Study Guide and Review - Chapter 4
Locate the vertical asymptotes, and sketch the graph of each function.
45. y = 3 tan x
SOLUTION: The graph of y = 3 tan x is the graph of y = tan x expanded vertically. The period is
or . Find the location of
two consecutive vertical asymptotes.
and Create a table listing the coordinates of key points for y = 3 tan x for one period on
.
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = tan x
y = 3 tan x
(0, 0)
(0, 0)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
46. SOLUTION: eSolutions Manual - Powered by Cognero
The graph of
Page 1
is the graph of y = tan x compressed vertically and translated
units right. The Study Guide and Review - Chapter 4
46. SOLUTION: is the graph of y = tan x compressed vertically and translated
The graph of
units right. The or . Find the location of two consecutive vertical asymptotes.
period is
and Create a table listing the coordinates of key points for
for one period on [0,
].
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = tan x
x =0
(0, 0)
Intermediate
Point
Vertical
Asymptote
x=π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
47. SOLUTION: eSolutions Manual - Powered by Cognero
The graph of
Page 2
is the graph of y = cot x shifted
location of two consecutive vertical asymptotes.
units to the left. The period is
or . Find the
Study Guide and Review - Chapter 4
47. SOLUTION: is the graph of y = cot x shifted
The graph of
units to the left. The period is
or . Find the
location of two consecutive vertical asymptotes.
and Create a table listing the coordinates of key points for
for one period on
.
Function
y = cot x
Vertical
Asymptote
Intermediate
Point
x-int
x =0
Intermediate
Point
Vertical
Asymptote
x=π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
48. y = −cot (x –
)
SOLUTION: The graph of
period is
is the graph of y = cot x reflected in the x-axis and shifted π units to the left. The
or . Find the location of two consecutive vertical asymptotes.
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and Page 3
Study Guide and Review - Chapter 4
48. y = −cot (x –
)
SOLUTION: is the graph of y = cot x reflected in the x-axis and shifted π units to the left. The
The graph of
or . Find the location of two consecutive vertical asymptotes.
period is
and Create a table listing the coordinates of key points for
for one period on [ , 2 ].
Function
y = cot x
Vertical
Asymptote
Intermediate
Point
x-int
x =0
Intermediate
Point
Vertical
Asymptote
x=π
x = 2π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
49. SOLUTION: The graph of
is the graph of y = sec x expanded vertically and expanded horizontally. The period is
or 4 . Find the location of two consecutive vertical asymptotes.
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and Page 4
Study Guide and Review - Chapter 4
49. SOLUTION: is the graph of y = sec x expanded vertically and expanded horizontally. The period is
The graph of
or 4 . Find the location of two consecutive vertical asymptotes.
and Create a table listing the coordinates of key points for
for one period on [−π, 3π].
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = sec x
(0, 1)
(0, 1)
x=π
Intermediate
Point
Vertical
Asymptote
x = 3π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
50. y = –csc (2x)
SOLUTION: The graph of
is the graph of y = csc x compressed horizontally and reflected in the x-axis. The period is
or . Find the location of two consecutive vertical asymptotes.
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and Page 5
Study Guide and Review - Chapter 4
50. y = –csc (2x)
SOLUTION: is the graph of y = csc x compressed horizontally and reflected in the x-axis. The period is
The graph of
or . Find the location of two consecutive vertical asymptotes.
and Create a table listing the coordinates of key points for
for one period on
.
Function
y = csc x
Vertical
Asymptote
Intermediate
Point
x-int
y = –csc
2x
x = −π
Intermediate
Point
Vertical
Asymptote
x =0
x =0
x=π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
51. y = sec (x –
)
SOLUTION: The graph of y = sec (x −
) is the graph of y = sec x translated π units to the right. The period is
or 2 . Find
the location of two vertical asymptotes.
and eSolutions Manual - Powered by Cognero
Page 6
Study Guide and Review - Chapter 4
51. y = sec (x –
)
SOLUTION: The graph of y = sec (x −
) is the graph of y = sec x translated π units to the right. The period is
or 2 . Find
the location of two vertical asymptotes.
and Create a table listing the coordinates of key points for y = sec (x − π) for one period on
.
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = sec x
y = sec (x −
π)
(0, 1)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
52. eSolutions
Manual - Powered by Cognero
SOLUTION: The graph of
Page 7
is the graph of y = csc x compressed vertically and translated
units to the left. Study Guide and Review - Chapter 4
52. SOLUTION: is the graph of y = csc x compressed vertically and translated
The graph of
The period is
units to the left. or 2 . Find the location of two consecutive vertical asymptotes.
and Create a table listing the coordinates of key points for
for one period on
.
Function
Vertical
Asymptote
Intermediate
Point
x-int
Intermediate
Point
Vertical
Asymptote
y=
csc x
x =0
x=π
Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one
more cycle to the left and right of the first curve.
Find the exact value of each expression, if it exists.
53. sin−1 (−1)
eSolutions Manual - Powered by Cognero
Page 8
SOLUTION: Find a point on the unit circle on the interval
with a y-coordinate of –1.
Study Guide and Review - Chapter 4
Find the exact value of each expression, if it exists.
53. sin−1 (−1)
SOLUTION: with a y-coordinate of –1.
Find a point on the unit circle on the interval
When t =
, sin t = –1. Therefore, sin
–1
–1 =
.
54. SOLUTION: Find a point on the unit circle on the interval
with a x-coordinate of
.
When t =
, cos t =
–1
. Therefore, cos
=
.
55. SOLUTION: Find a point on the unit circle on the interval
eSolutions Manual - Powered by Cognero
such that =
Page 9
–1
Study
Guide
- Chapter
When
t = ,and
cos t Review
=
. Therefore,
cos 4
=
.
55. SOLUTION: such that Find a point on the unit circle on the interval
When t =
, tan t =
. Therefore, tan
–1
=
= .
56. arcsin 0
SOLUTION: Find a point on the unit circle on the interval
with a y-coordinate of 0.
When t = 0, sin t = 0. Therefore, arcsin 0 = 0.
57. arctan −1
SOLUTION: Find a point on the unit circle on the interval
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such that =
Page 10
Study Guide and Review - Chapter 4
When t = 0, sin t = 0. Therefore, arcsin 0 = 0.
57. arctan −1
SOLUTION: such that Find a point on the unit circle on the interval
When t =
, tan t =
= . Therefore, arctan
=
.
58. arccos
SOLUTION: Find a point on the unit circle on the interval
with a x-coordinate of
When t =
=
, cos t =
. Therefore, arccos
.
.
59. SOLUTION: The inverse property applies, because
lies on the interval [–1, 1]. Therefore,
=
.
60. SOLUTION: The inverse property applies, because cos –3π lies on the interval [–1, 1]. Therefore,
eSolutions Manual - Powered by Cognero
Page 11
SOLUTION: Study
and Review
- Chapter lies on the interval [–1,
4
TheGuide
inverse property
applies, because
1]. Therefore,
=
.
60. SOLUTION: The inverse property applies, because cos –3π lies on the interval [–1, 1]. Therefore,
eSolutions Manual - Powered by Cognero
Page 12