15: more transformations © christine crisp “teach a level maths” vol. 2: a2 core modules
TRANSCRIPT
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15: More 15: More TransformationsTransformations
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
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Trig Transformations
Combined Translations
• x axis translation of –a and
y axis translation of b
a
b
y f x y f x a b( ) ( )
• x axis translation of +a and
y axis translation of b
a
b
y f x y f x a b( ) ( )
Note
Opposite sign
Note
Opposite sign
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Trig Transformations
Stretches
The function
)(xkfy is obtained from )(xfy
by a stretch of scale factor ( s.f. ) k,parallel to the y-axis.
The function
)(kxfy is obtained from )(xfy
by a stretch of scale factor ( s.f. ) ,parallel to the x-axis.
k1
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Trig Transformations
Reflections
Reflections in the axes • Reflecting in the x-axis changes the
sign of y )()( xfyxfy
)()( xfyxfy
• Reflecting in the y-axis changes the sign of x
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Trig Transformations
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
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Trig Transformations
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.
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Trig Transformations
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
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Trig Transformations
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
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Trig Transformations
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
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Trig Transformations
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
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Trig Transformations
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
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Trig Transformations
Reflection in the x-axis Every y-value changes sign when we reflect in the x-axis e.g.
So, xyxy sinsin
xy sin
xy sin
x
x
In general, a reflection in the x-axis is given by
)()( xfyxfy
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Trig Transformations
then (iii) a reflection in the x-axis
(i) a stretch of s.f. 2 parallel to the x-axis
then (ii) a translation of
2
0
e.g.3 Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).
xy cos
20 x
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Trig Transformations
xcos
Solution:(i) a stretch of s.f. 2 parallel to the x-
axis x21cos
xy 21cos
xy cos2
stretch
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Trig Transformations
Brackets aren’t essential here but they make it clearer.
(ii) a translation of :
2
0 x21cos 2cos 2
1 x
2cos 21 xy
2
translate
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Trig Transformations
2cos 21 xy
(ii) a translation of :
2
0 x21cos 2cos 2
1 x
2cos 21 x 2cos 2
1 x
2cos 21 xy
2
translate reflect
x
x
(iii) a reflection in the x-axis
2cos 21 x
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Trig Transformations
Exercises1. Describe the transformations that map
the graphs of the 1st of each function given below onto the 2nd. Sketch the graphs at each stage.
xy sin xy 2sin1(a) to
( Draw for )xsin 20 x
(b) y = cosx to y = 2cos(x – 30)
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Trig Transformations
xy sinSolution
s:
xy 2sin1(a) to
xy sin xy 2sin
Translatio
n
1
0xyxy 2sin12sin Stretch s.f. parallel to the x-
axis21
xy sin
xy 2sin xy 2sin
xy 2sin1
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Trig Transformations
y xcos( 30)
Vertical stretch factor
2
y x y xcos( 30) 2cos( 30)
(b) toSolution
s:
y xcos
Translation parallel to the x-
axis
30
0
-1
-2
1
2
30 60 90 120 150 180 210 240 270 300 3300X->
|̂Y
Y=cos(x)
Y=cos(x-30)
-1
-2
1
2
30 60 90 120 150 180 210 240 270 300 3300X->
|̂Y
Y=cos(x-30)
Y=2cos(x-30)
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Trig Transformations
(i) (ii))(xfy )2( xfy
The diagram shows part of the curve with equation
.)(xfy
Copy the diagram twice and on each diagram sketch one of the following:
)(xfy x
y
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Trig Transformations
Solution:
)2( xfy (ii))(xfy
)2( xfy
x
y
)(xfy
)(xfy
x
y
(i)
)(xfy
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Trig Transformations
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
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Trig Transformations
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 )
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Trig Transformations
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
![Page 25: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/25.jpg)
Trig Transformations
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.
![Page 26: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/26.jpg)
Trig Transformations
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
![Page 27: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/27.jpg)
Trig Transformations
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
![Page 28: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/28.jpg)
Trig Transformations
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
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Trig Transformations
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
![Page 30: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/30.jpg)
Trig Transformations
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
![Page 31: 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader031.vdocuments.us/reader031/viewer/2022031921/56649c935503460f9494ecbd/html5/thumbnails/31.jpg)
Trig Transformations