Download - 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 ( , | | ] [| | , )


| |

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


| |

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

2The period of ( ) 3cos is

3

3 4)2 ) )3 )4 )

4 3

xf x

A B C D E

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


, 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


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

a

a

y

b2

b2

2

b2

3b2

b2

2

b2

3

x- intercepts

0

0cos abxay

b2

4

b2

4 x

a

a

y

b2

b2

2b2

3b2

b2

2

b2

3 0

0cos abxay

b2

4b2

4

x

x- intercepts

3

3

x

y

4

2

4

3 4

2

4

3

0

232cos3 baxy

2

2

x

y

6

3

2

3

2

6

3

2

3

2

0

323cos2 baxy

x

y

0

b2

b2

2b2

3

b2

4b2

5

0sin abxay

a

a

x

y

0b2

b2

3

0sin abxay

a

a

0sin abxay

b2

2b2

4b2

5

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


Phase shift = – c/b

= –π/6

1

–1

π/2 π

3π/2 2π

x

y

–π/6


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


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

x

y

0 3

3

2 3

43

5

322

33

2

3sin3 bbaxy

3

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

b2

b2

2b2

3b2

b2

2

b2

3 0

b2

4

b2

4 5

2b

5

2b

sec( )y a bx

a

a

b2

b2

2b2

3b2

b2

2

b2

3 0

b2

4

b2

4 5

2b

5

2b

a

a

csc( )y a bx

x

y

b4

b4

b4

2b4

2

a

a

0tan abxay

x

y

b4

b4

b4

2b4

2 a

a

0tan abxay

x

y

b4

b4

3b4

2b4

4a

a

0cot abxay

x

y

b4

b4

2

a

a

0cot abxay

b4

3b4

4

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


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


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


–π/2 π/2

Graph one full period of 2cscx

x

ya =2, b = 1


2

–2

π/2 π3π/2

2π


Graph one full period of -3sec(2x/3)

x

ya = –3 , b = 2/3


3

–3

3π/4 3π/2 9π/4 3π


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


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


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


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


Sketch the graph of y = cos2(x)

1

π/2-π/2 π-π π/4 3π/2-3π/2 -π/4


1/2

Sketch the graph of y = sin|x|

π 2π-π-2π

-1

1

0


y = sin|x| =

sin(x) if x ≥ 0

−sin(x) if x ≤ 0