Functions and their curves 195
Hence y=mx+c=± 1 x+0, i.e.y=x and y=−x
are asymptotes.
To determine any asymptotes parallel to thex-and
y-axes for the functionx^3 −xy^2 + 2 x− 9 =0:
Equating the coefficient of the highest power ofxterm
to zero gives 1=0 which is not an equation of a line.
Hence there is no asymptote parallel with thex-axis.
Equating the coefficient of the highest power ofyterm
to zero gives−x=0 from which,x=0.
Hencex= 0 ,y=xandy=−xare asymptotes for the
functionx^3 −xy^2 + 2 x− 9 = 0.
Problem 12. Find the asymptotes for the function
y=
x^2 + 1
x
and sketch a graph of the function.
Rearrangingy=
x^2 + 1
x
givesyx=x^2 +1.
Equating the coefficient of the highest powerxterm to
zero gives 1=0, hence there is no asymptote parallel to
thex-axis.
Equating the coefficient of the highest poweryterm to
zero givesx=0.
Hence there is an asymptote at x=0(i.e.the
y-axis).
To determine any other asymptotes we substitute
y=mx+cintoyx=x^2 +1 which gives
(mx+c)x=x^2 + 1
i.e. mx^2 +cx=x^2 + 1
and (m− 1 )x^2 +cx− 1 = 0
Equating the coefficient of the highest powerxterm to
zero givesm− 1 =0, from whichm=1.
Equatingthecoefficient ofthenext highest powerxterm
to zero givesc=0. Hencey=mx+c= 1 x+0, i.e.y=x
is an asymptote.
Asketchofy=
x^2 + 1
x
is shown in Fig. 18.35.
It is possible to determine maximum/minimum points
on the graph (see Chapter 28).
Since y=
x^2 + 1
x
=
x^2
x
+
1
x
=x+x−^1
then
dy
dx
= 1 −x−^2 = 1 −
1
x^2
= 0
for a turning point.
Hence 1=
1
x^2
andx^2 =1, from which,x=±1.
Whenx=1,
y=
x^2 + 1
x
=
1 + 1
1
= 2
and whenx=−1,
y=
(− 1 )^2 + 1
− 1
=− 2
i.e.(1,2)and(−1,−2)aretheco-ordinatesoftheturning
points.
d^2 y
dx^2
= 2 x−^3 =
2
x^3
;whenx=1,
d^2 y
dx^2
is positive,
which indicates a minimum point and whenx=−1,
d^2 y
dx^2
is negative, which indicates a maximum point, as
shown in Fig. 18.35.
Now try the following exercise
Exercise 80 Further problems on
asymptotes
In Problems 1 to 3, determine the asymptotes
parallel to thex-andy-axes.
- y=
x− 2
x+ 1
[y= 1 ,x=−1]
- y^2 =
x
x− 3
[x= 3 ,y=1andy=−1]
- y=
x(x+ 3 )
(x+ 2 )(x+ 1 )
[x=− 1 ,x=−2andy=1]
In Problems 4 and 5, determine all the asymptotes.
- 8x− 10 +x^3 −xy^2 = 0
[x= 0 ,y=xandy=−x] - x^2 (y^2 − 16 )=y
[y= 4 ,y=−4andx=0]
In Problems 6 and 7, determine the asymptotes and
sketch the curves.
- y=
x^2 −x− 4
x+ (^1) [
x=− 1 ,y=x− 2 ,
see Fig 18. 40 ,page 202
]
- xy^2 −x^2 y+ 2 x−y= 5
[
x= 0 ,y= 0 ,y=x,
see Fig. 18. 41 ,page 202
]