write the equation of the conic section shown below.

Explanation:

Step1: Identify the type of conic

The graph is a hyperbola centered at the origin $(0,0)$ with a vertical transverse - axis. The general equation of a hyperbola with a vertical transverse - axis centered at the origin is $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1$.

Step2: Determine the value of (a)

The vertices of the hyperbola are at $(0,2)$ and $(0, - 2)$. The distance from the center $(0,0)$ to a vertex is (a). So, (a = 2), and (a^{2}=4).

Step3: Determine the value of (b)

The asymptotes of a hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ have the equations (y=\pm\frac{a}{b}x). The asymptotes of the given hyperbola have a slope of (\pm1). Since (a = 2) and the slope of the asymptote is (\frac{a}{b}=1), then (b = 2) and (b^{2}=4).

Answer:

$\frac{y^{2}}{4}-\frac{x^{2}}{4}=1$