Higher Engineering Mathematics, Sixth Edition

Functions and their curves 183

For example, if f(x)=(x− 1 )^2 ,anda=

1

2

,then

f(ax)=

(x

2

− 1

) 2

.

Both of these curves are shown in Fig. 18.17(a).

Similarly, y=cosx and y=cos2x are shown in
Fig. 18.17(b).

(v)y=−f(x)

The graph of y=−f(x) is obtained by reflecting
y=f(x)in thex-axis. For example, graphs ofy=ex
andy=−exare shown in Fig. 18.18(a) and graphs of
y=x^2 +2andy=−(x^2 + 2 )areshowninFig.18.18(b).

1.0

2 1.0

0

4

(a)

(b)

2

22 042 6

y 5 ( 2 x^21 )^2

x

y 5 cosx y 5 cos 2x

2 2

 3  2 

x

y 5 (x 2 1)^2

y

y

Figure 18.17

(a)

y

21 x

y 5 ex

y 52 ex

1

Figure 18.18

(b)

y

(^22) x
24
28
211
4
8
0 2
y 52 (x^21 2)
y 5 x^212
Figure 18.18(Continued)
(vi)y=f(−x)
The graph of y=f(−x)is obtained by reflecting
y=f(x)in they-axis. For example, graphs ofy=x^3
and y=(−x)^3 =−x^3 are shown in Fig. 18.19(a)
(a)
10
0
20
220
210
23 2223
(b)
2110
y 5 ( 2 x)^3
y 5 x^3
y 52 Inx
y 5 Inx
y
x
y
x
Figure 18.19