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 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

–3/2Phase 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 (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

Section 5.5 Question 38

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 2sinx and sinx

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

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

Section 5.6 Question 30

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

Section 5.6 Question 32

8

28

4

Graph one full period of 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 3/2sin(x/4). Now click to see the phase shift and to get 3/2sin(x/4+3π/4)

Graph one full period of sec(x − π/2 )+1

x

y

a = 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 2π

Section 5.7 Question 50

π/2

2

cos(x)

sec(x) sec(x − π/2 )

Click to shift π/2 unit to right

Click to shift 1 unit up

sec(x − π/2 )+1

| | | |

Graph one full period of 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

Section 5.7 Question 48

| | | |π/4

5

sin(x/3)

csc(x/3−π/12)

csc(x/3)

csc(x/3-π/12)+4

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)|

Click to see y = |(1/2)sin(3x)|

π/3 2π/3-π/3-2π/3

-1/2

1/2

Ran

ge o

f 1/

2sin

(3x)

Ran

ge o

f |1

/2si

n(3

x)|

0

One period of 1/2sin(3x)

One period of |1/2sin(3x)|

Section 5.5 Question 48

Sketch the graph of y = cos2(x)

1

π/2-π/2 π-π π/4 3π/2-3π/2 -π/4

Section 5.5 Question 65

1/2

Sketch the graph of y = sin|x|

π 2π-π-2π

-1

1

0

Section 5.5 Question 68

y = sin|x| =

sin(x) if x ≥ 0

−sin(x) if x ≤ 0