“Teach A Level Maths”
Vol. 1: AS Core Modules
44: Stretches of the
Trigonometric Functions
© Christine Crisp
Stretches of Trig Functions
Module C2
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Stretches of Algebraic Functions
In an earlier section, we met stretches.
Reminder:
 y  f ( x)  y  k f ( x)
( f ( x ) multiplied by k )
is a stretch of scale factor ( s.f. ) k, parallel
to the y-axis
y  x3  y  2 x3
e.g.
is a stretch of s.f. 2, parallel to the y-axis
y  2x 3
2
y  x3
Stretches of Algebraic Functions
y  f ( x )  y  f ( kx)
( x multiplied by k )
is a stretch of scale factor ( s.f. ) 1 , parallel
k
to the x-axis.
1
1
y
 y
e.g.
x
2x
is a stretch of s.f. 1 parallel to the x-axis.
2

 12
y
2
x
y
1
x
Stretches of Trig Functions
e.g. 1 Sketch the graph of the function y  2 sin x
Solution: We can use the fact that y  2 sin x is
a stretch of y  sin x .
y  sin x  y  2 sin x
is a stretch of s.f. 2, parallel to the y-axis.
y  sin x
Stretches of Trig Functions
e.g. 1 Sketch the graph of the function y  2 sin x
Solution: We can use the fact that y  2 sin x is
a stretch of y  sin x .
y  sin x  y  2 sin x
is a stretch of s.f. 2, parallel to the y-axis.
y  2 sin x
y  sin x
The scale factor of the stretch gives the amplitude
of the function.
Stretches of Trig Functions
e.g. 2 Sketch the graph of the function y  cos 2 x
Solution:
y  cos x
is a stretch of s.f.
So,

1
2
y  cos 2 x
, parallel to the x-axis.
y  cos x
Stretches of Trig Functions
e.g. 2 Sketch the graph of the function y  cos 2 x
Solution:
y  cos x
is a stretch of s.f.
So,

1
2
y  cos 2 x
, parallel to the x-axis.
y  cos x
y  cos 2 x
The period of cos2x is 180 or

radians.
Stretches of Trig Functions
Exercises
1. Give the equation of the function that is shown
on the sketch below.
y
4
y
x
x
180
Ans: y  3 cos x
360
Stretches of Trig Functions
Exercises
2. Describe in words the transformation
y  sin x

y  sin 12 x
Sketch both functions on the same axes for the
interval   x  2
Solution: A stretch of s.f. 2 parallel to the x-axis.
y  sin 12 x
y  sin x
Stretches of Trig Functions
Exercises
3. Sketch the graph of y  2 cos 3 x for
 180  x  180 showing the scales clearly.
What is the period of the function?
Solution:
y  2 cos 3 x
The period
is
360
 120
3
Stretches of Trig Functions
Stretches of Trig Functions
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Stretches of Trig Functions
 y  f ( x)  y  k f ( x)
is a stretch of scale factor ( s.f. ) k, parallel
to the y-axis e.g.
y  sin x  y  2 sin x
is a stretch of s.f. 2, parallel to the y-axis.
The scale factor gives the amplitude of the function.
 y  f ( x )  y  f ( kx)
is a stretch of scale factor ( s.f. ) 1 , parallel
k
to the x-axis e.g.
y  cos x
stretch of s.f.

1
2
y  cos 2 x
, parallel to the x-axis.
Stretches of Trig Functions
A useful application of stretches occurs with the
trigonometric functions
e.g. 1 Sketch the graph of the function y  2 sin x
Solution: We can use the fact that y  2 sin x is
a stretch of y  sin x .
y  sin x  y  2 sin x
is a stretch of s.f. 2, parallel to the y-axis.
y  2 sin x
y  sin x
Stretches of Trig Functions
e.g. 2 Sketch the graph of the function y  cos 2 x
Solution:
y  cos x
is a stretch of s.f.
So,

1
2
y  cos 2 x
, parallel to the x-axis.
y  cos 2 x
y  cos x
The period of cos2x is 180 or

radians.