Functions and their curves 191

A y

B

C D

2

3

4

5

24 23 22 0 1 2 3 4

21

22

23

24

25

21

x 12

y (^5) x 11
x 12
y (^5) x 11
1
x
Figure 18.32
With the above exampley=
x+ 2
x+ 1
, rearranging gives:
y(x+ 1 )=x+ 2
i.e. yx+y−x− 2 =0(1)
and x(y− 1 )+y− 2 = 0
The coefficient of the highest power ofx(in this casex^1 )
is(y− 1 ).Equatingtozerogives:y− 1 = 0
From which,y= 1 , which is an asymptote ofy=
x+ 2
x+ 1
as shown in Fig. 18.32.
Returning to equation (1): yx+y−x− 2 = 0
from which, y(x+ 1 )−x− 2 = 0.
The coefficient of the highest power ofy(in this case
y^1 )is(x+ 1 ). Equating to zero gives:x+ 1 =0 from
which,x=− 1 , whichis another asymptote ofy=
x+ 2
x+ 1
as shown in Fig. 18.32.
Problem 8. Determine the asymptotes for the
functiony=
x− 3
2 x+ 1
and hence sketch the curve.
Rearrangingy=
x− 3
2 x+ 1
gives:y( 2 x+ 1 )=x− 3
i.e. 2 xy+y=x− 3
or 2 xy+y−x+ 3 = 0
and x( 2 y− 1 )+y+ 3 = 0
Equating the coefficient of the highest power ofxto
zero gives: 2y− 1 =0 from which,y=^12 which is an
asymptote.
Sincey( 2 x+ 1 )=x−3 then equating the coefficient of
the highest power ofyto zero gives: 2x+ 1 =0 from
which,x=−^12 which is also an asymptote.
Whenx=0,y=
x− 3
2 x+ 1

− 3
1
=−3andwheny=0,
0 =
x− 3
2 x+ 1
from which,x− 3 =0andx=3.
Asketchofy=
x− 3
2 x+ 1
is shown in Fig. 18.33.