Algebra 2 - Problem Drill 20: Trigonometric Graphs
Question No. 1 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
1. The image shows the graph of one period of which of the following functions?
y
x
π/2
Question
0
π
(A) cos x
(B) cot x
(C) sec x
(D) sin x
(E) tan x
A. Incorrect!
The graph shows two asymptotes but the graph of cos x does not contain any
asymptotes.
B. Correct!
The trigonometric function cot x decreases as x increases, has a period of π, and
has vertical asymptotes located at integer multiples of π.
C. Incorrect!
The graph shows an x-intercept but the graph of sec x does not contain an xintercept.
Feedback
D. Incorrect!
The graph shows two asymptotes but the graph of sin x does not contain any
asymptotes.
E. Incorrect!
The vertical asymptotes of tan x are located at odd integer multiples of
π
2
.
The graph has the following characteristics:
1) vertical asymptotes at x = 0 and x = π
2) an x-intercept at
x =
π
2
3) decreases from left to right
Solution
The only listed function whose graph possesses all of these characteristics is cot x.
The correct answer is (B).
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Question No. 2 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
2. What is the period of the function cos (4x)?
(A)
(B)
Question
(C)
π
8
π
4
π
2
(D) 4π
(E) 8π
A. Incorrect!
The period of cos (4x) is equal to
2π
b
.
2π
b
.
B. Incorrect!
The period of cos (4x) is equal to
C. Correct!
The period of cos (4x) is equal to
Feedback
2π
2π
π
=
=
b
4
2
.
D. Incorrect!
The period of cos (4x) is equal to
2π
b
.
2π
b
.
E. Incorrect!
The period of cos (4x) is equal to
In cos (4x), the value of b is 4. Therefore, the period of cos (4x) is:
2π
2π
π
=
=
b
4
2
The correct answer is (C).
Solution
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Question No. 3 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
3. What is the period of the function tan
?
π
8
π
4
(A)
(B)
Question
x
8
(C)
π
2
(D) 4π
(E) 8π
A. Incorrect!
The period of
tan
x
8
is equal to
π
b
. Please try again.
tan
x
8
is equal to
π
b
. Please try again.
tan
x
8
is equal to
π
b
. Please try again.
tan
x
8
is equal to
π
b
. Please try again.
tan
x
8
is equal to
π
1
= π ÷
= 8π
b
8
B. Incorrect!
The period of
C. Incorrect!
The period of
Feedback
D. Incorrect!
The period of
E. Correct!
The period of
In
tan
x
8
, the value of b is
1
8
.
. Therefore, the period of
tan
x
8
is:
π
1
= π ÷
= 8π
b
8
The correct answer is (E).
Solution
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Question No. 4 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
4. The image shows the graph of one period of which functions?
Question
(A) -3 cos x
(B) 3 cos x
(C) cos (3x)
(D) -3 sin x
(E) 3 sin x
A. Incorrect!
The trigonometric function corresponding to this graph will be equal to zero at x =
0. This is not the graphed function because -3 cos 0 = -3.
B. Incorrect!
The trigonometric function corresponding to this graph will be equal to zero at x =
0. This is not the graphed function because 3 cos 0 = 3.
C. Incorrect!
The trigonometric function corresponding to this graph will be equal to zero at x =
0. This is not the graphed function because because cos(3 · 0) = cos 0 = 1.
Feedback
D. Correct!
The graph resembles the graph of 3 sin x except for the locations of maxima and
minima of the given graph are reversed. This means that the factor multiplying sin
x must be negative.
E. Incorrect!
The graph resembles the graph of 3 sin x except for the locations of maxima and
minima of the given graph are reversed. This means that the factor multiplying sin
x must be negative.
The given graph has:
1) x-intercepts at x = 0, x = π, and x = 2π
2) a minimum at
3) a maximum at
π
2
3π
x =
2
x =
4) the amplitude is 3
Solution
The graph resembles the graph of 3 sin x except for the locations of maxima and
minima of the given graph are reversed. This means that the factor multiplying sin
x must be negative.
The correct answer is (D).
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Question No. 5 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
5. Which of the following values of x does not correspond to an asymptote of the
trigonometric function tan (3x)?
(A)
(B)
(C)
Question
(D)
(E)
π
6
π
3
π
2
3π
2
5π
2
A. Incorrect!
π
6
This is an odd multiple of
. Therefore, this corresponds to an asymptote of the
trigonometric function tan (3x).
B. Correct!
π
6
This is an even multiple of
, so it is not an asymptote of tan (3x).
c. Incorrect!
π
6
This is an odd multiple of
. Therefore, this corresponds to an asymptote of the
trigonometric function tan (3x).
Feedback
D. Incorrect!
π
6
This is an odd multiple of
. Therefore, this corresponds to an asymptote of the
trigonometric function tan (3x).
E. Incorrect!
π
6
This is an odd multiple of
. Therefore, this corresponds to an asymptote of the
trigonometric function tan (3x).
The asymptotes of the tangent function occur when its argument is equal to an odd
multiple of
π
2
. It follows that the asymptotes of tan (3x) occur when the following
relation is obeyed:
Solution
3x =
±
π
2
x=
π
6
,
±
3π
5π
, ± ,...
2
2
3π
5π
±
, ± ,...
6
6
,
±
The asymptotes of tan (3x) occur at odd multiples of
an odd multiples of
π
6
is
π
3
because
π
2π
=
3
6
π
6
. The only value that is not
.
The correct answer is (B).
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Question No. 6 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
6. Choose the answer choice that fills the two blanks.
The graph of
⎛x π⎞
sin ⎜ − ⎟
⎝3 2⎠
is shifted to the _____ by _____ units with respect to the
graph of sin x.
π
6
3π
(B) left,
2
π
(C) right,
6
3π
(D) right,
2
1
(E) right,
3
(A) left,
Question
A. Incorrect!
Recall that a horizontal shift is determined using
c
b
.
c
b
.
c
b
.
B. Incorrect!
Recall that a horizontal shift is determined using
C. Incorrect!
Recall that a horizontal shift is determined using
Feedback
D. Correct!
You used the fact that a horizontal shift is determined using
c
b
.
E. Incorrect!
Recall that a horizontal shift is determined using
c
b
.
Recall that a horizontal shift is determined using
c
b
. In this function
c =
Solution
π
2
.
c
π
1
=
÷
b
2
3
π 3
=
⋅
2 1
3π
=
2
Since the ratio is positive, the graph is shifted to the right.
The correct answer is (D).
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b =
1
3
and
Question No. 7 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
7. Which of the following trigonometric functions correspond to the graph shown in
the figure?
Question
(A) f(x)
(B) f(x)
(C) f(x)
(D) f(x)
(E) f(x)
=
=
=
=
=
-5 cos x
-2 – 3 cos x
2 – 3 cos x
-2 + 3 cos x
2 + 3 cos x
A. Incorrect!
This function corresponds to a cosine graph with amplitude of 5 that has been
reflected through the x-axis (i.e., "turned upside down ").
B. Incorrect!
This function corresponds to a cosine graph with amplitude of 3 that has been
reflected through the x-axis (i.e., "turned upside down ") and shifted down by 2
units.
C. Incorrect!
This function corresponds to a cosine graph with amplitude of 3 that has been
reflected through the x-axis (i.e., "turned upside down ") and shifted up by 2 units.
Feedback
D. Correct!
This function corresponds to a cosine graph with amplitude of 3 that has been
shifted down by 2 units.
E. Incorrect!
This function corresponds to a cosine graph with amplitude of 3 that has been
shifted up by 2 units.
The graph contains maxima at x = 0 and x = 2π. The graph as a minimum at x = π
and no vertical asymptotes. These facts suggest that the graph is a cosine graph
that has not been horizontally shifted.
The vertical distance between the maxima and minimum is 6. The amplitude equals
one-half of this vertical distance. Therefore, the amplitude equals 3.
Solution
The value of the maximum is 1 and not 3. Therefore, the graph has been shifted
down by 2 units.
The correct answer is (D).
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Question No. 8 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
8. Which of the following pairs of trigonometric functions do not intersect the xaxis?
Question
(A) cos x and sin x
(B) tan x and cot x
(C) sec x and csc x
(D) cot x and sec x
(E) sin x and csc x
A. Incorrect!
Cos x intersects the x-axis at odd integer multiples of
π
2
while sin x intersects the
x-axis at integer multiples of π.
B. Incorrect!
Tan x intersects the x-axis at integer multiples of π while cot x intersects the x-axis
π
2
at odd integer multiples of .
C. Correct!
Neither sec x and csc x intersect the x-axis.
Feedback
D. Incorrect!
π
2
Cot x intersects the x-axis at odd integer multiples of .
E. Incorrect!
Sin x intersects the x-axis at integer multiples of π.
Sec x and csc x are the only basic trigonometric functions that do not intersect the
x-axis.
The correct answer is (C).
Solution
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Question No. 9 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
9. Which of the following trigonometric functions correspond to the graph shown in
the figure?
Question
f ( x ) = sec
x
2
f(x) = sec (2x)
f(x) = cot x
f ( x ) = csc
x
2
f(x) = csc(2x)
A. Correct!
The graph of
f ( x ) = sec
x
2
has a period of 4π and vertical asymptotes at odd integer
multiples of π.
B. Incorrect!
The graph of f(x) = sec (2x) has a period of π and vertical asymptotes at odd
integer multiples of
π
4
.
C. Incorrect!
The graph of f(x) = cot x has a period of π and vertical asymptotes at integer
multiples of π.
Feedback
D. Incorrect!
The graph of
f ( x ) = csc
x
2
f ( x ) = csc
x
2
has the same period as the graph within the figure, but
has vertical asymptotes at even integer multiples of π.
E. Incorrect!
The graph of f(x) = csc(2x) has a period of π and vertical asymptotes at integer
multiples of
π
2
.
The primary two characteristics of the graph presented within the figure are:
1) the graph has a period of 4π
2) the graph has vertical asymptotes at odd integer multiples of π
Solution
The only listed trigonometric function with these characteristics is
The correct answer is (A).
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f ( x ) = sec
x
2
.
Question No. 10 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
10. The presence of the factor e − x within the trigonometric expression 4e − x sin(2 x )
causes:
Question
(A) The
(B) The
(C) The
(D) The
(E) The
amplitude to increase as x increases.
amplitude to decrease as x increases.
graph to shift downward.
graph to shift upward.
graph to shift to the left.
A. Incorrect!
The negative sign within the exponent of
increases.
e
−x
causes the amplitude to decrease as x
B. Correct!
The negative sign within the exponent of
increases.
e
−x
causes the amplitude to decrease as x
C. Incorrect!
Vertical shifts are controlled by the constant that is added to the trigonometric
function.
Feedback
D. Incorrect!
Vertical shifts are controlled by the constant that is added to the trigonometric
function.
E. Incorrect!
Horizontal shifts are determined using the b and c values of the trigonometric
function.
The negative sign within the exponent of e − x causes the amplitude to decrease as x
increases. The amplitude is equal to 4 when x = 0 and decreases as we travel to
the right.
The correct answer is (B).
Solution
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