Final answer:
The standard form equation of the hyperbola with vertices at (0,2) and (0,-2) and asymptote y=1/2x is x^2/16 - y^2/4 = 1.
Step-by-step explanation:
To solve the mathematical problem of finding the standard form equation of a hyperbola with the given vertices and asymptote, we need to understand the relationship between these elements and the standard form equation of a hyperbola.
The vertices provided, (0,2) and (0,-2), suggest that the hyperbola is centered at the origin (0,0), and since the vertices are on the y-axis, the transverse axis is vertical. The distance between the center and either vertex is the length of the semi-major axis, a. Because the vertices are 2 units from the center, a = 2.
The equation of the asymptote y = 1/2x gives us the slope of the asymptotes. In the standard form of a hyperbola's equation, the slope of the asymptotes is a/b, where b is the distance from the center to the co-vertices on the transverse axis. From y = 1/2x, we have that a/b = 1/2. Knowing a = 2, we can solve for b to get b = 4.
Putting this information together, the equation of a vertical hyperbola centered at the origin is x^2/b^2 - y^2/a^2 = 1. Substituting a = 2 and b = 4 we get x^2/16 - y^2/4 = 1, which is the standard form equation of the hyperbola.