8.3, Problem 12

Determine the vertices, foci and asymptotes of the hyperbola $\displaystyle \frac{x^2}{2} - y^2 = 1$. Then sketch its graph

Notice that the equation has the form $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Since the $x^2$-term is positive, then the hyperbola has a horizontal transverse axis;, its vertices and foci are located on the $x$-axis. Since $a^2 = 2$ and $b^2 = 1$, we get $a = \sqrt{2}$ and $b = 1$ and $c = \sqrt{a^2 + b^2} = \sqrt{3}$. Thus, we obtain

vertices $(\pm a, 0) \to (\pm \sqrt{2}, 0)$

foci $(\pm c, 0) \to (\pm \sqrt{3}, 0)$

asymptote $\displaystyle y = \pm \frac{b}{a} x \to y = \pm \frac{1}{\sqrt{2}} x$