Kenneth G. answered • 02/04/14
Tutor
New to Wyzant
Experienced Tutor of Mathematics and Statistics
The denominator is -3(x-2)(x+3). This means the graph has vertical asymptotes at x=2 and x=-3. So the domain consists of three pieces (-infinity,-3), (-3,2) and (2,infinity).
If you do the division (x3-x2-6x)/(-3x2-3x+18) you get -x/3 + 2/3 with a remainder 2x-12. This means that the function is assymptotic to the line -x/3 + 2/3 as x approaches infinity or -infinity.
Once you have the asymptotes you are almost done. There are 3 parts of the graph, one on each of the three parts of the domain.
I'm assuming that you don't know calculus in the following. With calculus you could determine the slope of the tangent to the graph in the various regions and it would help you graph the function.
For the region (-infinity, -3) make a table of x and y values for the function to get an idea of the shape of the graph in that region.
x y
-10 4.1
-6 3
-4 3,1
-3.2 7.6
Do the same for the intervals (-3,2) and (2, infinity). The result should be three graphs as follows.
1. The graph on the interval (-infinity, -3) goes to +infinity as x approaches -3, and is asymptotic to the line -x/3 + 2/3 as x approaches - infinity. In this region f(x) is always greater than 2.
2. The graph in the region (-3,2) is an inverted-U shape. The graph approaches -infinity as x approaches -3 or 2. The graph is always less than 1 in this region.
3. The graph in the region (2, infinity) goes to +infinity as x approaches 2, and is asymptotic to the line -x/3 + 2/3 as x goes to infinity.
