204 GRAPHS

A y

5

4

3

2

1

(^01234) x
D
x + 2
y = x + 1
x + 2
y = x + 1
− 4 − 3 − 2 − 1
C
− 2
− 3
− 4
− 5
B
− 1
Figure 19.32
The coefficient of the highest power ofx(in this
casex^1 )is(y−1). Equating to zero gives:y− 1 = 0
From which, y= 1 , which is an asymptote of
y=
x+ 2
x+ 1
as shown in Fig. 19.32.
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
casey^1 )is(x+1). Equating to zero gives:x+ 1 = 0
from which,x=− 1 , which is another asymptote
ofy=
x+ 2
x+ 1
as shown in Fig. 19.32.
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(2x+1)=x− 3
i.e. 2 xy+y=x− 3
or 2 xy+y−x+ 3 = 0
and x(2y−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(2x+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
=−3 and when
y=0, 0=
x− 3
2 x+ 1
from which,x− 3 =0 andx=3.
A sketch ofy=
x− 3
2 x+ 1
is shown in Fig. 19.33.