Section 4.3

To graph a rational function:
1)      Find the x-intercept(s) by plugging in zero for y.
2)      Find the y-intercept by plugging in zero for x.
3)      Find the equations of the vertical asymptotes by setting the denominator equal to zero.
4)      Find

If this limit equals L, then y = L is the equation of the horizontal asymptote.
If this limit does not exist, then the function has no horizontal asymptote.
5)      Use the first derivative to determine the intervals where f is increasing and decreasing.  From the first derivative, you can also find the critical points, relative minima, and relative maxima.
6)      Use the second derivative to determine the intervals where f is concave up and concave down.  From this you can find the inflection points.  (Any change in the sign of will be an inflection point since polynomial functions can’t have discontinuities.)
7)      Plot all intercepts, asymptotes, relative extrema, and inflection points.  Then use the information in the table above to complete the graph.
8)      Look at your picture and see if it agrees with the information you obtained in steps 1-6.