Final answer:
To find the vertices and covertices of a hyperbola, compare the equation with the standard equation and identify the values of a, b, h, and k. The center of the hyperbola is (h, k), and the vertices are located at (h ± a, k), while the covertices are located at (h, k ± b).
Step-by-step explanation:
To find the vertices and covertices of the hyperbola given by the equation ((y+4)^(2))/(36)-((x-1)^(2))/(81)=1, we need to identify the values of a, b, h, and k. Comparing the given equation with the standard equation for a hyperbola, we have (y-k)^(2)/a^(2) - (x-h)^(2)/b^(2) = 1. From this, we can determine that h = 1 and k = -4. Therefore, the center of the hyperbola is (1, -4). The values of a and b can be found by taking the square root of the denominators of the y and x terms, respectively. Therefore, a = 6 and b = 9. Using these values, we can determine the vertices and covertices. The vertices are located at (h ± a, k), which gives us the vertices (7, -4) and (-5, -4). The covertices are located at (h, k ± b), which gives us the covertices (1, 5) and (1, -13).