1. Graphing Rational Functions Example #6 End ShowEnd Show Slide #1 NextNext We want to graph this rational function showing all relevant characteristics.

2. Graphing Rational Functions Example #6 PreviousPreviousSlide #2 NextNext First we must factor both numerator and denominator, but don’t reduce the fraction yet. Both factor into 2 binomials.

3. Graphing Rational Functions Example #6 PreviousPreviousSlide #3 NextNext Note the domain restrictions, where the denominator is 0.

4. Graphing Rational Functions Example #6 PreviousPreviousSlide #4 NextNext Now reduce the fraction. In this case, it doesn't reduce.

5. Graphing Rational Functions Example #6 PreviousPreviousSlide #5 NextNext Any places where the reduced form is undefined, the denominator is 0, forms a vertical asymptote. Remember to give the V. A. as the full equation of the line and to graph it as a dashed line.

6. Graphing Rational Functions Example #6 PreviousPreviousSlide #6 NextNext Any values of x that are not in the domain of the function but are not a V.A. form holes in the graph. In other words, any factor that reduced completely out of the denominator would create a hole in the graph where it is 0. Thus, there are no holes in this case.

7. Graphing Rational Functions Example #6 PreviousPreviousSlide #7 NextNext Next look at the degrees of both the numerator and the denominator. Because both the denominator's and the numerator's degrees are the same, 2, there will be a horizontal asymptote at y=(the ratio of the leading coefficients) and there is no oblique asymptote.

8. Graphing Rational Functions Example #6 PreviousPreviousSlide #8 NextNext Next we need to find where the graph of f(x) would intersect the H.A. To do this we set the reduced form equal to the number from the H.A., and solve for x.

9. Graphing Rational Functions Example #6 PreviousPreviousSlide #9 NextNext Since the equation has a solution, the intersection will be the point with x- coordinate of the solution of the equation, and the y-coordinate will be the number from the H.A.

10. Graphing Rational Functions Example #6 PreviousPreviousSlide #10 NextNext We find the x-intercepts by solving when the function is 0, which would be when the numerator is 0. Thus, when x+1=0.

11. Graphing Rational Functions Example #6 PreviousPreviousSlide #11 NextNext Now find the y-intercept by plugging in 0 for x.

12. Graphing Rational Functions Example #6 PreviousPreviousSlide #12 NextNext Plot any additional points needed. Here I only plotted one more point at x=2 since a point hadn't been plotted to the right of that V.A. You can always choose to plot more points than required to help you find the graph.

13. Graphing Rational Functions Example #6 PreviousPreviousSlide #13 NextNext Finally draw in the curve. For the part to the right of the V.A., x=1, we use that it can't cross the H.A. and it has to approach the V.A. and the H.A.

14. Graphing Rational Functions Example #6 PreviousPreviousSlide #14 NextNext For the section between the V.A.'s, we use that it can't cross the H.A., it has to approach both V.A.'s and the multiplicity of the x-int. of -1 is 2, so the graph bounces of the x-axis.

15. Graphing Rational Functions Example #6 PreviousPreviousSlide #15 NextNext For -3<x<-2, we use that the graph has to approach the V.A. of x=-2, and the graph can't cross the x-axis in this interval.

16. Graphing Rational Functions Example #6 PreviousPreviousSlide #16 NextNext For x<-3, we have to use that the graph has to approach the H.A., and to find out that the graph crosses the H.A., we can use either that the intersection w/ the H.A. has a multiplicity of 1, or we could just plot another point at x=-4.

17. Graphing Rational Functions Example #6 PreviousPreviousSlide #17 End ShowEnd Show This finishes the graph.