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```4-5 Graphing Other Trigonometric Functions
Locate the vertical asymptotes, and sketch the graph of each function.
2.
SOLUTION:
is the graph of y = tan x shifted
The graph of
y = a tan (bx + c), so a = 1, b = 1, and c =
units to the left. The period is
or
.
. Use the tangent asymptote equations to find the location of the
asymptotes.
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = tan x
for one period on
.
y = tan (x +
4)
(0, 0)
Intermediate
Point
Vertical
Asymptote
(0, 1)
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
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4-5 Graphing Other Trigonometric Functions
4. y = –3 tan
SOLUTION:
The graph of y = –3 tan
axis. The period is
is the graph of y = tan x expanded vertically, expanded horizontally, and reflected in the x-
or 3 .
y = a tan (bx + c), so a = –3, b = , and c = 0. Use the tangent asymptote equations to find the location of two
consecutive vertical asymptotes.
Create a table listing the coordinates of key points for y = tan 2x for one period on
Function
.
y = tan x
Vertical
Asymptote
Intermediate
Point
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4-5 Graphing Other Trigonometric Functions
x-int
(0, 0)
(0, 0)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
6. y = –tan 3x
SOLUTION:
The graph of y = – tan 3x is the graph of y = tan x compressed horizontally and reflected in the x-axis. The period is
or
.
y = a tan (bx + c), so a = –1, b = 3, and c = 0. Use the tangent asymptote equations to find the location of two
consecutive vertical asymptotes.
Create a table listing the coordinates of key points for y = tan 2x for one period on
Function
y = tan x
.
y = 2 tan
x
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4-5 Graphing Other Trigonometric Functions
Vertical
Asymptote
Intermediate
Point
x-int
(0, 0)
(0, 0)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
8.
SOLUTION:
The graph of
is the graph of y = cot x expanded horizontally. The period is
or 2 .
y = a cot (bx + c), so a = 1, b = , and c = 0. Use the tangent asymptote equations to find the location of the
asymptotes.
Find the location of two consecutive vertical asymptotes.
Create a table listing the coordinates of key points for
for one period on
.
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4-5 Graphing Other Trigonometric Functions
Function
y = cot x
Vertical
Asymptote
Intermediate
Point
x-int
x=0
x=0
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
10.
SOLUTION:
is the graph of y = csc x compressed horizontally and shifted units to the left. The
The graph of
period is
or
.
y = a csc (bx + c), so a = 1, b = 4, and c =
asymptotes.
. Use the asymptote equations to find the location of two vertical
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4-5 Graphing Other Trigonometric Functions
Create a table listing the coordinates of key points for
Function
for one period on
.
y = csc x
Vertical
Asymptote
Intermediate
Point
x-int
Intermediate
Point
Vertical
Asymptote
x = –π
x=
x=0
x=
x=π
x=
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
12. y = –2 csc 3x
SOLUTION:
is the graph of y = csc x expanded vertically, compressed horizontally, and reflected in the
The graph of
x-axis. The period is
or
.
y = a csc (bx + c), so a = –2, b = 3, and c = 0. Use the asymptote equations to find the location of two vertical
asymptotes.
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4-5 Graphing Other Trigonometric Functions
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
y = csc x
y = –2 csc 3x
x=0
x=0
for one period on
.
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
14.
SOLUTION:
The graph of
is the graph of y = sec x expanded horizontally and shifted π units to the left. The
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4-5 Graphing Other Trigonometric Functions
period is
or 10 .
y = a sec (bx + c), so a = 1, b = , and c = . Use the asymptote equations to find the location of two vertical
asymptotes.
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
for one period on
.
y = sec x
(0, 1)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
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4-5 Graphing Other Trigonometric Functions
16. y = –sec
SOLUTION:
The graph of
is the graph of y = sec x expanded horizontally and reflected in the x-axis. The period is
or 16 .
y = a sec (bx + c), so a = –1, b = , and c = 0. Use the asymptote equations to find the location of two vertical
asymptotes.
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
for one period on [−4 , 12 ].
y = sec x
(0, 1)
(0, –1)
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4-5 Graphing Other Trigonometric Functions
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
Locate the vertical asymptotes, and sketch the graph of each function.
29. y = sec x + 3
SOLUTION:
The graph of y = sec x + 3 is the graph of y = sec x shifted 3 units up. The period is
or 2 .
y = a sec (bx + c), so a = 1, b = 1, and c = 3. Use the asymptote equations to find the location of two vertical
asymptotes.
Create a table listing the coordinates of key points for y = sec x + 3 for one period on
Function
y = sec x
.
y = sec x
+3
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4-5 Graphing Other Trigonometric Functions
Vertical
Asymptote
Intermediate
Point
x-int
(0, 1)
(0, 4)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
30.
SOLUTION:
The graph of
is the graph of y = sec x shifted
to the right and 4 units up. The period is
or
2 .
y = a sec (bx + c), so a = 1, b = 1, and c =
. Use the asymptote equations to find the location of two vertical
asymptotes.
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4-5 Graphing Other Trigonometric Functions
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
for one period on
.
y = sec x
x =0
(0, 1)
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
31. y = csc
−2
SOLUTION:
The graph of
is the graph of y = csc x expanded horizontally and shifted 2 units down. The period is
or 6π.
y = a tan (bx + c), so a = 1, b = , and c = 2. Use the asymptote equations to find the location of two vertical
asymptotes.
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4-5 Graphing Other Trigonometric Functions
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
for one period on
.
y = csc x
x=0
x=0
Intermediate
Point
Vertical
Asymptote
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
32.
SOLUTION:
The graph of
is the graph of y = csc x compressed horizontally, shifted
shifted 3 units up. The period is
.
y = a csc (bx + c), so a = 1, b = 3, and c =
asymptotes.
units to the left, and
. Use the asymptote equations to find the location of two vertical
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4-5 Graphing Other Trigonometric Functions
Create a table listing the coordinates of key points for
Function
for one period on
.
y = csc x
Vertical
Asympt
Interm
Point
x-int
x=0
Interm
Point
Vertical
Asympt
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
33. y = cot (2x + π) − 3
SOLUTION:
The graph of
is the graph of y = cot x compressed horizontally, shifted
units to the left, and
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4-5 Graphing Other Trigonometric Functions
shifted 3 units down. The period is
.
y = a cot (bx + c), so a = 1, b = 2, and c = . Use the asymptote equations to find the location of two consecutive
vertical asymptotes.
Create a table listing the coordinates of key points for
Function
Vertical
Asymptote
Intermediate
Point
x-int
for one period on
.
y = cot x
x=0
Intermediate
Point
Vertical
Asymptote
x=0
Sketch the curve through the indicated key points for the function. Then repeat the pattern.
34.
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4-5 Graphing Other Trigonometric Functions
SOLUTION:
is the graph of y = cot x expanded horizontally, shifted π units to the left, and shifted
The graph of
1 unit down. The period is
or 2π.
y = a cot (bx + c), so a = 1, b = , and c = . Use the asymptote equations to find the location of two consecutive
vertical asymptotes.
Create a table listing the coordinates of key points for
Function
Vertical
Asympt
Interm
Point
x-int
for one period on
.
y = cot x
x=0
Interm
Point
Vertical
Asympt
Sketch the curve through the indicated key points for the function. Then repeat the pattern.