Higher Engineering Mathematics

194 GRAPHS

(xi) Polar Curves

The equation of a polar curve is of the formr=f(θ).
An example of a polar curve,r=asinθ, is shown in
Fig. 19.12.

Figure 19.12

19.2 Simple transformations

From the graph ofy=f(x) it is possible to deduce

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 of

y=af(x), y=f(x)+a, y=f(x+a), y=f(ax),

y=−f(x) andy=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 ofy=af(x). Thus

the graph ofy=af(x) can be obtained by stretching

y=f(x) parallel to they-axis by a scale factor ‘a’.

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

Fig. 19.13(a) and graphs ofy=sinθandy=2 sinθ

are shown in Fig. 19.13(b).

(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. For exam-
ple, iff(x)=x,y=f(x)+3 becomesy=x+3, as
shown in Fig. 19.14(a). Similarly, iff(θ)=cosθ,
theny=f(θ)+2 becomesy=cosθ+2, as shown
in Fig. 19.14(b). Also, iff(x)=x^2 , theny=f(x)+ 3
becomesy=x^2 +3, as shown in Fig. 19.14(c).
(iii)y=f(x+a)

The graph ofy=f(x) is translated by ‘a’ units par-
allel to thex-axis to obtainy=f(x+a). If ‘a’> 0

0 π

2

π 3 π

2

2 π

1

y

2

θ

y = 2 sinθ

y = sinθ

(b)

Figure 19.13

it movesy=f(x) in the negative direction on the

x-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, iff(x)=sinx,y=f

(

x−

π

3

)

becomesy=sin

(

x−

π

3

)

as shown in Fig. 19.15(a)

andy=sin

(

x+

π

4

)

is shown in Fig. 19.15(b).

Similarly graphs of y=x^2 , y=(x−1)^2 and

y=(x+2)^2 are shown in Fig. 19.16.

(iv)y=f(ax)

For each point (x 1 ,y 1 ) on the graph ofy=f(x), there

exists a point

(x

1

a

,y 1

)

on the graph ofy=f(ax).

Thus the graph ofy=f(ax) can be obtained by

stretchingy=f(x) parallel to thex-axis by a scale

factor

1

a