1. Graphing Rational Functions Example #5 End ShowEnd ShowSlide #1 NextNext We want to graph this rational function showing all relevant characteristics.

2. Graphing Rational Functions Example #5 PreviousPreviousSlide #2 NextNext First we must factor both numerator and denominator, but don’t reduce the fraction yet. Numerator: Prime. Denominator: Factor out the GCF of x.

3. Graphing Rational Functions Example #5 PreviousPreviousSlide #3 NextNext Note the domain restrictions, where the denominator is 0. Note that x^2 +1 is never 0 at any real numbers.

4. Graphing Rational Functions Example #5 PreviousPreviousSlide #4 NextNext Now reduce the fraction. In this case, we cancel the x.

5. Graphing Rational Functions Example #5 PreviousPreviousSlide #5 NextNext Any places where the reduced form is undefined, the denominator is 0, forms a vertical asymptote. Since x^2+1 is never 0 for any real number, there are no V.A.

6. Graphing Rational Functions Example #5 PreviousPreviousSlide #6 NextNext Any values of x that are not in the domain of the function but are not 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.

7. Graphing Rational Functions Example #5 PreviousPreviousSlide #7 NextNext Since this example is undefined at 0, but there are no V.A., there is a hole at x=0. To find the y-coordinate of the hole, plug 0 into the reduced form.

8. Graphing Rational Functions Example #5 PreviousPreviousSlide #8 NextNext Next look at the degrees of both the numerator and the denominator. Because the denominator's degree, 3, is larger than the numerator's, 1, the line y=0 is automatically the horizontal asymptote and there is no oblique asymptote.

9. Graphing Rational Functions Example #5 PreviousPreviousSlide #9 NextNext Since the H.A. is the x-axis, the intersections with the H.A. are also the x- int. We find the x-int. by solving when the function is 0 which would be when the numerator is 0. Thus, when 1=0. Since 1 is never 0, there are no x-int.

10. Graphing Rational Functions Example #5 PreviousPreviousSlide #10 NextNext Now find the y-intercept by plugging in 0 for x. In this case, since the function is undefined at x=0, there isn't a y-intercept, but remember that there is a hole on the y-axis.

11. Graphing Rational Functions Example #5 PreviousPreviousSlide #11 NextNext Plot any additional points needed. Here we don’t need any other points, but you can find some other points if you want to.

12. Graphing Rational Functions Example #5 PreviousPreviousSlide #12 NextNext Finally draw in the curve. For the part to the right of the y-axis, we use that it has to start at (0,1), it can't cross the x-axis and it has to approach the H.A.

13. Graphing Rational Functions Example #5 PreviousPreviousSlide #13 NextNext For the part to the left of the y-axis, we use that it has to start at (0,1), it can't cross the x-axis and it has to approach the H.A.

14. Graphing Rational Functions Example #5 PreviousPreviousSlide #14 End ShowEnd Show This finishes the graph.