1. Graphing Rational Functions Example #8 PreviousPreviousSlide #1 NextNext We want to graph this rational function showing all relevant characteristics.

2. Graphing Rational Functions Example #8 PreviousPreviousSlide #2 NextNext First we must factor both numerator and denominator, but don’t reduce the fraction yet. Numerator: Factor out the negative, then difference of squares twice. Denominator: It's prime.

3. Graphing Rational Functions Example #8 PreviousPreviousSlide #3 NextNext Note the domain restrictions, where the denominator is 0.

4. Graphing Rational Functions Example #8 PreviousPreviousSlide #4 NextNext Now reduce the fraction. In this case, there are no common factors. So it doesn't reduce.

5. Graphing Rational Functions Example #8 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. and the full equation of the line and to graph it as a dashed line.

6. Graphing Rational Functions Example #8 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. Since this example didn't reduce, it has no holes.

7. Graphing Rational Functions Example #8 PreviousPreviousSlide #7 NextNext Next look at the degrees of both the numerator and the denominator. Because the denominator's degree, 1, is less than the numerator's, 4,by more than 1, there is neither a horizontal asymptote nor an oblique asymptote.

8. Graphing Rational Functions Example #8 PreviousPreviousSlide #8 NextNext Optional step: Even though there isn't a H.A. or an O.A. we can find out the end behavior of the graph. By dividing the leading terms, -x 4 and 2x, we get -0.5x 3. So the end behavior of the graph of f(x) will be like that of y= -0.5x 3, a vertical S that opens down on the right and up on the left.

9. Graphing Rational Functions Example #8 PreviousPreviousSlide #9 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 and x-1=0. Note that x 2 +1=0 would lead to imaginary solutions which is why we ignored it.

10. Graphing Rational Functions Example #8 PreviousPreviousSlide #10 NextNext Now find the y-intercept by plugging in 0 for x, but in this case that would lead to a 0 in the denominator. Thus, there can't be a y-intercept.

11. Graphing Rational Functions Example #8 PreviousPreviousSlide #11 NextNext Plot any additional points needed. In this case, we don't need any other points to determine the graph. Though, you can always plot more points if you want to.

12. Graphing Rational Functions Example #8 PreviousPreviousSlide #12 NextNext Finally draw in the curve. For x>1, we can use the right end behavior to know the graph has to curve down to the right of x=1. You could also plot more points to determine this.

13. Graphing Rational Functions Example #8 PreviousPreviousSlide #13 NextNext For -1/2<x<1, we can use the y-intercept, that the graph has to approach the V.A, and that there are no x-intercepts for -1/2<x<0.

14. Graphing Rational Functions Example #8 PreviousPreviousSlide #14 NextNext For x<-1, we can use the left end behavior to know the graph has to curve up to the left of x=-1. You could also plot more points to determine this.

15. Graphing Rational Functions Example #8 PreviousPreviousSlide #15 NextNext For -1<x<-1/2, we can use that the graph has to approach the V.A, and that there are no more x-intercepts for -1<x <-1/2.

16. Graphing Rational Functions Example #8 PreviousPreviousSlide #16 NextNext This finishes the graph.