Higher Engineering Mathematics, Sixth Edition

186 Higher Engineering Mathematics

Now try the following exercise

Exercise 77 Further problems on simple

transformations with curve sketching

Sketch the following graphs, showing relevant

points:

(Answers on page 200, Fig. 18.39)

1. y= 3 x− 5


2. y=− 3 x+ 4


3. y=x^2 + 3


4. y=(x− 3 )^2


5. y=(x− 4 )^2 + 2


6. y=x−x^2


7. y=x^3 + 2


8. y= 1 +2cos3x


9. y= 3 −2sin

(

x+

π

4

)

10. y=2lnx

18.3 Periodic functions

A functionf(x)is said to beperiodiciff(x+T)=

f(x)for all values ofx,whereT is some positive

number.Tis the interval between two successive repe-

titions and is called the period of the functionf(x).For

example,y=sinxis periodic inxwith period 2πsince

sinx=sin(x+ 2 π)=sin(x+ 4 π), and so on. Similarly,

y=cosxis a periodic function with period 2πsince

cosx=cos(x+ 2 π)=cos(x+ 4 π), and so on. In gen-

eral, ify=sinωtory=cosωtthen the period of the

waveform is 2π/ω. The function shown in Fig. 18.24 is

1

 1

 2   0  2 

f(x)

x

Figure 18.24

also periodic of period 2πand is defined by:

f(x)=

{

− 1 , when−π≤x≤ 0

1 , when 0≤x≤π

18.4 Continuous and discontinuous functions

If a graph of a functionhas no sudden jumps or breaks it

is called acontinuous function, examples being the

graphs of sine and cosine functions. However, other

graphs make finitejumps at a point or pointsin the inter-

val. The square wave shown in Fig. 18.24 hasfinite

discontinuitiesasx=π,2π,3π, and so on, and is

therefore a discontinuous function.y=tanxis another

example of a discontinuous function.

18.5 Even and odd functions

Even functions

A functiony=f(x)is said to be even iff(−x)=f(x)

for all values ofx. Graphs of even functions are always

symmetrical about they-axis (i.e. is a mirror image).

Twoexamplesofevenfunctionsarey=x^2 andy=cosx

as shown in Fig. 18.25.

2322210

(a)

y

2

123

4

6

8

y 5 x^2

0

(b)

y

2   2  x

y 5 cosx

2 

2

x

Figure 18.25