y x- intercepts x.
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Transcript of y x- intercepts x.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
sin( )If , then
cos( )
Domain is ( , ) and Range is [ | | , | | ]
2 is called the period of function
| |
| | is called Amplitude
is called Phas
a bx c dy
a bx c d
a d a d
b
a
c
b
e Shift
is called vertical translationd
If sec( ) , then
Domain is all real numbers except (2 1)2
Range is ( , | | ] [| | , )
2 is called the period of function
| |
No Amplitude for the functions sec, csc, tan and cot
is
y a bx c d
cx n
b ba d a d
b
c
b
called Phase Shift
is called vertical translation
(2 1) are the asymptotes of the function2
d
cx n
b b
If csc( ) , then
Domain is all real numbers except
Range is ( , | | ] [| | , )
are the asymptotes of the function
2 is called the period of function
| |
is called Phase Sh
y a bx c d
n cx
b ba d a d
n cx
b b
b
c
b
ift
is called vertical translationd
If tan( ) , then
Domain is all real numbers except (2 1)2
Range is ( , ), Period is | |
(2 1) are the asymptotes of the function2
If cot( ) , then
Domain is all real numb
y a bx c d
cx n
b b
b
cx n
b b
y a bx c d
ers except 2
Range is ( , ), Period is | |
are the asymptotes of the function2
n cx
b b
b
n cx
b b
EXAM QUESTION
Let ( ) sin( ), where 0.
If the periond of is 12 and (3) 4, then (25) ?
)2 )6 )4 )0 )8
f x a bx b
f f f
A B C D E
EXAM QUESTION
Let ( ) tan( ), where a 0, 0.
3If the periond of is 3, then ( ) is
4
)equal to )undefined ) equal to
)equal to )equal to
f x a bx b
f f
a aA B C
b bD a E b
EXAM QUESTION
Let ( ) cos( ).
If the periond of is 8 and (4) 3, then (12) ?
)3 )4 )12 )8 )0
f x a bx
f f f
A B C D E
EXAM QUESTION
The range of ( ) 1 4sec is
)( , 3] [5, )
)( , 3) (5, )
)( , 1) (1, )
)( , 1] [1, )
)( 1,4) (5, )
f x x
A
B
C
D
E
EXAM QUESTION
Let n be any integer, then the equation of
the vertical asymptote of the function
( ) 2csc is2
) 2
) 2 1
) 4
) (2 1)
) 2
xf x
A x n
B x n
C x n
D x n
E x n
EXAM QUESTION
If ( ) 2cot 2 , then the number of
the vertical asymptotes over the interval
3, is equal to
4 4
)2 )1 )3 )4 )0
f x x
A B C D E
EXAM QUESTION
2If ( ) 3cot , then the number of
3the vertical asymptotes over the interval
3 15, is equal to
4 4
)3 )2 )4 )5 )6
xf x
A B C D E
EXAM QUESTION
If ( ) 3cot , then the number of 2
the vertical asymptotes over the interval
, is equal to2
)1 )2 )0 )3 )4
xf x
A B C D E
EXAM QUESTION
3If ( ) 2cot , then the number of
2the vertical asymptotes over the interval
,3 is equal to6
)5 )9 )3 )2 )4
xf x
A B C D E
EXAM QUESTION
If ( ) 3 2cot , then the number of 3
the vertical asymptotes over the interval
4,4 is equal to
)6 )3 )2 )1 )4
xf x
A B C D E
EXAM QUESTION
The number of the intercepts of the graph
( ) 2cot 2 on the interval ( , ) is
)4
)3
)2
)1
)5
x
f x x
A
B
C
D
E
Period = 2π/|b| = 2π/3
2
-2
π/6 5π/6 π/2
Draw one full period of y=2sin(3x–π/2)
2π/3
This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x–π/2)
π/3
Amplitude = |a| = 2
Phase shift = -c/b = π/6
Graph one full period of sin(x–π /2) –1/2
a =1, b =1,c = – π/2 and d = –1/2
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b = π/2
Vertical translation: 1/2 units down
2
y = sin(x)
y = sin(x–π /2)
y = sin(x–π /2) –1/2
1
1/2
–1
–3/2
Phase shift
π/2 units right
Vertical translation½ units down
Section 5.7 Question 43
Graph one full period of 2sin(3x–π /2) +1
a =2, b =3,c = – π/2 and
d = 1
Amplitude = |a| =2
Period = 2π/b = 2π/3
Phase shift = – c/b = π/6
Vertical translation:
1 unit up
2
3
–1
–2
π/6
π/2 2π/3
x
y
π/3
This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation and to get 2sin(3x–π/2)+1
1
Graph one full period of sin(x+π /6)
a =1, b =1,c = π/6
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b
= –π/6
1
–1
π/2 π
3π/2 2π
x
y
–π/6
Section 5.7 Question 18
This is the graph of sin(x). Now click to see the phase shift and to get sin(x+π/6)
Graph one full period of cos(2x–π/3)
a =1, b = 2,c = – π/3
Amplitude = |a| =1
Period = 2π/b = π
Phase shift = – c/b
= π/6
1
–1
x
y
π/2ππ/6 π/4 3π/4
Section 5.7 Question 20
This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3)
7π/6
Graph one full period of y=(1/2)sin(πx/3)
a =1/2, b = π/3
Amplitude = |a| =1/2
Period = 2π/b = 6
1/2
–1/2
3/2
9/2 6
x
y
3
2
–2
1
–1
2
2
32
In [0 , π] ,
In [π , 2π] ,
0≤ sinx ≤ 2sinx
2sinx ≤ sinx ≤ 0
Graph one full period of y=2sinx and y= sinx
x
y
2
2
2
–2
Draw one full period of y = 2tan(x/2)
a = 2 and b = 1/2 , 4b = 2
Asymptotes:
Lets draw asymptotes
Mark 2 and –2 on the y-axis
and ±π/4b = ±π/2 on the x-axis
x = ±2π/4b = ± 2π/2 = ± π
Now we can draw the graph
Section 5.6 Question 29
Graph one full period of (3/2)csc(3x)
a =3/2, b = 3
Period = 2π/b = 2π/3
π/6 π/3π/2
2π/3
Section 5.6 Question 34
3/2
–3/2
Graph one full period of (1/3)tanx
a =1/3, b =1→4b = 4
Period = π/|b| = π
π/4–π/4
1/3
–1/3
Section 5.6 Question 22
–π/2 π/2
Graph one full period of 2cscx
x
ya =2, b = 1
Period = 2π/b = 2π
2
–2
π/2 π3π/2
2π
Section 5.6 Question 28
Graph one full period of -3sec(2x/3)
x
ya = –3 , b = 2/3
Period = 2π/b = 3π
3
–3
3π/4 3π/2 9π/4 3π
Section 5.6 Question 36
x
y
12
6
12
6
3
–3
Draw one full period of y = –3tan(3x)
a = –3 and b = 3 , 4b = 12
Asymptotes:
Lets draw asymptotes Mark 3 and –3 on the y-axis
and ±π/4b = ±π/12 on the x-axis
x = ±2π/4b = ± 2π/12 = ± π/6Period = π/b = π/2
Now we can draw the graph
x
y
8
8
3
0
1/2
–1/2
Draw one full period of y = (1/2)cot(2x)
a = 1/2 and b = 2 , 4b = 8
Asymptotes:
Lets draw asymptotes Mark 1/2 and –1/2 on the y-axis
and π/8, 2π/8, 3π/8 and 4π/8 on the
x-axis
x = π/b = π/2 and x = 0(y-axis)Period = π/b = π/2
Now we can draw the graph
8
28
4
Graph one full period of y=3/2sin(x /4+3π /4)
a =3/2, b = 1/4,c = 3π/4
Amplitude = |a| = 3/2
Period = 2π/b = 8π
Phase shift = – c/b = –3π
2π
6π 8π
3/2
–3/2
–3π x
y
4π
This is the graph of y=3/2sin(x/4). Now click to see the phase shift and to get y=3/2sin(x/4+3π/4)
Graph one full period of y= sec(x − π/2 )+1
x
ya = 1 , b = 1, c = −π/2, d = 1
Period = 2π/|b| = 2π
Phase shift = −c/b = π/2
Vertical translation :
(d =) 1 unit up
1
–1
π
3π/2 2ππ/2
2
cos(x)
sec(x)
Click to shift π/2 unit to right
Click to shift 1 unit up
| | | |
Graph one full period of y= csc(x/3-π/12)+4
x
y
a = 1, b = 1/3, c = π/3, d = 4
Period = 2π/|b| = 6π
Phase shift = -c/b = π/4
Vertical translation: 4 unit up
1
–1
3π/2 3π
9π/2
6π| | | |
π/4
5
3
sin(x/3)
csc(x/3)
1/2sin(3x) ≥ 0 1/2sin(3x) ≥ 0
1/2sin(3x) ≤ 01/2sin(3x) ≤ 0
Sketch the graph of y = |(1/2)sin(3x)|
π/3 2π/3-π/3-2π/3
-1/2
1/2
0
Section 5.5 Question 48