1. Aims: To be able to use graphical calc to investigate graphs of To be able to use graphical calc to investigate graphs of rational functions rational functions To be able to draw graphs of rational functions of the To be able to draw graphs of rational functions of the forms: and forms: and Graphs Lesson 2

2. 2 Starter

3. 3 Summary of Steps 1. Find the intercepts with the axes 2. Find the vertical asymptotes 3. Examine the behaviour as x tends to   Properties of today’s types of graphs! ; only 1 vertical and 1 horizontal asymptote. ; will have no vertical asymptote and 1 horizontal. Where the denominator is an irreducible quadratic Often this graph cuts the horizontal asymptote Denominator is a repeated factor

4. 1. Find the intercepts with the axes Sketch If x = 0, y = y = 0, x = 2. Find the vertical asymptotes (x – 2) 2 = 0 when x = this is the only vertical asymptote 3. Find the Examine the behaviour as x tends to   x    y   To see which side of 1 it approaches at, enter the graph in your calculator and see So +∞ tends to 1 from -∞tends to 1 from

5. 5 So y = 1 is a horizontal asymptote Notice the graph has to cut the horizontal asymptote. This happens at the point where y = and x =

6. 1. Find the intercepts with the axes Sketch If x = 0, y = y = 0, ( )( ) = 0, so x = 2. Find the vertical asymptotes 3. Examine the behaviour as x tends to   x    y   So +∞ tends to 1 from -∞tends to 1 from x² + 2x + 6 = 0 has no real solutions so there are no vertical asymptotes To see which side of 1 it approaches at, enter the graph in your calculator and see

7. 7 So y = 1 is a horizontal asymptote

8. 8 2. Sketch On w/b Do Exercise 5D page 63 And Exercise 5E page 64

9. 9