How do I sketch accurate graphs for rational functions in a short amount of time? (I.e. A step by step guide of sketching graphs)

Often in Further Pure modules for various exam boards a common question in the exam paper will be to sketch a graph for a rational function. The step by step aproach I used is as follows:

1. Change the form (if required) of the function to a form where the denominators of fractions have greater order than numerators. Find points of intersection with the axis and min or max points and labelling them on the graph.

2. Find the limits of the function (by looking at denominators) and drawing dotted lines on the graph to represent these limits as linear functions (i.e. lines on the graph)

3. See which side of limit the graph will travel along on by testing values in function. Then drawing appropriate short lines so you know where the connect the graph.

4. Finally, mark any points where the function intersects the limits and connect the graphs lines together smoothly.