Question

write an equation for the hyperbola shown in the graph. the equation for the hyperbola above is \\( \square = 1 \\). (simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the standard form of the hyperbola

The hyperbola has a vertical transverse axis. The standard form is $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1$.

Step2: Find the value of \(a\)

From the graph, the distance from the center to a vertex is \(a\). The vertices are at \(y=\pm 4\) (assuming the units from the graph - by counting the grid - if we assume each small square is 1 unit, and from the center \((0,0)\) to the vertex on \(y\) - axis). So \(a = 4\), and \(a^{2}=16\).

Step3: Find the value of \(b\)

The equations of the asymptotes for a hyperbola \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) are \(y=\pm\frac{a}{b}x\). Given the asymptotes \(y = \pm2x\), we have \(\frac{a}{b}=2\). Since \(a = 4\), then \(b = 2\), and \(b^{2}=4\).

Answer:

\(\frac{y^{2}}{16}-\frac{x^{2}}{4}\)