College Algebra, Chapter 8, 8.3, Section 8.3, Problem 16

Determine the vertices, foci and asymptotes of the hyperbola $\displaystyle x^2 - y^2 + 4 = 0$. Then sketch its graph

We can rewrite the equation as


$\begin{equation} \begin{aligned} x^2 - y^2 + 4 =& 0 && \text{Subtract } 4 \\ \\ x^2 - y^2 =& -4 && \text{Divide both sides by } -4 \\ \\ \frac{y^2}{4} - \frac{x^2}{4} =& 1 && \end{aligned} \end{equation}$


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 $a^2 = 4$ and $b^2 = 4$, we get $a = 2$ and $b = 2$ and $c = \sqrt{a^2 + b^2} = 2 \sqrt{2}$. Thus, we obtain

vertices $(0, \pm a) \to (0, \pm 2)$

foci $(0, \pm c) \to (0, \pm 2 \sqrt{2})$

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