Hyperbola: Graph from description: 3rd Quadrant: Find the standard equation and draw the graph for

the hyperbola described as follows:

The vertices are at (-6, -2) and (-2, -2). The asymptotes have slopes of +3/2 and -3/2 .

We need to determine the general equation in the form , with asymptotes of slopes +b/a and -b/a. (h, k) are the coordinates of the center of the hyperbola.

In this case, the vertices have the same y-coordinate, and must lie on the line y = -2. The center (h, k) will be midway between, or at (-4, -2). Hence, we have h = -4 and k = -2.

From the slope information we determine b = 3 and a = 2.

Hence the equation is , or

We can check our results by substituting the x-coordinate of a vertex into the original equation and determining the corresponding y-coordinate. It should agree with the value of k, the y-coordinate of the center and the vertices. Using x = -2, we get

, or

, or

, or

, and the value of y agrees with the y-coordinate of the vertex, k = -2. You can check in a similar way with the x-coordinate of the other vertex: x = -6.

Now draw your graph, and then check it.