Higher Engineering Mathematics

208 GRAPHS

Hence y=mx+c=± 1 x+0, i.e. y=x and

y=−xare asymptotes.

To determine any asymptotes parallel to the x-

andy-axes for the functionx^3 −xy^2 + 2 x− 9 =0:

Equating the coefficient of the highest power ofx

term to zero gives 1=0 which is not an equation of

a line. Hence there is no asymptote parallel with the

x-axis.

Equating the coefficient of the highest power ofy

term 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 func-

tion 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 atx=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.

Equating the coefficient of the next highest powerx

term to zero givesc=0. Hencey=mx+c= 1 x+0,

i.e.y=xis an asymptote.

A sketch ofy=

x^2 + 1

x

is shown in Fig. 19.35.

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) are the co-ordinates of the

turning 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. 19.35.

Now try the following exercise.

Exercise 88 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=1 andy=−1]

3.y=

x(x+3)

(x+2)(x+1)

[x=−1,x=−2 andy=1]

In Problems 4 and 5, determine all the asymp-

totes

4. 8x− 10 +x^3 −xy^2 = 0
   [x=0,y=xandy=−x]
   5.x^2 (y^2 −16)=y
   [y=4,y=−4 andx=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. 19 .40, page 215
]
7.xy^2 −x^2 y+ 2 x−y= 5
[
x=0,y=0,y=x,
see Fig. 19 .41, page 215
]