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).