Write the standard form equation for a hyperbola with center at the origin, vertices at $(0,3)$ and $(0,-3)$, and foci at $(0,4)$ and $(0,-4)$.

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Step 1 ：The standard form equation for a hyperbola with center at the origin is given by \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) if the vertices are on the x-axis and \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) if the vertices are on the y-axis. Here, \( a \) is the distance from the center to a vertex and \( b \) is the distance from the center to a co-vertex. The relationship between \( a \), \( b \), and \( c \) (the distance from the center to a focus) is \( c^2 = a^2 + b^2 \).

Step 2 ：Since the vertices are on the y-axis, we know that \( a = 3 \) and \( c = 4 \). We can use these values to find \( b \).

Step 3 ：Using the relationship \( c^2 = a^2 + b^2 \), we find that \( b = \sqrt{c^2 - a^2} = \sqrt{4^2 - 3^2} = \sqrt{7} \approx 2.6457513110645907 \).

Step 4 ：Now that we have the values of \( a \) and \( b \), we can substitute them into the equation of the hyperbola to get the final equation.

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

Step 6 ：Final Answer: The standard form equation for the hyperbola with center at the origin, vertices at \( (0,3) \) and \( (0,-3) \), and foci at \( (0,4) \) and \( (0,-4) \) is \( \boxed{\frac{y^2}{9} - \frac{x^2}{7} = 1} \).