44: stretches of the trigonometric functions © christine crisp “teach a level maths” vol. 1: as...
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44: Stretches of the 44: Stretches of the Trigonometric Trigonometric
FunctionsFunctions
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Stretches of Trig Functions
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Module C2
Stretches of Algebraic Functions
In an earlier section, we met stretches.
is a stretch of scale factor ( s.f. ) k, parallel to the y-axis
33 2 xyxy e.g. is a stretch of s.f. 2, parallel to the y-axis
)()( xfkyxfy Reminder:
3xy
32xy 2
( multiplied by k ))(xf
Stretches of Algebraic Functions
is a stretch of scale factor ( s.f. ) , parallel to the x-axis.
k1
)()( kxfyxfy
e.g. x
yx
y2
11
is a stretch of s.f. parallel to the x-axis. 2
1
xy
1
xy
2
21
( x multiplied by k )
Stretches of Trig Functions
xy sin
e.g. 1 Sketch the graph of the function xy sin2
xyxy sin2sin is a stretch of s.f. 2, parallel to the y-axis.
Solution: We can use the fact that is a stretch of .xy sin
xy sin2
Stretches of Trig Functions
e.g. 1 Sketch the graph of the function xy sin2
xyxy sin2sin
Solution: We can use the fact that is a stretch of .xy sin
xy sin2
xy sin2
xy sin
is a stretch of s.f. 2, parallel to the y-axis.
The scale factor of the stretch gives the amplitude of the function.
Stretches of Trig Functions
xy cos
e.g. 2 Sketch the graph of the function xy 2cos
Solution: xyxy 2coscos
is a stretch of s.f. , parallel to the x-axis. So,
21
Stretches of Trig Functions
e.g. 2 Sketch the graph of the function xy 2cos
Solution: xyxy 2coscos
is a stretch of s.f. , parallel to the x-axis. So,
21
xy 2cos
xy cos
The period of is or radians.x2cos 180
Stretches of Trig Functions
Exercises
1. Give the equation of the function that is shown on the sketch below.
x
y
360
4
180
Ans: xy cos3
y
x
Stretches of Trig Functions
Solution:
A stretch of s.f. 2 parallel to the x-axis.
Sketch both functions on the same axes for the interval 2 x
2. Describe in words the transformation xyxy
21sinsin
xy sin
xy21sin
Exercises
Stretches of Trig Functions
Solution:
180180 x3. Sketch the graph of for
showing the scales clearly. What is the period of the function?
xy 3cos2
xy 3cos2The period is
1203
360
Exercises
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
is a stretch of scale factor ( s.f. ) k, parallel to the y-axis e.g.
)()( xfkyxfy
xyxy sin2sin is a stretch of s.f. 2, parallel to the y-axis.
is a stretch of scale factor ( s.f. ) , parallel to the x-axis e.g.
k1
)()( kxfyxfy
xyxy 2coscos
stretch of s.f. , parallel to the x-axis.21
The scale factor gives the amplitude of the function.
Stretches of Trig Functions
e.g. 1 Sketch the graph of the function xy sin2
xyxy sin2sin
Solution: We can use the fact that is a stretch of .xy sin
xy sin2
xy sin2
xy sin
A useful application of stretches occurs with the trigonometric functions
is a stretch of s.f. 2, parallel to the y-axis.
Stretches of Trig Functions
e.g. 2 Sketch the graph of the function xy 2cos
Solution: xyxy 2coscos
is a stretch of s.f. , parallel to the x-axis. So,
21
xy 2cos
xy cos
The period of is or radians.x2cos 180