Higher Engineering Mathematics, Sixth Edition

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.

1. y=

x− 2

x+ 1

[y= 1 ,x=−1]

2. y^2 =

x

x− 3

[x= 3 ,y=1andy=−1]

3. y=

x(x+ 3 )

(x+ 2 )(x+ 1 )

[x=− 1 ,x=−2andy=1]

In Problems 4 and 5, determine all the asymptotes.

4. 8x− 10 +x^3 −xy^2 = 0
   [x= 0 ,y=xandy=−x]


5. x^2 (y^2 − 16 )=y
   [y= 4 ,y=−4andx=0]

In Problems 6 and 7, determine the asymptotes and

sketch the curves.

6. y=

x^2 −x− 4

x+ (^1) [
x=− 1 ,y=x− 2 ,
see Fig 18. 40 ,page 202
]

7. xy^2 −x^2 y+ 2 x−y= 5
   [
   x= 0 ,y= 0 ,y=x,
   see Fig. 18. 41 ,page 202

]