Hmwk 9 over sec. 2.3
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School
Western Kentucky University *
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Course
117
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
27
Uploaded by CountThunderYak33
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Hmwk 9 over sec. 2.3-Rachel Chrisman
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1.
2.
3.
4.
5.
Student: Rachel Chrisman
Date: 01/01/24
Instructor: Robin Ayers
Course: MATH 117 Trigonometry On Demand
July 2023 - July 2025
Assignment: Hmwk 9 over sec. 2.3
Which of the following statements is true?
Choose the correct answer.
A.
The x-intercepts of are the same as the x-coordinates of the center points of .
y =
tan
x
y =
tan
x
B.
The domain of is all real numbers except for values of x for which .
y =
tan
x
sin
x = 0
C.
The zeros of are all values of x for which .
y =
tan
x
cos
x = 0
D.
The function is one-to-one.
y =
tan
x
When sketching the graph of which of the following best describes how to determine the x-coordinates of the halfway points of the principal cycle?
y = A
tan
(
) + D
Bx + C
Choose the correct answer.
A.
The x-coordinate of each halfway point is located halfway between the x-coordinate of a local maximum point and the x-coordinate B.
The x-coordinate of the halfway points are and .
x =
B
C
x = −
B
C
C.
The x-coordinate of each halfway point is located halfway between the x-coordinate of the center point and a vertical asymptote.
D.
The x-coordinate of each halfway point is located halfway between the two vertical asymptotes.
Which of the following statements is true?
Choose the correct answer.
A.
The domain of is all real numbers except for values of x for which .
y =
cot
x
cos
x = 0
B.
The principal cycle of the graph of occurs on the interval .
y =
cot
x
−
2
,
2
C.
The function has infinitely many vertical asymptotes with equations where n is an integer.
y =
cot
x
x = n
D.
The zeros of are of the form where n is an integer.
y =
cot
x
n
2
Which of the following statements is not
true?
Choose the correct answer.
A.
The function obtains a relative maximum at where n is an integer.
y =
csc
x
x =
+ 2 n
3
2
B.
The graph of has infinitely many vertical asymptotes of the form where n is an integer.
y =
csc
x
x = n
C.
The range of is .
y =
csc
x
(
[
]
−
, − 1
)
1,
D.
The graph of is symmetric about the y-axis.
y =
csc
x
Which of the following statements is not
true?
Choose the correct answer.
A.
The domain of is .
y =
sec
x
x
x ≠
(
)
2n + 1
2
, where n is an integer
B.
The range of is identical to the range of .
y =
sec
x
y =
csc
x
C.
The function obtains a local maximum at values of x for which the function obtains a local maximum.
y =
sec
x
y =
cos
x
D.
The period of is .
y =
sec
x
P = 2
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
2/27
6.
7.
8.
9.
Sketch the graph of y
, and then identify as many of the following properties that apply. Answer (a)
through (e)
.
=
tan
x
Choose the correct graph of the function below.
A.
-4
4
x
y
B.
-4
4
x
y
C.
-4
4
y
(a)
Is the function an function?
odd
no
yes
(b)
Is the y-intercept equal to ?
0
no
yes
(c)
Is the ?
range ( −
,
)
no
yes
(d)
?
Does every halfway point have an x coordinate of − 1 or 1
yes
no
(e)
the ?
Is
interval for the graph of the principal cycle (0,
)
no
yes
List all points on the graph of y
on the interval that have a y-coordinate of .
=
tan
x
−
2
,4
−
3
(Type an ordered pair. Use a comma to separate answers as needed. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Use the periodic property of y
to determine which of the following expressions is equivalent to .
=
tan
x
tan
5
4
i. tan
−
3
4
ii. tan
3
4
iii. tan
−
4
iv. tan
7
4
Choose the correct answer below.
A.
tan
3
4
B.
tan
−
4
C.
tan
−
3
4
D.
tan
7
4
Given the following function, determine the interval for the principal cycle.
y = 3
tan
x +
6
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
3
2
−
3
2
3
2
−
3
2
−
3
2
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
3/27
10.
11.
12.
13.
14.
15.
Given the following function, determine the interval for the principal cycle.
y = 4
tan
( − 3x +
) + 2
Which of the following functions is equivalent to y
?
= 4
tan
( − 3x +
) + 2
A.
y = 4
tan
(3x −
) + 2
B.
y = − 4
tan
(3x −
) + 2
C.
y = 4
tan
(3x +
) − 2
D.
y = − 4
tan
(3x +
) − 2
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Determine the period of the function.
y = 4
tan
x −
3
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the following function, determine the equations of the vertical asymptotes of the principal cycle.
y = 5
tan
x +
3
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Type equations. Simplify your answers. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
Given the following function, determine the equations of the vertical asymptotes of the principal cycle.
y = 4
tan
( − 3x +
) − 1
Which of the following functions is equivalent to y
?
= 4
tan
( − 3x +
) − 1
A.
y = 4
tan
(3x +
) + 1
B.
y = − 4
tan
(3x −
) − 1
C.
y = 4
tan
(3x −
) − 1
D.
y = − 4
tan
(3x +
) + 1
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Type equations. Simplify your answers. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
Given the following function, determine the coordinates of the center point of the principal cycle.
y = 4
tan
x +
3
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the following function, determine the coordinates of the center point of the principal cycle.
y = 3
tan
( − 4x −
) − 1
Which of the following functions is equivalent to y
?
= 3
tan
( − 4x −
) − 1
A.
y = 3
tan
(4x −
) + 1
B.
y = 3
tan
(4x +
) − 1
C.
y = − 3
tan
(4x +
) − 1
D.
y = − 3
tan
(4x −
) + 1
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
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Hmwk 9 over sec. 2.3-Rachel Chrisman
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16.
17.
Determine the coordinates of the two halfway points of the principal cycle of the function.
y = 2
tan
x −
3
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y =
tan
x +
6
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period of is .
y =
tan
x +
6
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y =
tan
x +
6
A.
-3
3
x
y
B.
-3
3
x
y
C.
-3
3
−
2
3
3
−
2
3
3
−
2
3
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
5/27
18.
19.
Given the following function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 4
tan
x −
6
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Choose the correct graph below. A.
0
-8
8
x
y
B.
0
-8
8
x
y
C.
0
-8
8
Given the function y
, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
=
tan
(6x −
)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote of the principal cycle is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of y
. Choose the correct graph below.
=
tan
(6x −
)
6
6
A.
0
-2
-1
1
2
x
y
B.
0
-2
-1
1
2
x
y
C.
0
-2
-1
1
2
y
2
3
−
3
−
2
3
3
−
2
3
−
4
−
12
−
4
−
12
12
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
6/27
20.
21.
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y =
tan
(4x)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y =
tan
(4x)
A.
-2
2
x
y
B.
-2
2
x
y
C.
-2
2
y
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 4
tan
( − 3x)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is . (Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y = 4
tan
( − 3x)
A.
-8
8
x
y
B.
-8
8
x
y
C.
-8
8
y
−
8
8
− 2
2
−
8
−
6
6
−
6
6
−
3
2
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Hmwk 9 over sec. 2.3-Rachel Chrisman
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22.
23.
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 3
tan
(2x −
) − 2
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period of is .
y = 3
tan
(2x −
) − 2
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y = 3
tan
(2x −
) − 2
A.
0
-12
-6
6
12
x
y
B.
0
-6
-3
3
6
x
y
C.
0
-6
-3
3
6
y
Given the function y
, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
= 3
tan
( − 4x +
) − 1
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identity the graph of the principal cycle of y
. Choose the correct graph below.
= 3
tan
( − 4x +
) − 1
A.
0
-12
-6
6
12
x
y
B.
0
-6
-3
3
6
x
y
C.
0
-12
-6
6
12
y
2
3
2
4
3
4
4
2
3
2
8
3
8
2
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
8/27
24.
25.
26.
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 2 −
tan
x −
3
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period of is .
y = 2 −
tan
x −
3
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of . Choose the correct graph below.
y = 2 −
tan
x −
3
A.
0
-12
6
x
y
B.
0
-6
6
x
y
C.
0
-6
6
y
Sketch the following function.
y =
tan
x +
3
Choose the correct graph below.
A.
-2
-1
1
2
x
y
B.
-2
-1
0
1
2
x
y
C.
Sketch the following function.
y = 3
tan
(2x +
) + 2
Choose the correct graph below.
A.
-8
-4
0
4
8
x
y
B.
-8
-4
0
4
8
x
y
C.
2
3
2
−
6
5
6
−
6
−
5
6
6
6
7
6
−
5
6
−
3
4
−
4
−
3
4
−
4
−
3
4
−
4
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
9/27
27.
28.
29.
Sketch the following function.
y = − 3
tan
(
− 2x) + 1
Choose the correct graph below.
A.
-12
-6
0
6
12
x
y
B.
-6
-3
0
3
6
x
y
C.
-6
-3
0
3
6
y
Sketch the graph of y
, and then identify as many of the following properties that apply. Answer (a)
through (e)
.
=
cot
x
Choose the correct graph of the function below.
A.
-4
4
x
y
B.
-4
4
x
y
C.
-4
4
y
(a)
Is the function an function?
even
no
yes
(b)
Is the y-intercept equal to ?
0
yes
no
(c)
Is the ?
range all real numbers except odd integer multiples of 2
yes
no
(d)
?
Are the zeros of the form where n is an odd integer
n
2
no
yes
(e)
the ?
Does
function have a period of P =
no
yes
List all points on the graph of y
on the interval that have a y-coordinate .
=
cot
x
−
2
,3
of −
3
The points are .
(Type an ordered pair. Use a comma to separate answers as needed. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
4
3
4
4
3
4
4
2
−
2
−
−
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
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30.
31.
32.
33.
34.
35.
Use the periodic property of y
to determine which of the following expressions is equivalent to .
=
cot
x
cot
16
7
i. cot
−
7
ii. cot
−
5
7
iii. cot
15
7
iv. cot
13
7
Choose the correct answer below.
A.
cot
−
7
B.
cot
15
7
C.
cot
13
7
D.
cot
−
5
7
Given the following function, determine the interval for the principal cycle.
y =
cot
(3x)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the following function, determine the interval for the principal cycle.
y =
cot
(
− 2x)
Which of the following functions is equivalent to ?
y =
cot
(
− 2x)
A.
y =
cot
(2x −
)
B.
y = −
cot
(2x +
)
C.
y = −
cot
(2x −
)
D.
y =
cot
(2x +
)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Determine the period of the function.
y =
cot
(4x)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the following function, determine the equations for the vertical asymptotes of the principal cycle.
y =
cot
(2x)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Type equations. Simplify your answers. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
Given the following function, determine the equations for the vertical asymptotes of the principal cycle.
y =
cot
(
− 3x)
Which of the following functions is equivalent to ?
y =
cot
(
− 3x)
A.
y = −
cot
(
− 3x)
B.
y =
cot
(3x +
)
C.
y =
cot
(3x −
)
D.
y = −
cot
(3x −
)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Type equations. Simplify your answers. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
11/27
36.
37.
38.
Given the following function, determine the coordinates of the center point of the principal cycle.
y =
cot
(2x)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Determine the coordinates of the two halfway points of the principal cycle of the function.
y =
cot
(4x)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Given the function , determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y =
cot
x +
6
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period of is .
y =
cot
x +
6
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most halfway point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y =
cot
x +
6
A.
0
-2
2
x
y
B.
0
-2
2
x
y
C.
0
-2
2
y
−
5
6
6
−
6
5
6
−
6
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
12/27
39.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 2
cot
x +
4
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
0
-4
4
x
y
B.
0
-4
4
x
y
C.
-4
4
−
3
4
4
3
4
−
4
−
3
4
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
13/27
40.
41.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y =
cot
(10x)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period of is .
y =
cot
(10x)
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Identify the graph of the principal cycle of . Choose the correct graph below.
y =
cot
(10x)
A.
-2
2
x
y
B.
-2
2
x
y
C.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 4
cot
( − 11x)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
-8
-4
4
8
x
y
B.
-8
-4
4
8
x
y
C.
-8
-4
4
8
y
0
10
0
−
10
− 10
0
11
0
11
0
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
14/27
42.
43.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = − 3
cot
( − 4x −
)
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The left vertical asymptote of the principal cycle is and the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answers, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
0
-6
-3
3
6
x
y
B.
0
-6
-3
3
6
x
y
C.
0
-6
-3
3
6
y
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 2
cot
(3x −
) + 1
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is
.
(Simplify your answers. Type equations. Type exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
0
-6
-3
3
6
x
y
B.
0
-6
-3
3
6
x
y
C.
0
-6
-3
3
6
y
−
4
−
4
3
2
3
−
2
3
−
3
3
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
15/27
44.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = 1 −
cot
x +
3
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
0
-1
1
2
3
x
y
B.
0
-1
1
2
3
x
y
C.
0
-1
1
2
3
y
2
3
−
3
3
−
2
3
−
3
Your preview ends here
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
16/27
45.
46.
47.
Given the function, determine the interval for the principal cycle. Determine the period. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
y = − 4
cot
(
− 3x) + 1
The interval of the principal cycle is .
(Type your answer in interval notation. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The period is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The equation of the left vertical asymptote of the principal cycle is and the equation of the right vertical asymptote is .
(Simplify your answers. Type equations. Type exact answer, using as needed. Use integers or fractions for any numbers in the equations.)
The coordinates of the center point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the left-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
The coordinates of the right-most half way point of the principal cycle are .
(Type an ordered pair. Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Sketch the graph. Choose the correct graph below.
A.
0
-8
-4
4
8
x
y
B.
0
-8
-4
4
8
x
y
C.
0
-8
-4
4
8
y
Sketch the function.
y =
cot
x +
6
Choose the correct graph below.
A.
-2
-1
1
2
B.
-2
-1
1
2
C.
-2
-1
0
1
2
Sketch the function.
y = 4
cot
(2x −
) − 2
Choose the correct graph below.
A.
-10
-5
0
5
10
x
y
B.
-10
-5
0
5
10
x
y
C.
3
2
3
−
3
−
2
3
3
−
6
5
6
11
6
−
6
5
6
11
6
5
6
−
−
2
2
−
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
17/27
48.
49.
Sketch the function.
y = − 3
cot
(
− 2x) − 1
Choose the correct graph below.
A.
-6
-3
0
3
6
x
y
B.
-6
-3
0
3
6
x
y
C.
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
tan
(Bx − C) + D
>
-6
-4
-2
2
4
6
x
y
The period of this function is . (Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of B?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of C?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of D?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of A?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What function of the form , where B
0, is represented by the given graph?
y = A
tan
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
2
−
−
2
−
,0
2
, − 5
4
,5
3
4
x = 0
x =
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
18/27
50.
51.
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
tan
(Bx − C) + D
>
-8
-6
-4
-2
2
4
6
8
x
y
The period of this function is .
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of B?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of C?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of D?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of A?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What function of the form where B
0 is represented by the given graph?
y = A
tan
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
tan
(Bx − C) + D
>
-4
-2
2
x
y
The period of this function is . (Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of B?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of C?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of D?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What is the value of A?
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
What function of the form for B
0 is represented by the given graph?
y = A
tan
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
( ,3)
, − 1
3
4
,7
5
4
x =
2
x =
3
2
−
, − 1
4
, − 3
− 5
16
,1
− 3
16
x =
− 3
8
x = −
8
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
19/27
52.
53.
54.
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
tan
(Bx − C) + D
>
-4
-2
2
4
x
y
What function of the form , where B
0, is represented by the given graph?
y = A
tan
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
tan
(Bx − C) + D
>
-8
-6
-4
-2
2
4
6
8
x
y
What function of the form where B
0 is represented by the given graph?
y = A
tan
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
The principal cycle of the graph of a trigonometric function of the form for B
0 is given. Determine the equation of the function represented by the graph.
y = A
cot
(Bx − C) + D
>
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
x
y
What function of the form for B
0 is represented by the given graph?
y = A
cot
(Bx − C) + D
>
(Type an equation. Type your answer in standard form. Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)
,0
2
,3
4
, − 3
3
4
x = 0
x =
( −
,1)
, − 4
− 5
4
,6
− 3
4
x =
− 3
2
x = −
2
,0
3
4
,5
5
8
, − 5
7
8
x =
2
x =
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
20/27
55.
56.
57.
58.
59.
Sketch the graph of y
, and then identify as many of the following properties that apply. Answer (a)
through (e)
.
=
csc
x
Choose the correct graph of the function below.
A.
-4
4
x
y
B.
-4
4
x
y
C.
-4
4
y
(a) Is the function an function?
even
yes
no
(b) Does the function obtain a relative maximum at x
where n is an ?
=
n
odd integer
yes
no
(c) Is the ?
range ( −
, − 1]
[1,
)
no
yes
(d) ?
Are the equations of the vertical asymptotes of the form x = n
where n is an integer
yes
no
(e) the ?
Does
function have a period of P = 2
no
yes
Determine the equations of the vertical asymptotes of the graph of y
on the interval .
5
= 3
csc
(x −
)
[
]
−
,3
Choose the correct answer below.
A.
, x =
−
, x =
, x =
2
2
3
2
x = 2
, and x = 3
B.
, x =
−
, x = 0, x =
x = 2
, and x = 3
C.
, x =
−
, x = 0, x =
2
3
2
x = 2
, and x = 3
D.
, x =
−
, x = 0, x =
2
2
x = 2
, and x = 3
Determine the coordinates of the relative minimum points of the graph of y
on the interval .
three
= 5
sec
(x +
)
[
]
− 3 ,
The coordinates of the relative minimum point(s) of y
on the interval is(are) .
= 5
sec
(x +
)
[
]
− 3 ,
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
Determine the coordinates of the relative minimum points of the graph of y
on the interval .
three
= 2
sec
(4x +
) + 3
−
3
4
,
4
The coordinates of the relative minimum point(s) of y
on the interval is(are) . = 2
sec
(4x +
) + 3
−
3
4
,
4
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
Determine the coordinates of the two relative maximum points of the graph of y
on the interval .
= 5
sec
(3x +
) + 3
−
,
3
The coordinates of the relative maximum point(s) of y
on the interval
is(are) . = 5
sec
(3x +
) + 3
−
,
3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
2
−
− 2
2
−
− 2
−
− 2
1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
21/27
60.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y = 3
csc
(x +
)
a) First sketch two cycles of the corresponding reciprocal function. Which of the following graphs represents the graph of the corresponding reciprocal function?
A.
-2
2
-4
4
x
y
B.
2
-2
-4
4
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
on the two cycles selected in part a above?
= 3
sin
(x +
)
= 3
csc
(x +
)
A.
, x =
−
, x =
−
, x =
−
, x =
5
2
3
2
2
2
and x =
B.
x =
− 3
, x =
− 2
, and x =
−
C.
, x =
− 3
, x =
− 2
, x =
−
, x = 0
and x =
D.
x =
−
, x =
−
, x =
−
, and x =
5
2
3
2
2
2
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of y
on the two cycles selected in part a above is (are) .
y = 3
csc
(x +
)
= 3
csc
(x +
)
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of y
on the two cycles selected in part a above is (are) .
y = 3
csc
(x +
)
= 3
csc
(x +
)
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
= 3
csc
(x +
)
A.
-2
2
-4
4
x
y
B.
2
-2
-4
4
x
y
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
22/27
61.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y =
sec
x −
3
a) First sketch two cycles of the corresponding reciprocal function. Which of the following graphs represents the graph of the corresponding reciprocal function?
A.
-1
1
-2
2
x
y
B.
-1
1
-2
2
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
on the two cycles selected in part a above?
=
cos
x −
3
=
sec
x −
3
A.
x =
−
, x =
−
, and x =
7
6
6
5
6
B.
x =
−
, x =
−
, x =
, and x =
5
3
2
3
3
4
3
C.
, x =
−
, x =
−
, x =
, x =
5
3
2
3
3
4
3
and x =
11
6
D.
, x =
−
, x =
−
, x =
7
6
6
5
6
and x =
11
6
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of on the two cycles selected in part a above is(are) . y =
sec
x −
3
y =
sec
x −
3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of on the two cycles selected in part a above is(are) .
y =
sec
x −
3
y =
sec
x −
3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
=
sec
x −
3
A.
-1
1
-2
2
x
y
B.
1
-1
-2
2
x
y
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
23/27
62.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y = −
sec
x +
3
a) First sketch two cycles of the corresponding reciprocal function. Which of the following graphs represents the graph of the corresponding reciprocal function?
A.
-1
1
-2
2
x
y
B.
1
-1
-2
2
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
on the two cycles selected in part a above?
= −
cos
x +
3
= −
sec
x +
3
A.
x =
−
, x =
−
, and x =
11
6
5
6
6
B.
x =
−
, x =
−
, x =
−
, and x =
7
3
4
3
3
2
3
C.
, x =
−
, x =
−
, x =
11
6
5
6
6
and x =
7
6
D.
, x =
−
, x =
−
, x =
−
, x =
7
3
4
3
3
2
3
and x =
7
6
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of on the two cycles selected in part a above is(are) . y = −
sec
x +
3
y = −
sec
x +
3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of on the two cycles selected in part a above is(are) .
y = −
sec
x +
3
y = −
sec
x +
3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
= −
sec
x +
3
A.
1
-1
-2
2
x
y
B.
-1
1
-2
2
x
y
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
− 4
4
− 2
2
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
24/27
63.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y = 5
sec
(3x −
) + 3
a) First sketch two cycles of the corresponding reciprocal function
. Which of the following graphs represents the graph
of the corresponding reciprocal function
?
s
s
s
A.
-12
12
x
y
B.
-12
12
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
on the two cycles selected in part a above?
= 5
cos
(3x −
)
= 5
sec
(3x −
) + 3
A.
, x =
−
, x =
, x =
6
6
2
and x =
5
6
B.
x =
−
, x = 0, x =
, and x =
3
3
2
3
C.
x =
−
, x =
, and x =
6
6
2
D.
, x =
−
, x = 0, x =
, x =
3
3
2
3
and x =
5
6
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of on the two cycles selected in part a above is(are) . y = 5
sec
(3x −
) + 3
y = 5
sec
(3x −
) + 3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of on the two cycles selected in part a above is(are) .
y = 5
sec
(3x −
) + 3
y = 5
sec
(3x −
) + 3
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
= 5
sec
(3x −
) + 3
A.
-12
12
x
y
B.
-12
12
x
y
−
2
−
2
−
2
−
2
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
25/27
64.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y = 3 − 4
csc
(x −
)
a) First sketch two cycles of the corresponding reciprocal function
. Which of the following graphs represents the graph
of the corresponding reciprocal function
?
s
s
s
A.
-10
10
x
y
B.
-10
10
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
?
= − 4
sin
(x −
)
= 3 − 4
csc
(x −
)
A.
x =
−
, x = 0, and x =
B.
, x =
−
, x =
, x =
, x =
2
2
3
2
5
2
x = 2
, and x = 3
C.
x =
−
, x =
, x =
, and x =
2
2
3
2
5
2
D.
, x =
−
, x = 0, x =
x = 2
, and x = 3
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of y
is(are) . 3 − 4
csc
(x −
)
= 3 − 4
csc
(x −
)
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of y
is(are) .
3 − 4
csc
(x −
)
= 3 − 4
csc
(x −
)
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
= 3 − 4
csc
(x −
)
A.
-10
10
x
y
B.
-10
10
x
y
− 2
4
− 2
4
− 2
4
− 2
4
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
26/27
65.
66.
Determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of the function and sketch its graph by answering parts a) through e) below.
y =
sec
(
− 3x) + 4
a) First sketch two cycles of the corresponding reciprocal functions. Which of the following graphs represents the graphs of the corresponding reciprocal functions?
A.
-8
8
x
y
B.
-8
8
x
y
b) The vertical asymptotes occur at the x-intercepts of the reciprocal function y
. What are all of the vertical asymptotes of y
?
=
=
sec
(
− 3x) + 4
A.
x =
−
, x = 0,
3
and x =
5
6
B.
x =
−
, x =
, x =
,
6
6
2
and x =
5
6
C.
x =
−
, x =
, x =
, and x =
6
6
2
2
3
D.
x =
−
, x = 0, x =
, and x =
3
3
2
3
c) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative maximum point(s) of y
is(are) .
sec
(
− 3x) + 4
=
sec
(
− 3x) + 4
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
d) Using the graph of the reciprocal function in part a, sketch the graph of the function . The coordinates of the relative minimum point(s) of y
is(are) .
sec
(
− 3x) + 4
=
sec
(
− 3x) + 4
(Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.)
e) Which of the following represents the graph of y
?
=
sec
(
− 3x) + 4
A.
-8
8
x
y
B.
-8
8
x
y
Sketch the following function.
y =
csc
x +
6
Which of the following represents the graph of y
?
=
csc
x +
6
A.
1
-1
-2
2
x
y
B.
1
-1
-2
2
x
y
−
2
−
2
−
2
−
2
− 4
4
− 2
2
− 4
4
− 2
2
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1/1/24, 5:14 PM
Hmwk 9 over sec. 2.3-Rachel Chrisman
https://tdx.acs.pearsonprd.tech/api/v1/print/highered
27/27
67.
68.
Sketch the following function.
y = 5
csc
(3x +
) + 2
Which of the following represents the graph of y
?
= 5
csc
(3x +
) + 2
A.
-10
10
x
y
B.
-10
10
x
y
Sketch the following function.
y =
csc
(
− 4x) + 2
Which of the following represents the graph of y
?
=
csc
(
− 4x) + 2
A.
-6
6
x
y
B.
-6
6
x
y
C.
-6
6
x
y
− 2
− 2
−
2
−
2
−
2
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