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b) and have no characteristicsin common except for their y-intercept and zeros.2. a)
b)
c) i) The graph of intersects the -axisat 0,
ii) The maximum value occurs at and every
since the period is
iii) The minimum value occurs at and every
since the period is
3. a) The graph of intersects the -axis at
b) The maximum values occur at 0 and every since the period is
c) The minimum value occurs at and every since the period is
4. Here is the graph of
Here is the graph of
The two graphs appear to be identical.5. a) The graph of intersects the -axisat 0,
b) The graph of has vertical asymptotes
at
6.4 Transformations of TrigonometricFunctions, pp. 343–346
1. a) period:
amplitude:horizontal translation:equation of the axis:
b) period:
amplitude:
horizontal translation:
equation of the axis:
c) period:
amplitude:horizontal translation:equation of the axis: y 5 21
d 5 0
0 a 0 5 0 2 0 5 2
2p
0 k 05
2p
3
y 5 3
d 5p
4
0 a 0 5 0 1 0 5 1
2p
0 k 05
2p
0 1 05 2p
y 5 0
d 5 0
0 a 0 5 0 0.5 0 5 0.5
2p
0 k 05
2p
0 4 05
p
2
nPItn 5p
21 np,
63p
2, c6
p
2,
y 5 tan u
nPItn 5 np,
62p, c6p,
uy 5 tan u
y 5 tan x:
y 5sin x
cos x:
nPItn 5 2p 1 2np,
2p.
2p,p
nPItn 5 2np,
2p.
2p,
nPItn 5p
21 np,
63p
2, c6
p
2,
uy 5 cos u
nPItn 53p
21 2np,
2p.
2p,3p
2
nPItn 5p
21 2np,
2p.
2p,p
2
nPItn 5 np,
62p, c6p,
uy 5 sin u
u 5 3.93
u 5 0.79
u 5 22.36
u 5 25.50
y 5 tan uy 5 sin u
6-13Advanced Functions Solutions Manual
08-035_06_AFSM_C06_001-034.qxd 7/22/08 4:11 PM Page 13
d) period:
amplitude:
horizontal translation:
equation of the axis:2. For
For
For
For
Only the last one is cut off.3.
period:
amplitude:
horizontal translation: units to the left
equation of the axis:
4.a)
period:
b)
period:
c)
period:
d)
period:
5. a)equation of the axis is
b)equation of the axis is
c)equation of the axis is
d)equation of the axis is
y 5 22 cos a1
2xb 2 1
y 5 21;
amplitude 5 2,period 5 4p,
y 5 22.5 cos a1
3xb 1 6.5
y 5 6.5;
amplitude 5 2.5,period 5 6p,
y 5 26 sin (0.5x) 2 2
y 5 22;
amplitude 5 6,period 5 4p,
y 5 18 sin xy 5 0;
amplitude 5 18,period 5 2p,
f(x) 5 11 sin (4px)
k 5 4p
2p
0 k 05
1
2
a 5 11
f(x) 5 80 sin a1
3xb 2
9
10
k 51
3
2p
0 k 05 6p
a 5 80
f(x) 52
5 sina
p
5xb 1
1
15
k 5p
5
2p
0 k 05 10
a 52
5
f(x) 5 25 sin(2x) 2 4
k 5 2
2p
0 k 05 p
a 5 25
y 5 a sin(k(x 2 d)) 1 c
y 5 4
d 5 2p
4
0 a 0 5 022 0 5 2
2p
0 k 05
2p
0 4 05
p
2
y 5 22 cosa4ax 1p
4bb 1 4
x246
0
y
4p
43p
2p– 4
p– 43p– 2
p
y 5 5 cos a22x 1p
3b 2 2
y 5 2 sin(3x) 2 1
y 5 sin ax 2p
4b 1 3
y 5 0.5 cos (4x)
y 5 22
d 5p
6
0 a 0 5 0 5 0 5 5
2p
0 k 05
2p
0 22 05 p
6-14 Chapter 6: Trigonometric Functions
08-035_06_AFSM_C06_001-034.qxd 7/22/08 4:11 PM Page 14
6. a) vertical stretch by a factor of 4, verticaltranslation 3 units up
b) reflection in the x-axis, horizontal stretch by afactor of 4
c) horizontal translation to the right, verticaltranslation 1 unit down
d) horizontal compression by a factor of horizontaltranslation to the left
7. a)
b)
c)
d)
8. a)
b)
0
y
x
42
–4–2
–6
2p
23p 2pp
0
y
x
42
–2
6
2p
23p 2pp
f(x) 5 cos a2ax 1p
2bb
f(x) 5 3 cos ax 2p
2b
f(x) 5 cos a21
2xb
f(x) 51
2 cos x 1 3
p
6
14,
p
6-15Advanced Functions Solutions Manual
08-035_06_AFSM_C06_001-034.qxd 7/22/08 4:11 PM Page 15
c)
d)
e)
f)
9. a) period:
The period of the function is This represents the time between one beat of aperson’s heart and the next beat.
b)
c)
d) The range for the function is between 80 and120. The range means the lowest blood pressure is80 and the highest blood pressure is 120.10. a)
b) There is a vertical stretch by a factor of 20. Theperiod is 0.8 s.
There is a horizontal compression by a factor
of
There is a horizontal translation 0.2 to the left.
c) y 5 20 sin a5p
2(x 1 0.2)b
1
0 k 05
2
5p.
k 55p
2
2p
k5 0.8
Time (s)
Hor
izon
tal d
ista
nce
from
cent
re (c
m)
x0
y
0.2 0.4 0.6 0.8 1.0 1.2 1.4
10
20
30
–10
–20
–30
1.6
0
y
1101009080
–20
120
1 2 3 4
x
P(60) 5 220 cos a5p
3(60)b 1 100 5 80
65.
k 56
5
2p
0 k 05
5p
3
0
y x
–4–2
–6
2p
23p 2pp
0
y x
–4–2
–6
2p
23p 2pp
0
y
x
42
–2
6
p2p
23p 2p
0
y
x
42
–2
6
p2p
23p 2p
6-16 Chapter 6: Trigonometric Functions
08-035_06_AFSM_C06_001-034.qxd 7/23/08 11:11 AM Page 16
11. a)
b) vertical stretch by a factor of 25, reflection in thex-axis, vertical translation 27 units up; the period is 3 s.
horizontal compression by a factor of
c)
12. By looking at the difference in the x-values of
the two maximums, and we see that the
period is
13. Answers may vary. For example,
Since the maximum is 4 units above theminimum would be at If the period of thefunction is , then the minimum would be at
of
14. a) This is a cosine function with
b) This is a sine function with a reflection in the x-axis and an
c) The y-axis is and the amplitude is 4. Thefunction is shifted horizontally to the right by 10.
15.
16. a) The car starts at the closest distance to thepole which is 100 m.b) The centre of the track is 400 m from the polebecause it is half the distance between the closestand furthest point.c) The radius is m.d) The period of the function is 80 s. This is howlong it takes to complete one lap.
e)
Mid-Chapter Review, p. 349
1. a)
b)
c)
d)11p
12 radians 3 a
180°
p radiansb 5 165°
5 radians 3 a180°
p radiansb 8 286.5°
4p radians 3 a180°
p radiansb 5 720°
p
8 radians 3 a
180°
p radiansb 5 22.5°
2p(300)
80 m/s 8 23.561 94 m/s
400 2 100 5 300
y 5 4 sin ap
20(x 2 10)b 2 1
period 52p
405
p
20
y 5 21
y 5 22 sin ap
4xb
period 52p
85
p
4
amplitude 5 2.
y 5 cos (4px)
period 52p
0.55 4p
amplitude 5 1.
14p
13.
p
131 p
2p
y 5 5.
y 5 9,
a14p
13, 5b.
2p
7.
23p
7,2
5p
7
y 5 225 cos a2p
3xb 1 27
1
0 k 05
3
2p
k 52p
3
2p
k5 3
0x
102030405060
y
21 43 65
Time (s)
Dis
tanc
e ab
ove
the
grou
nd (c
m)
6-17Advanced Functions Solutions Manual
Start with graph of y 5 sin x.
Reflect in the x-axis and stretchvertically by a factor of 2 to
produce graph of y 5 22 sin x.
Stretch horizontally by a factorof 2 to produce graph of
y 5 22 sin (0.5 x).
Translate units to the right to
produce graph of
.y 5 22 sin Q0.5 Qx 2p
4RR
p
4
Translate 3 units up to producegraph of
.y 5 22 sin Q0.5 Qx 2p
4RR 1 3
08-035_06_AFSM_C06_001-034.qxd 7/23/08 11:12 AM Page 17
