College Algebra, Chapter 8, 8.3, Section 8.3, Problem 10

Determine the vertices, foci and asymptotes of the hyperbola $\displaystyle \frac{y^2}{9} - \frac{x^2}{16} = 1$. Then sketch its graph

Notice that the equation has the form $\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. Since the $y^2$-term is positive, then the hyperbola has a vertical transverse axis, its vertices and foci are located on the $y$-axis. Since $a^2 = 9$ and $b^2 = 16$, we get $a = 3$ and $b = 4$ and $c = \sqrt{a^2 + b^2} = 5$. Thus,

vertices $(0, \pm a) \to (0, \pm 3)$

foci $(0, \pm c) \to (0, \pm 5)$

asymptote $\displaystyle y = \pm \frac{a}{b} x \to y = \pm \frac{3}{4} x$