Higher Engineering Mathematics, Sixth Edition

Functions and their curves 181

0

1

y 5 ex

y

x

Figure 18.11

O

a

a

r 5 a sin

Figure 18.12

18.2 Simple transformations

From the graph of y=f(x)it is possible to ded-
uce the graphs of other functions which are transfor-
mations ofy=f(x). For example, knowing the graph
ofy=f(x), can help us draw the graphs ofy=af(x),
y=f(x)+a,y=f(x+a),y=f(ax),y=−f(x)and
y=f(−x).
(i)y=af(x)
For each point (x 1 ,y 1 ) on the graph ofy=f(x)there
exists a point (x 1 ,ay 1 ) on the graph of y=af(x).
Thus the graph of y=af(x) can be obtained by
stretchingy=f(x)parallel to they-axis by a scale
factor ‘a’.

Graphs ofy=x+1andy= 3 (x+ 1 )are shown in

Fig. 18.13(a) and graphs ofy=sinθandy=2sinθare

shown in Fig. 18.13(b).

0

2

3 

2

2  

1

2

(b)

(a)

8

6

4

2

0 12

y

y

y 5 2 sin

y 5 sin



y 5 3(x 1 1)

y 5 x 11

x

Figure 18.13

(ii)y=f(x)+a

The graph ofy=f(x)is translated by ‘a’ units par-

allel to they-axis to obtainy=f(x)+a.Forexam-

ple, if f(x)=x, y=f(x)+3 becomesy=x+3, as

shown in Fig. 18.14(a). Similarly, if f(θ )=cosθ,

theny=f(θ )+2 becomesy=cosθ+2, as shown in

Fig. 18.14(b). Also, if f(x)=x^2 ,theny=f(x)+ 3

becomesy=x^2 +3, as shown in Fig. 18.14(c).

(iii)y=f(x+a)

The graph ofy=f(x)is translated by ‘a’ units parallel

to thex-axis to obtainy=f(x+a).If‘a’>0 it moves

y=f(x)in the negative direction on thex-axis (i.e. to

the left), and if ‘a’<0 it movesy=f(x)in the positive

direction on thex-axis (i.e. to the right).For example, if

f(x)=sinx,y=f

(

x−

π

3

)

becomesy=sin

(

x−

π

3

)

as shown in Fig. 18.15(a) andy=sin

(

x+

π

4

)

is shown

in Fig. 18.15(b).