08-035_06_AFSM_C06_001-034.qxd
7/22/08
4:11 PM
Page 13
b) y 5 sin u and y 5 tan u have no characteristics
in common except for their y-intercept and zeros.
2. a)
b) The maximum values occur at 0 and every 2p,
since the period is 2p.
tn 5 2np, nPI
c) The minimum value occurs at p and every 2p,
since the period is 2p.
tn 5 2p 1 2np, nPI
sin x
4. Here is the graph of y 5
:
cos x
b)
u 5 25.50
Here is the graph of y 5 tan x:
u 5 22.36
The two graphs appear to be identical.
5. a) The graph of y 5 tan u intersects the u-axis
at 0, 6p, 62p, c
tn 5 np, nPI
b) The graph of y 5 tan u has vertical asymptotes
p
2
at 6 , 6
u 5 0.79
tn 5
3p
,c
2
p
1 np, nPI
2
6.4 Transformations of Trigonometric
Functions, pp. 343–346
u 5 3.93
c) i) The graph of y 5 sin u intersects the u-axis
at 0, 6p, 62p, c
tn 5 np, nPI
p
ii) The maximum value occurs at and every 2p,
2
since the period is 2p.
p
tn 5 1 2np, nPI
2
3p
iii) The minimum value occurs at
and every 2p,
2
since the period is 2p.
3p
1 2np, nPI
tn 5
2
3. a) The graph of y 5 cos u intersects the u-axis at
p
3p
6 ,6 ,c
2
2
p
tn 5 1 np, nPI
2
Advanced Functions Solutions Manual
1. a) period:
2p
2p
p
5
5
0k0
040
2
amplitude: 0 a 0 5 0 0.5 0 5 0.5
horizontal translation: d 5 0
equation of the axis: y 5 0
b) period:
2p
2p
5
5 2p
0k0
010
amplitude: 0 a 0 5 0 1 0 5 1
horizontal translation: d 5
p
4
equation of the axis: y 5 3
2p
2p
5
c) period:
0k0
3
amplitude: 0 a 0 5 0 2 0 5 2
horizontal translation: d 5 0
equation of the axis: y 5 21
6-13
08-035_06_AFSM_C06_001-034.qxd
d) period:
7/22/08
4:11 PM
2p
2p
5
5p
0k0
0 22 0
Page 14
p
4
horizontal translation: d 5 2 units to the left
amplitude: 0 a 0 5 0 5 0 5 5
p
horizontal translation: d 5
6
equation of the axis: y 5 22
2. For y 5 0.5 cos (4x)
equation of the axis: y 5 4
4. y 5 a sin(k(x 2 d)) 1 c
a) a 5 25
period:
For y 5 sin ax 2
p
b13
4
2p
5p
0k0
k52
f(x) 5 25 sin(2x) 2 4
2
b) a 5
5
period:
2p
5 10
0k0
p
5
p
1
2
f(x) 5 sina xb 1
5
5
15
c) a 5 80
k5
For y 5 2 sin(3x) 2 1
period:
2p
5 6p
0k0
1
3
1
9
f(x) 5 80 sin a xb 2
3
10
d) a 5 11
k5
For y 5 5 cos a22x 1
p
b22
3
period:
Only the last one is cut off.
3.
y
6
4
2
0
– 3p – p –p
4 2 4
x
p p 3p
4 2 4
y 5 22 cosa4ax 1
period:
p
bb 1 4
4
2p
2p
p
5
5
0k0
040
2
amplitude: 0 a 0 5 0 22 0 5 2
6-14
2p
1
5
0k0
2
k 5 4p
f(x) 5 11 sin (4px)
5. a) period 5 2p, amplitude 5 18,
equation of the axis is y 5 0;
y 5 18 sin x
b) period 5 4p, amplitude 5 6,
equation of the axis is y 5 22;
y 5 26 sin (0.5x) 2 2
c) period 5 6p, amplitude 5 2.5,
equation of the axis is y 5 6.5;
y 5 22.5 cos a xb 1 6.5
1
3
d) period 5 4p, amplitude 5 2,
equation of the axis is y 5 21;
y 5 22 cos a xb 2 1
1
2
Chapter 6: Trigonometric Functions
08-035_06_AFSM_C06_001-034.qxd
7/22/08
4:11 PM
Page 15
6. a) vertical stretch by a factor of 4, vertical
translation 3 units up
c) f(x) 5 3 cos ax 2
b) reflection in the x-axis, horizontal stretch by a
factor of 4
d) f(x) 5 cos a2ax 1
c) horizontal translation p to the right, vertical
translation 1 unit down
8. a)
6
4
2
p
b
2
p
bb
2
y
0
–2
x
p
2
p
3p
2
2p
d) horizontal compression by a factor of 14, horizontal
p
translation to the left
6
b)
7. a) f(x) 5
1
cos x 1 3
2
4 y
2
0
–2
–4
–6
x
p
2
p
3p
2
2p
1
b) f(x) 5 cos a2 xb
2
Advanced Functions Solutions Manual
6-15
08-035_06_AFSM_C06_001-034.qxd
c)
6
4
2
7/23/08
11:11 AM
Page 16
y
x
0
–2
p
2
p
3p
2
2p
9. a) period:
2p
5p
5
0k0
3
6
5
The period of the function is 65.
This represents the time between one beat of a
person’s heart and the next beat.
5p
b) P(60) 5 220 cos a (60)b 1 100 5 80
3
c)
y
120
110
100
90
80
x
0
1
2
3
4
–20
k5
6
4
2
y
x
0
–2
p
2
p
3p
2
y
e)
0
–2
–4
–6
2p
d) The range for the function is between 80 and
120. The range means the lowest blood pressure is
80 and the highest blood pressure is 120.
y
10. a)
30
x
p
2
p
3p
2
2p
Horizontal distance from
centre (cm)
d)
20
10
0
–10
x
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
–20
–30
Time (s)
y
f)
0
–2
–4
–6
x
p
2
p
3p
2
2p
b) There is a vertical stretch by a factor of 20. The
period is 0.8 s.
2p
5 0.8
k
5p
k5
2
There is a horizontal compression by a factor
of
1
2
5 .
0k0
5p
There is a horizontal translation 0.2 to the left.
c) y 5 20 sin a
6-16
5p
(x 1 0.2)b
2
Chapter 6: Trigonometric Functions
08-035_06_AFSM_C06_001-034.qxd
11. a)
Distance above the ground (cm)
60
50
40
30
20
10
0
7/23/08
11:12 AM
Page 17
y
15.
Reflect in the x-axis and stretch
vertically by a factor of 2 to
produce graph of y 5 22 sin x.
x
1
2 3 4 5 6
Stretch horizontally by a factor
of 2 to produce graph of
y 5 22 sin (0.5 x).
Time (s)
b) vertical stretch by a factor of 25, reflection in the
x-axis, vertical translation 27 units up; the period
is 3 s.
2p
53
k
2p
k5
3
3
1
5
horizontal compression by a factor of
0k0
2p
2p
c) y 5 225 cos a xb 1 27
3
12. By looking at the difference in the x-values of
5p
3p
the two maximums, 2 and 2 , we see that the
period is
2p
.
7
7
14p
, 5b.
13
Since the maximum is 4 units above y 5 9, the
minimum would be at y 5 5. If the period of the
function is 2p, then the minimum would be at
of
p
Translate units to the right to
4
produce graph of
y 5 22 sin Q 0.5 Qx 2 RR .
p
4
Translate 3 units up to produce
graph of
y 5 22 sin Q 0.5 Qx 2 RR 1 3.
4
p
7
13. Answers may vary. For example, a
p
1p
13
Start with graph of y 5 sin x.
14p
.
13
14. a) This is a cosine function with amplitude 5 1.
2p
5 4p
period 5
0.5
y 5 cos (4px)
b) This is a sine function with a reflection in the
x-axis and an amplitude 5 2.
p
2p
5
period 5
8
4
p
y 5 22 sin a xb
4
c) The y-axis is y 5 21 and the amplitude is 4. The
function is shifted horizontally to the right by 10.
p
2p
5
period 5
40
20
p
y 5 4 sin a (x 2 10)b 2 1
20
Advanced Functions Solutions Manual
16. a) The car starts at the closest distance to the
pole which is 100 m.
b) The centre of the track is 400 m from the pole
because it is half the distance between the closest
and furthest point.
c) The radius is 400 2 100 5 300 m.
d) The period of the function is 80 s. This is how
long it takes to complete one lap.
2p(300)
m /s 8 23.561 94 m/s
e)
80
Mid-Chapter Review, p. 349
p
180°
radians 3 a
b 5 22.5°
8
p radians
180°
b 5 720°
b) 4p radians 3 a
p radians
180°
b 8 286.5°
c) 5 radians 3 a
p radians
11p
180°
radians 3 a
b 5 165°
d)
12
p radians
1. a)
6-17