Find the expanded form of the hyperbola with center at origin, foci at (0, ±4), and vertices at (0, ±3).

Step 1 ：The equation of a hyperbola centered at the origin with vertices along the y-axis is given by \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where a is the distance from the center to a vertex and b is the distance from the center to a co-vertex.

Step 2 ：Given that the vertices are at (0, ±3), we can say that \(a = 3\).

Step 3 ：The foci of a hyperbola are given by the equation \(c = \sqrt{a^2 + b^2}\), where c is the distance from the center to a focus. Given that the foci are at (0, ±4), we can say that \(c = 4\).

Step 4 ：We can then solve for b using the equation \(c = \sqrt{a^2 + b^2}\), which gives us \(b = \sqrt{c^2 - a^2} = \sqrt{4^2 - 3^2} = \sqrt{7}\).

Step 5 ：Substituting a and b into the equation of the hyperbola, we get the expanded form: \(\frac{y^2}{3^2} - \frac{x^2}{\sqrt{7}^2} = 1\) or \(\frac{y^2}{9} - \frac{x^2}{7} = 1\).