Final answer:
The equations of the asymptotes of the hyperbola (y - 2)^2/9 - (x + 2)^2 = 1 are y = 3x + 8 and y = -3x - 4.
Step-by-step explanation:
The given equation of a hyperbola is (y - 2)^2/9 - (x + 2)^2 = 1. To find the equations of the asymptotes of the hyperbola, you need to consider the standard form of the hyperbola equation, which is (y - k)^2/a^2 - (x - h)^2/b^2 = 1, where the center of the hyperbola is at (h, k) and the slopes of the asymptotes are ±a/b. In this case, a^2 = 9 and b^2 = 1, so a = 3 and b = 1.
The asymptotes of a hyperbola are straight lines that pass through the center of the hyperbola, which can be found by setting the equation to zero and solving for y. The slopes of the asymptotes can be found using ±a/b, which yields ±(3/1) = ±3. Hence, the equations of the asymptotes are y = 3(x + 2) + 2 and y = -3(x + 2) + 2, simplifying to y = 3x + 8 and y = -3x - 4, which corresponds to answer option C.