College Algebra, Chapter 8, 8.3, Section 8.3, Problem 26

Find the equation for the hyperbola whose graph is shown below.







The hyperbola $\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ has a vertical transverse axis, its vertices $(0, \pm a)$ and foci $(0, \pm c)$ are located on the $y$-axis. Notice from the graph that the hyperbola have vertices on $(0, \pm 3)$ which gives $a = 3$. Also, if the asymptotes are $y = \pm 3x$, then $\displaystyle \frac{a}{b} = 3$


$
\begin{equation}
\begin{aligned}

\frac{3}{b} =& 3
\\
\\
b =& 1

\end{aligned}
\end{equation}
$


Therefore, the equation is


$
\begin{equation}
\begin{aligned}

& \frac{y^2}{3^2} - \frac{x^2}{1^2} = 1
&&
\\
\\
& \text{or}
\\
\\
& \frac{y^2}{9 } - x^2 = 1

\end{aligned}
\end{equation}
$