Vertices: \( (0, \pm 8) \); foci: \( (0, \pm 4) \)
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Did you know that the given vertices and foci represent a hyperbola? In this case, the vertices (0, ±8) indicate that the hyperbola opens vertically, as they lie on the y-axis. The distance from the center to the vertices shows the distance of 8 units, while the foci at (0, ±4) tell us that the hyperbola’s shape is defined by the relationship between the distance to the foci and the vertices, adhering to the hyperbola equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). If you're looking to sketch this hyperbola, start with plotting the center, which in this case is at the origin (0,0). From there, mark the vertices at (0, 8) and (0, -8). Then, plot the foci at (0, 4) and (0, -4). Finally, remember that the branches of the hyperbola will curve outward from each vertex, getting closer towards the foci but never touching them. This will give you a nice visual representation of this fascinating curve!