We know that the y-coordinates of the vertices are (0, -2, 2) and the y-coordinates of the foci are (0, -5, 5). The hyperbola is vertical since the vertices and foci are along the y-axis.
The standard form of a vertical hyperbola centered at the origin is y²/a² - x²/b² = 1.
Let's first calculate a and c.
1. "a" represents the distance from the center to a vertex. Since our hyperbola is centered at the origin (0,0) and one of the vertices is at (0, -2), this distance is simply |0 - (-2)| = 2.
2. "c" represents the distance from the center to a focus. Again, since our hyperbola is centered at the origin and one of the foci is at (0, -5), this distance is simply |0 - (-5)| = 5.
Next, we calculate "b". The relationship between a, b, and c in a hyperbola is defined by the equation c² = a² + b². Solving for b gives us b = sqrt(c² - a²), so we simply plug in our already known values; b = sqrt(5² - 2²) which approximates to 4.5826.
With these values, we can state the standard form of the equation of the hyperbola as y²/4 - x²/21 = 1.
So, the a, b, c values for this vertical hyperbola are approximately 2, 4.5826, and 5 respectively. And the equation in standard form is y²/4 - x²/21 = 1.