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)
- y= 3 x− 5
- y=− 3 x+ 4
- y=x^2 + 3
- y=(x− 3 )^2
- y=(x− 4 )^2 + 2
- y=x−x^2
- y=x^3 + 2
- y= 1 +2cos3x
- y= 3 −2sin
(
x+
π
4
)
- 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