(aqt73) to Drill 8 seckin (54330)
This print-out should have 13 questions.
Multiple-choice questions may continue on
the next column or page nd all choices
before answering.
1
on [4, 4], what is b?
1. b = 4
2. b = 2 correct
001
10.0 points
Find the fundamental period of f when
4. b =
f (x) = 2 cos(3x) .
2
correct
3
1. fundamental period =
2. fundamental period = 3
3. fundamental period =
2
3
5. fundamental period =
3
2
6. fundamental period = 3
Explanation:
As
3 2 1
1
2
3
2
3
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
3
y = + 2 cos x ,
2
in other words, the graph of
2
.
3
fundamental period =
y = 2 cos x
3
shifted vertically by a term y = 2 . Thus b
is given by
10.0 points
When
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
1
shows, the given graph is that of
Explanation:
Since cos x has fundamental period 2 , f
has 3 cycles in any interval of length 2 . Consequently, f has
002
7
4
5. b = 4
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
3
2
4. fundamental period =
3. b = 2
b=2 .
3 2 1
1
1
2
3
003 (part 1 of 2) 10.0 points
Consider
1
y = cos 3x
5
2
3
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
is the graph of
y = a + b cos mx,
keywords: trig graph, trig function, cos function, amplitude, phase shift, vertical shift,
(m > 0),
a) Find the period.
2
1. correct
3
to (aqt73) Drill 8 seckin (54330)
a) Find the period.
2.
6
1.
3. 5
4.
2
3
2
2. none of these
3
3. 10 correct
5. 3
10
5.
5
4.
6. none of these
Explanation:
The period of the function y (x) is
T=
2
2
= .
3
3
6. 5
Explanation:
The period of the function y (x) is
T = 2 5 = 10 .
004 (part 2 of 2) 10.0 points
b) Find the amplitude.
006 (part 2 of 2) 10.0 points
b) Find the amplitude.
1. None of these
2. 3
1. 5
1
correct
5
5
4.
2
1
5.
5
3.
2.
2
3
3. None of these
3
correct
2
3
5.
4
3
6.
10
Explanation:
The function y (x) is of the general form
4.
6. 5
Explanation:
The function y (x) is of the general form
A cos
2t
T
,
where T is the period, A is the amplitude, and
is the phase shift. Thus the amplitude is
1
.
A=
5
A cos
,
where T is the period, A is the amplitude and
is the phase. Thus the amplitude is
A=
Note that in this case = accounts for
the minus sign in the equation for y (x).
007
005 (part 1 of 2) 10.0 points
Consider
x
3
y = cos
2
5
2t
T
3
.
2
10.0 points
Consider
f (x) = cos x
g (x) = cos(x + 9 )
to (aqt73) Drill 8 seckin (54330)
Which statement best describes the relationship between the graphs of f and g ?
1. The graph of g is a horizontal shift to the
right of the graph of f by 9 units.
2. The graph of g is an upward vertical shift
of the graph of f by 9 units.
3. The graph of g is an downward vertical
shift of the graph of f by 9 units.
4. None of these.
5. The graph of g is a horizontal shift to the
left of the graph of f by 9 units. correct
Explanation:
The function g (x) is of the general form
g ( x) = A + f ( x C ) ,
where A is the vertical shift of the graph of
g (x) respectively to the graph of f (x) and C
is the horizontal shift.
g (x) = cos[x (9 )]
is obtained from f (x) by shifting f (x) to
the left 9 units.
008
10.0 points
Consider
f (x) = sin 3x
g (x) = sin(3x)
Which statement best describes the relationship between the graphs f of and g ?
1. The graph of g is a vertical downward
shift of the graph of f .
3
5. The graph of g is a horizontal shift to the
left by 3 units of the graph of f .
6. The graph of g and the graph of f are
identical.
Explanation:
Recall that
g (x) = sin(3x) = sin 3x = f (x)
Thus the graph of g is a reection over the
x-axis of the graph of f .
009
10.0 points
Consider
f (x) = cos 6x
g (x) = 6 + cos 6x
Which statement best describes the relationship between the graphs of f and g ?
1. None of these.
2. The graph of g has 6 times the amplitude
of the graph of f .
3. The graph of g is a vertical shift upward
by 6 units of the graph of f .
4. The graph of g has one-half the amplitude
of the graph of f .
5. The graph of g is a vertical shift downward
by 6 units of the graph of f . correct
Explanation:
The function f (x) is of the general form
g ( x) = A + f ( x) ,
2. The graph of g is a reection over the
x-axis of the graph of f . correct
where A denotes a vertical shift. Since in our
case A = 6 < 0, the graph of g (x) will be
a vertical shift downward by 6 units of the
graph of f .
3. The graph of g is a vertical upward shift
of the graph of f .
010 10.0 points
Consider the functions
4. The graph of g is a horizontal shift to the
right by 3 units of the graph of f .
f (x) = 2 cos 2x
g (x) = cos 3x
to (aqt73) Drill 8 seckin (54330)
Which statement best describes the relationship between the graphs of f (x) and g (x)?
4
Explanation:
By considering the graph
1. The periods of f and g are the same.
2. None of these.
3. The period of f is larger than the period
of g . correct
4. The period of f is smaller than the period
of g .
Explanation:
From the graph
4
2
2
4
2
3
we nd that the solutions of the equation are
x = 1 , 3 , . . . .
6
2
g
1
012 10.0 points
Consider the equation
2
6
3 2 1
1
Since only 1 are within the interval [2, 2]
we obtain
x = {1, 1} .
f
4
1
4
because the maximum values of f are farther
apart than g , the period of f (x) is larger then
the period of g (x).
sin(x) =
1
.
2
Find real number(s) x in the interval [2, 2]
such that the equation holds.
1
1. ,
6
1
2. ,
6
5
1
5
, ,
6
6
6
5
11
7
, , correct
6
6
6
3. None of these
011 10.0 points
Consider the equation
cos(x) = 1 .
Find real number(s) x in the interval [2, 2]
such that the equation holds.
1. 1, 1 correct
2. 1
3.
1
2
4. None of these
5. 0
11
6. ,
22
7
11
7 11
, ,
4. ,
66
6
6
7
11
1
5
5. , , ,
6
6
6
6
15
6. , only
66
1 5 7 11
7. , , ,
666 6
Explanation:
By considering the graph
1
3 2 1
1
1
2
3
to (aqt73) Drill 8 seckin (54330)
we nd that the solutions of the equation are
x=
1
5
2n or x = 2n ,
6
6
where n is some integer. The solutions that
fall within the interval [2, 2] are, consequently,
x=
7
11
15
, , ,
66
6
6
013
.
10.0 points
The gure
4
2
6
4
2
2
4
6
2
4
6
shows the graphs of the functions
f ( x) = 2 1 x 2 ,
and
g (x) = 2 cos x .
1. None of these
2. The gray graph is the graph of f (x) correct
3. The black graph is the graph of f (x)
Explanation:
The graph of f (x) = 2 1x2 is a parabola
opening downward. Thus f corresponds to
the gray graph. The graph of g (x) = 2 cos x is
periodic so g corresponds to the black graph.
5