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

The form of the standard equation is .

First let's group the x and y terms:

.

Next we factor as follows:

Now we complete the squares by adding and subtracting within the parentheses:

.

Next move the subtracted terms outside the parentheses (and remember the factor)

Now move the "outside"constants to the right hand side of the equation and write the parentheses as squares:

Since we need a "1" on the right-hand side in order to compare with the general equation, we now divide by 36 to get:

After canceling, we have

Now we can conclude that h = -4, k = -2, a is 2 and b is 3.

The standard graph has its center at (h, k). Here that is (-4, -2). The vertices of the hyperbola are "a" units to the right and left of the center, or at x = -6 and x = -2.

The asymptotes go through the center with slopes +b/a and -b/a. In this case the slopes are 3/2 and -3/2.

Draw your graph. When complete, check it.